<!DOCTYPE html><html lang="en"><head><meta charset="utf-8"><meta name="viewport" content="width=device-width, initial-scale=1.0"><meta name="generator" content="rustdoc"><meta name="description" content="Source to the Rust file `/home/mrh/.cargo/registry/src/github.com-1ecc6299db9ec823/nalgebra-0.18.0/src/geometry/quaternion.rs`."><meta name="keywords" content="rust, rustlang, rust-lang"><title>quaternion.rs.html -- source</title><link rel="stylesheet" type="text/css" href="../../../normalize.css"><link rel="stylesheet" type="text/css" href="../../../rustdoc.css" id="mainThemeStyle"><link rel="stylesheet" type="text/css" href="../../../dark.css"><link rel="stylesheet" type="text/css" href="../../../light.css" id="themeStyle"><script src="../../../storage.js"></script><noscript><link rel="stylesheet" href="../../../noscript.css"></noscript><link rel="shortcut icon" href="http://nalgebra.org/img/favicon.ico"><style type="text/css">#crate-search{background-image:url("../../../down-arrow.svg");}</style></head><body class="rustdoc source"><!--[if lte IE 8]><div class="warning">This old browser is unsupported and will most likely display funky things.</div><![endif]--><nav class="sidebar"><div class="sidebar-menu">☰</div><a href='../../../nalgebra/index.html'><div class='logo-container'><img src='../../../rust-logo.png' alt='logo'></div></a></nav><div class="theme-picker"><button id="theme-picker" aria-label="Pick another theme!"><img src="../../../brush.svg" width="18" alt="Pick another theme!"></button><div id="theme-choices"></div></div><script src="../../../theme.js"></script><nav class="sub"><form class="search-form js-only"><div class="search-container"><div><select id="crate-search"><option value="All crates">All crates</option></select><input class="search-input" name="search" autocomplete="off" spellcheck="false" placeholder="Click or press ‘S’ to search, ‘?’ for more options…" type="search"></div><a id="settings-menu" href="../../../settings.html"><img src="../../../wheel.svg" width="18" alt="Change settings"></a></div></form></nav><section id="main" class="content"><pre class="line-numbers"><span id="1"> 1</span>
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<span id="1490">1490</span>
</pre><div class="example-wrap"><pre class="rust ">
<span class="kw">use</span> <span class="ident">approx</span>::{<span class="ident">AbsDiffEq</span>, <span class="ident">RelativeEq</span>, <span class="ident">UlpsEq</span>};
<span class="kw">use</span> <span class="ident">num</span>::<span class="ident">Zero</span>;
<span class="kw">use</span> <span class="ident">std</span>::<span class="ident">fmt</span>;
<span class="kw">use</span> <span class="ident">std</span>::<span class="ident">hash</span>;
<span class="attribute">#[<span class="ident">cfg</span>(<span class="ident">feature</span> <span class="op">=</span> <span class="string">"abomonation-serialize"</span>)]</span>
<span class="kw">use</span> <span class="ident">std</span>::<span class="ident">io</span>::{<span class="prelude-ty">Result</span> <span class="kw">as</span> <span class="ident">IOResult</span>, <span class="ident">Write</span>};
<span class="attribute">#[<span class="ident">cfg</span>(<span class="ident">feature</span> <span class="op">=</span> <span class="string">"serde-serialize"</span>)]</span>
<span class="kw">use</span> <span class="kw">crate</span>::<span class="ident">base</span>::<span class="ident">storage</span>::<span class="ident">Owned</span>;
<span class="attribute">#[<span class="ident">cfg</span>(<span class="ident">feature</span> <span class="op">=</span> <span class="string">"serde-serialize"</span>)]</span>
<span class="kw">use</span> <span class="ident">serde</span>::{<span class="ident">Deserialize</span>, <span class="ident">Deserializer</span>, <span class="ident">Serialize</span>, <span class="ident">Serializer</span>};
<span class="attribute">#[<span class="ident">cfg</span>(<span class="ident">feature</span> <span class="op">=</span> <span class="string">"abomonation-serialize"</span>)]</span>
<span class="kw">use</span> <span class="ident">abomonation</span>::<span class="ident">Abomonation</span>;
<span class="kw">use</span> <span class="ident">alga</span>::<span class="ident">general</span>::<span class="ident">RealField</span>;
<span class="kw">use</span> <span class="kw">crate</span>::<span class="ident">base</span>::<span class="ident">dimension</span>::{<span class="ident">U1</span>, <span class="ident">U3</span>, <span class="ident">U4</span>};
<span class="kw">use</span> <span class="kw">crate</span>::<span class="ident">base</span>::<span class="ident">storage</span>::{<span class="ident">CStride</span>, <span class="ident">RStride</span>};
<span class="kw">use</span> <span class="kw">crate</span>::<span class="ident">base</span>::{<span class="ident">Matrix3</span>, <span class="ident">Matrix4</span>, <span class="ident">MatrixSlice</span>, <span class="ident">MatrixSliceMut</span>, <span class="ident">Unit</span>, <span class="ident">Vector3</span>, <span class="ident">Vector4</span>};
<span class="kw">use</span> <span class="kw">crate</span>::<span class="ident">geometry</span>::{<span class="ident">Point3</span>, <span class="ident">Rotation</span>};
<span class="doccomment">/// A quaternion. See the type alias `UnitQuaternion = Unit<Quaternion>` for a quaternion</span>
<span class="doccomment">/// that may be used as a rotation.</span>
<span class="attribute">#[<span class="ident">repr</span>(<span class="ident">C</span>)]</span>
<span class="attribute">#[<span class="ident">derive</span>(<span class="ident">Debug</span>)]</span>
<span class="kw">pub</span> <span class="kw">struct</span> <span class="ident">Quaternion</span><span class="op"><</span><span class="ident">N</span>: <span class="ident">RealField</span><span class="op">></span> {
<span class="doccomment">/// This quaternion as a 4D vector of coordinates in the `[ x, y, z, w ]` storage order.</span>
<span class="kw">pub</span> <span class="ident">coords</span>: <span class="ident">Vector4</span><span class="op"><</span><span class="ident">N</span><span class="op">></span>,
}
<span class="attribute">#[<span class="ident">cfg</span>(<span class="ident">feature</span> <span class="op">=</span> <span class="string">"abomonation-serialize"</span>)]</span>
<span class="kw">impl</span><span class="op"><</span><span class="ident">N</span>: <span class="ident">RealField</span><span class="op">></span> <span class="ident">Abomonation</span> <span class="kw">for</span> <span class="ident">Quaternion</span><span class="op"><</span><span class="ident">N</span><span class="op">></span>
<span class="kw">where</span> <span class="ident">Vector4</span><span class="op"><</span><span class="ident">N</span><span class="op">></span>: <span class="ident">Abomonation</span>
{
<span class="kw">unsafe</span> <span class="kw">fn</span> <span class="ident">entomb</span><span class="op"><</span><span class="ident">W</span>: <span class="ident">Write</span><span class="op">></span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">writer</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">W</span>) <span class="op">-</span><span class="op">></span> <span class="ident">IOResult</span><span class="op"><</span>()<span class="op">></span> {
<span class="self">self</span>.<span class="ident">coords</span>.<span class="ident">entomb</span>(<span class="ident">writer</span>)
}
<span class="kw">fn</span> <span class="ident">extent</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">usize</span> {
<span class="self">self</span>.<span class="ident">coords</span>.<span class="ident">extent</span>()
}
<span class="kw">unsafe</span> <span class="kw">fn</span> <span class="ident">exhume</span><span class="op"><</span><span class="lifetime">'a</span>, <span class="lifetime">'b</span><span class="op">></span>(<span class="kw-2">&</span><span class="lifetime">'a</span> <span class="kw-2">mut</span> <span class="self">self</span>, <span class="ident">bytes</span>: <span class="kw-2">&</span><span class="lifetime">'b</span> <span class="kw-2">mut</span> [<span class="ident">u8</span>]) <span class="op">-</span><span class="op">></span> <span class="prelude-ty">Option</span><span class="op"><</span><span class="kw-2">&</span><span class="lifetime">'b</span> <span class="kw-2">mut</span> [<span class="ident">u8</span>]<span class="op">></span> {
<span class="self">self</span>.<span class="ident">coords</span>.<span class="ident">exhume</span>(<span class="ident">bytes</span>)
}
}
<span class="kw">impl</span><span class="op"><</span><span class="ident">N</span>: <span class="ident">RealField</span> <span class="op">+</span> <span class="ident">Eq</span><span class="op">></span> <span class="ident">Eq</span> <span class="kw">for</span> <span class="ident">Quaternion</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {}
<span class="kw">impl</span><span class="op"><</span><span class="ident">N</span>: <span class="ident">RealField</span><span class="op">></span> <span class="ident">PartialEq</span> <span class="kw">for</span> <span class="ident">Quaternion</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="kw">fn</span> <span class="ident">eq</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">rhs</span>: <span class="kw-2">&</span><span class="self">Self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">bool</span> {
<span class="self">self</span>.<span class="ident">coords</span> <span class="op">=</span><span class="op">=</span> <span class="ident">rhs</span>.<span class="ident">coords</span> <span class="op">|</span><span class="op">|</span>
<span class="comment">// Account for the double-covering of S², i.e. q = -q</span>
<span class="self">self</span>.<span class="ident">as_vector</span>().<span class="ident">iter</span>().<span class="ident">zip</span>(<span class="ident">rhs</span>.<span class="ident">as_vector</span>().<span class="ident">iter</span>()).<span class="ident">all</span>(<span class="op">|</span>(<span class="ident">a</span>, <span class="ident">b</span>)<span class="op">|</span> <span class="kw-2">*</span><span class="ident">a</span> <span class="op">=</span><span class="op">=</span> <span class="op">-</span><span class="kw-2">*</span><span class="ident">b</span>)
}
}
<span class="kw">impl</span><span class="op"><</span><span class="ident">N</span>: <span class="ident">RealField</span> <span class="op">+</span> <span class="ident">hash</span>::<span class="ident">Hash</span><span class="op">></span> <span class="ident">hash</span>::<span class="ident">Hash</span> <span class="kw">for</span> <span class="ident">Quaternion</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="kw">fn</span> <span class="ident">hash</span><span class="op"><</span><span class="ident">H</span>: <span class="ident">hash</span>::<span class="ident">Hasher</span><span class="op">></span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">state</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">H</span>) {
<span class="self">self</span>.<span class="ident">coords</span>.<span class="ident">hash</span>(<span class="ident">state</span>)
}
}
<span class="kw">impl</span><span class="op"><</span><span class="ident">N</span>: <span class="ident">RealField</span><span class="op">></span> <span class="ident">Copy</span> <span class="kw">for</span> <span class="ident">Quaternion</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {}
<span class="kw">impl</span><span class="op"><</span><span class="ident">N</span>: <span class="ident">RealField</span><span class="op">></span> <span class="ident">Clone</span> <span class="kw">for</span> <span class="ident">Quaternion</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">fn</span> <span class="ident">clone</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="self">Self</span>::<span class="ident">from</span>(<span class="self">self</span>.<span class="ident">coords</span>.<span class="ident">clone</span>())
}
}
<span class="attribute">#[<span class="ident">cfg</span>(<span class="ident">feature</span> <span class="op">=</span> <span class="string">"serde-serialize"</span>)]</span>
<span class="kw">impl</span><span class="op"><</span><span class="ident">N</span>: <span class="ident">RealField</span><span class="op">></span> <span class="ident">Serialize</span> <span class="kw">for</span> <span class="ident">Quaternion</span><span class="op"><</span><span class="ident">N</span><span class="op">></span>
<span class="kw">where</span> <span class="ident">Owned</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">U4</span><span class="op">></span>: <span class="ident">Serialize</span>
{
<span class="kw">fn</span> <span class="ident">serialize</span><span class="op"><</span><span class="ident">S</span><span class="op">></span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">serializer</span>: <span class="ident">S</span>) <span class="op">-</span><span class="op">></span> <span class="prelude-ty">Result</span><span class="op"><</span><span class="ident">S</span>::<span class="prelude-val">Ok</span>, <span class="ident">S</span>::<span class="ident">Error</span><span class="op">></span>
<span class="kw">where</span> <span class="ident">S</span>: <span class="ident">Serializer</span> {
<span class="self">self</span>.<span class="ident">coords</span>.<span class="ident">serialize</span>(<span class="ident">serializer</span>)
}
}
<span class="attribute">#[<span class="ident">cfg</span>(<span class="ident">feature</span> <span class="op">=</span> <span class="string">"serde-serialize"</span>)]</span>
<span class="kw">impl</span><span class="op"><</span><span class="lifetime">'a</span>, <span class="ident">N</span>: <span class="ident">RealField</span><span class="op">></span> <span class="ident">Deserialize</span><span class="op"><</span><span class="lifetime">'a</span><span class="op">></span> <span class="kw">for</span> <span class="ident">Quaternion</span><span class="op"><</span><span class="ident">N</span><span class="op">></span>
<span class="kw">where</span> <span class="ident">Owned</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">U4</span><span class="op">></span>: <span class="ident">Deserialize</span><span class="op"><</span><span class="lifetime">'a</span><span class="op">></span>
{
<span class="kw">fn</span> <span class="ident">deserialize</span><span class="op"><</span><span class="ident">Des</span><span class="op">></span>(<span class="ident">deserializer</span>: <span class="ident">Des</span>) <span class="op">-</span><span class="op">></span> <span class="prelude-ty">Result</span><span class="op"><</span><span class="self">Self</span>, <span class="ident">Des</span>::<span class="ident">Error</span><span class="op">></span>
<span class="kw">where</span> <span class="ident">Des</span>: <span class="ident">Deserializer</span><span class="op"><</span><span class="lifetime">'a</span><span class="op">></span> {
<span class="kw">let</span> <span class="ident">coords</span> <span class="op">=</span> <span class="ident">Vector4</span>::<span class="op"><</span><span class="ident">N</span><span class="op">></span>::<span class="ident">deserialize</span>(<span class="ident">deserializer</span>)<span class="question-mark">?</span>;
<span class="prelude-val">Ok</span>(<span class="self">Self</span>::<span class="ident">from</span>(<span class="ident">coords</span>))
}
}
<span class="kw">impl</span><span class="op"><</span><span class="ident">N</span>: <span class="ident">RealField</span><span class="op">></span> <span class="ident">Quaternion</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="doccomment">/// Moves this unit quaternion into one that owns its data.</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="attribute">#[<span class="ident">deprecated</span>(<span class="ident">note</span> <span class="op">=</span> <span class="string">"This method is a no-op and will be removed in a future release."</span>)]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">into_owned</span>(<span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="self">self</span>
}
<span class="doccomment">/// Clones this unit quaternion into one that owns its data.</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="attribute">#[<span class="ident">deprecated</span>(<span class="ident">note</span> <span class="op">=</span> <span class="string">"This method is a no-op and will be removed in a future release."</span>)]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">clone_owned</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="self">Self</span>::<span class="ident">from</span>(<span class="self">self</span>.<span class="ident">coords</span>.<span class="ident">clone_owned</span>())
}
<span class="doccomment">/// Normalizes this quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// let q_normalized = q.normalize();</span>
<span class="doccomment">/// relative_eq!(q_normalized.norm(), 1.0);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">normalize</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="self">Self</span>::<span class="ident">from</span>(<span class="self">self</span>.<span class="ident">coords</span>.<span class="ident">normalize</span>())
}
<span class="doccomment">/// The imaginary part of this quaternion.</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">imag</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">Vector3</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="self">self</span>.<span class="ident">coords</span>.<span class="ident">xyz</span>()
}
<span class="doccomment">/// The conjugate of this quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// let conj = q.conjugate();</span>
<span class="doccomment">/// assert!(conj.i == -2.0 && conj.j == -3.0 && conj.k == -4.0 && conj.w == 1.0);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">conjugate</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="self">Self</span>::<span class="ident">from_parts</span>(<span class="self">self</span>.<span class="ident">w</span>, <span class="op">-</span><span class="self">self</span>.<span class="ident">imag</span>())
}
<span class="doccomment">/// Inverts this quaternion if it is not zero.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// let inv_q = q.try_inverse();</span>
<span class="doccomment">///</span>
<span class="doccomment">/// assert!(inv_q.is_some());</span>
<span class="doccomment">/// assert_relative_eq!(inv_q.unwrap() * q, Quaternion::identity());</span>
<span class="doccomment">///</span>
<span class="doccomment">/// //Non-invertible case</span>
<span class="doccomment">/// let q = Quaternion::new(0.0, 0.0, 0.0, 0.0);</span>
<span class="doccomment">/// let inv_q = q.try_inverse();</span>
<span class="doccomment">///</span>
<span class="doccomment">/// assert!(inv_q.is_none());</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">try_inverse</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="prelude-ty">Option</span><span class="op"><</span><span class="self">Self</span><span class="op">></span> {
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">res</span> <span class="op">=</span> <span class="self">Self</span>::<span class="ident">from</span>(<span class="self">self</span>.<span class="ident">coords</span>.<span class="ident">clone_owned</span>());
<span class="kw">if</span> <span class="ident">res</span>.<span class="ident">try_inverse_mut</span>() {
<span class="prelude-val">Some</span>(<span class="ident">res</span>)
} <span class="kw">else</span> {
<span class="prelude-val">None</span>
}
}
<span class="doccomment">/// Linear interpolation between two quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// Computes `self * (1 - t) + other * t`.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let q1 = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// let q2 = Quaternion::new(10.0, 20.0, 30.0, 40.0);</span>
<span class="doccomment">///</span>
<span class="doccomment">/// assert_eq!(q1.lerp(&q2, 0.1), Quaternion::new(1.9, 3.8, 5.7, 7.6));</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">lerp</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">other</span>: <span class="kw-2">&</span><span class="self">Self</span>, <span class="ident">t</span>: <span class="ident">N</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="self">self</span> <span class="op">*</span> (<span class="ident">N</span>::<span class="ident">one</span>() <span class="op">-</span> <span class="ident">t</span>) <span class="op">+</span> <span class="ident">other</span> <span class="op">*</span> <span class="ident">t</span>
}
<span class="doccomment">/// The vector part `(i, j, k)` of this quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// assert_eq!(q.vector()[0], 2.0);</span>
<span class="doccomment">/// assert_eq!(q.vector()[1], 3.0);</span>
<span class="doccomment">/// assert_eq!(q.vector()[2], 4.0);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">vector</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">MatrixSlice</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">U3</span>, <span class="ident">U1</span>, <span class="ident">RStride</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">U4</span>, <span class="ident">U1</span><span class="op">></span>, <span class="ident">CStride</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">U4</span>, <span class="ident">U1</span><span class="op">></span><span class="op">></span> {
<span class="self">self</span>.<span class="ident">coords</span>.<span class="ident">fixed_rows</span>::<span class="op"><</span><span class="ident">U3</span><span class="op">></span>(<span class="number">0</span>)
}
<span class="doccomment">/// The scalar part `w` of this quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// assert_eq!(q.scalar(), 1.0);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">scalar</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">N</span> {
<span class="self">self</span>.<span class="ident">coords</span>[<span class="number">3</span>]
}
<span class="doccomment">/// Reinterprets this quaternion as a 4D vector.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # use nalgebra::{Vector4, Quaternion};</span>
<span class="doccomment">/// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// // Recall that the quaternion is stored internally as (i, j, k, w)</span>
<span class="doccomment">/// // while the crate::new constructor takes the arguments as (w, i, j, k).</span>
<span class="doccomment">/// assert_eq!(*q.as_vector(), Vector4::new(2.0, 3.0, 4.0, 1.0));</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">as_vector</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="kw-2">&</span><span class="ident">Vector4</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="kw-2">&</span><span class="self">self</span>.<span class="ident">coords</span>
}
<span class="doccomment">/// The norm of this quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// assert_relative_eq!(q.norm(), 5.47722557, epsilon = 1.0e-6);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">norm</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">N</span> {
<span class="self">self</span>.<span class="ident">coords</span>.<span class="ident">norm</span>()
}
<span class="doccomment">/// A synonym for the norm of this quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// Aka the length.</span>
<span class="doccomment">/// This is the same as `.norm()`</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// assert_relative_eq!(q.magnitude(), 5.47722557, epsilon = 1.0e-6);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">magnitude</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">N</span> {
<span class="self">self</span>.<span class="ident">norm</span>()
}
<span class="doccomment">/// The squared norm of this quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// assert_eq!(q.magnitude_squared(), 30.0);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">norm_squared</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">N</span> {
<span class="self">self</span>.<span class="ident">coords</span>.<span class="ident">norm_squared</span>()
}
<span class="doccomment">/// A synonym for the squared norm of this quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// Aka the squared length.</span>
<span class="doccomment">/// This is the same as `.norm_squared()`</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// assert_eq!(q.magnitude_squared(), 30.0);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">magnitude_squared</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">N</span> {
<span class="self">self</span>.<span class="ident">norm_squared</span>()
}
<span class="doccomment">/// The dot product of two quaternions.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let q1 = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// let q2 = Quaternion::new(5.0, 6.0, 7.0, 8.0);</span>
<span class="doccomment">/// assert_eq!(q1.dot(&q2), 70.0);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">dot</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">rhs</span>: <span class="kw-2">&</span><span class="self">Self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">N</span> {
<span class="self">self</span>.<span class="ident">coords</span>.<span class="ident">dot</span>(<span class="kw-2">&</span><span class="ident">rhs</span>.<span class="ident">coords</span>)
}
<span class="doccomment">/// Calculates the inner product (also known as the dot product).</span>
<span class="doccomment">/// See "Foundations of Game Engine Development, Volume 1: Mathematics" by Lengyel</span>
<span class="doccomment">/// Formula 4.89.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let a = Quaternion::new(0.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// let b = Quaternion::new(0.0, 5.0, 2.0, 1.0);</span>
<span class="doccomment">/// let expected = Quaternion::new(-20.0, 0.0, 0.0, 0.0);</span>
<span class="doccomment">/// let result = a.inner(&b);</span>
<span class="doccomment">/// assert_relative_eq!(expected, result, epsilon = 1.0e-5);</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">inner</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">other</span>: <span class="kw-2">&</span><span class="self">Self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
(<span class="self">self</span> <span class="op">*</span> <span class="ident">other</span> <span class="op">+</span> <span class="ident">other</span> <span class="op">*</span> <span class="self">self</span>).<span class="ident">half</span>()
}
<span class="doccomment">/// Calculates the outer product (also known as the wedge product).</span>
<span class="doccomment">/// See "Foundations of Game Engine Development, Volume 1: Mathematics" by Lengyel</span>
<span class="doccomment">/// Formula 4.89.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let a = Quaternion::new(0.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// let b = Quaternion::new(0.0, 5.0, 2.0, 1.0);</span>
<span class="doccomment">/// let expected = Quaternion::new(0.0, -5.0, 18.0, -11.0);</span>
<span class="doccomment">/// let result = a.outer(&b);</span>
<span class="doccomment">/// assert_relative_eq!(expected, result, epsilon = 1.0e-5);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">outer</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">other</span>: <span class="kw-2">&</span><span class="self">Self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
(<span class="self">self</span> <span class="op">*</span> <span class="ident">other</span> <span class="op">-</span> <span class="ident">other</span> <span class="op">*</span> <span class="self">self</span>).<span class="ident">half</span>()
}
<span class="doccomment">/// Calculates the projection of `self` onto `other` (also known as the parallel).</span>
<span class="doccomment">/// See "Foundations of Game Engine Development, Volume 1: Mathematics" by Lengyel</span>
<span class="doccomment">/// Formula 4.94.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let a = Quaternion::new(0.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// let b = Quaternion::new(0.0, 5.0, 2.0, 1.0);</span>
<span class="doccomment">/// let expected = Quaternion::new(0.0, 3.333333333333333, 1.3333333333333333, 0.6666666666666666);</span>
<span class="doccomment">/// let result = a.project(&b).unwrap();</span>
<span class="doccomment">/// assert_relative_eq!(expected, result, epsilon = 1.0e-5);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">project</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">other</span>: <span class="kw-2">&</span><span class="self">Self</span>) <span class="op">-</span><span class="op">></span> <span class="prelude-ty">Option</span><span class="op"><</span><span class="self">Self</span><span class="op">></span> {
<span class="self">self</span>.<span class="ident">inner</span>(<span class="ident">other</span>).<span class="ident">right_div</span>(<span class="ident">other</span>)
}
<span class="doccomment">/// Calculates the rejection of `self` from `other` (also known as the perpendicular).</span>
<span class="doccomment">/// See "Foundations of Game Engine Development, Volume 1: Mathematics" by Lengyel</span>
<span class="doccomment">/// Formula 4.94.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let a = Quaternion::new(0.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// let b = Quaternion::new(0.0, 5.0, 2.0, 1.0);</span>
<span class="doccomment">/// let expected = Quaternion::new(0.0, -1.3333333333333333, 1.6666666666666665, 3.3333333333333335);</span>
<span class="doccomment">/// let result = a.reject(&b).unwrap();</span>
<span class="doccomment">/// assert_relative_eq!(expected, result, epsilon = 1.0e-5);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">reject</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">other</span>: <span class="kw-2">&</span><span class="self">Self</span>) <span class="op">-</span><span class="op">></span> <span class="prelude-ty">Option</span><span class="op"><</span><span class="self">Self</span><span class="op">></span> {
<span class="self">self</span>.<span class="ident">outer</span>(<span class="ident">other</span>).<span class="ident">right_div</span>(<span class="ident">other</span>)
}
<span class="doccomment">/// The polar decomposition of this quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// Returns, from left to right: the quaternion norm, the half rotation angle, the rotation</span>
<span class="doccomment">/// axis. If the rotation angle is zero, the rotation axis is set to `None`.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # use std::f32;</span>
<span class="doccomment">/// # use nalgebra::{Vector3, Quaternion};</span>
<span class="doccomment">/// let q = Quaternion::new(0.0, 5.0, 0.0, 0.0);</span>
<span class="doccomment">/// let (norm, half_ang, axis) = q.polar_decomposition();</span>
<span class="doccomment">/// assert_eq!(norm, 5.0);</span>
<span class="doccomment">/// assert_eq!(half_ang, f32::consts::FRAC_PI_2);</span>
<span class="doccomment">/// assert_eq!(axis, Some(Vector3::x_axis()));</span>
<span class="doccomment">/// ```</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">polar_decomposition</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> (<span class="ident">N</span>, <span class="ident">N</span>, <span class="prelude-ty">Option</span><span class="op"><</span><span class="ident">Unit</span><span class="op"><</span><span class="ident">Vector3</span><span class="op"><</span><span class="ident">N</span><span class="op">></span><span class="op">></span><span class="op">></span>) {
<span class="kw">if</span> <span class="kw">let</span> <span class="prelude-val">Some</span>((<span class="ident">q</span>, <span class="ident">n</span>)) <span class="op">=</span> <span class="ident">Unit</span>::<span class="ident">try_new_and_get</span>(<span class="kw-2">*</span><span class="self">self</span>, <span class="ident">N</span>::<span class="ident">zero</span>()) {
<span class="kw">if</span> <span class="kw">let</span> <span class="prelude-val">Some</span>(<span class="ident">axis</span>) <span class="op">=</span> <span class="ident">Unit</span>::<span class="ident">try_new</span>(<span class="self">self</span>.<span class="ident">vector</span>().<span class="ident">clone_owned</span>(), <span class="ident">N</span>::<span class="ident">zero</span>()) {
<span class="kw">let</span> <span class="ident">angle</span> <span class="op">=</span> <span class="ident">q</span>.<span class="ident">angle</span>() <span class="op">/</span> <span class="kw">crate</span>::<span class="ident">convert</span>(<span class="number">2.0f64</span>);
(<span class="ident">n</span>, <span class="ident">angle</span>, <span class="prelude-val">Some</span>(<span class="ident">axis</span>))
} <span class="kw">else</span> {
(<span class="ident">n</span>, <span class="ident">N</span>::<span class="ident">zero</span>(), <span class="prelude-val">None</span>)
}
} <span class="kw">else</span> {
(<span class="ident">N</span>::<span class="ident">zero</span>(), <span class="ident">N</span>::<span class="ident">zero</span>(), <span class="prelude-val">None</span>)
}
}
<span class="doccomment">/// Compute the natural logarithm of a quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let q = Quaternion::new(2.0, 5.0, 0.0, 0.0);</span>
<span class="doccomment">/// assert_relative_eq!(q.ln(), Quaternion::new(1.683647, 1.190289, 0.0, 0.0), epsilon = 1.0e-6)</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">ln</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="kw">let</span> <span class="ident">n</span> <span class="op">=</span> <span class="self">self</span>.<span class="ident">norm</span>();
<span class="kw">let</span> <span class="ident">v</span> <span class="op">=</span> <span class="self">self</span>.<span class="ident">vector</span>();
<span class="kw">let</span> <span class="ident">s</span> <span class="op">=</span> <span class="self">self</span>.<span class="ident">scalar</span>();
<span class="self">Self</span>::<span class="ident">from_parts</span>(<span class="ident">n</span>.<span class="ident">ln</span>(), <span class="ident">v</span>.<span class="ident">normalize</span>() <span class="op">*</span> (<span class="ident">s</span> <span class="op">/</span> <span class="ident">n</span>).<span class="ident">acos</span>())
}
<span class="doccomment">/// Compute the exponential of a quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let q = Quaternion::new(1.683647, 1.190289, 0.0, 0.0);</span>
<span class="doccomment">/// assert_relative_eq!(q.exp(), Quaternion::new(2.0, 5.0, 0.0, 0.0), epsilon = 1.0e-5)</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">exp</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="self">self</span>.<span class="ident">exp_eps</span>(<span class="ident">N</span>::<span class="ident">default_epsilon</span>())
}
<span class="doccomment">/// Compute the exponential of a quaternion. Returns the identity if the vector part of this quaternion</span>
<span class="doccomment">/// has a norm smaller than `eps`.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let q = Quaternion::new(1.683647, 1.190289, 0.0, 0.0);</span>
<span class="doccomment">/// assert_relative_eq!(q.exp_eps(1.0e-6), Quaternion::new(2.0, 5.0, 0.0, 0.0), epsilon = 1.0e-5);</span>
<span class="doccomment">///</span>
<span class="doccomment">/// // Singular case.</span>
<span class="doccomment">/// let q = Quaternion::new(0.0000001, 0.0, 0.0, 0.0);</span>
<span class="doccomment">/// assert_eq!(q.exp_eps(1.0e-6), Quaternion::identity());</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">exp_eps</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">eps</span>: <span class="ident">N</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="kw">let</span> <span class="ident">v</span> <span class="op">=</span> <span class="self">self</span>.<span class="ident">vector</span>();
<span class="kw">let</span> <span class="ident">nn</span> <span class="op">=</span> <span class="ident">v</span>.<span class="ident">norm_squared</span>();
<span class="kw">if</span> <span class="ident">nn</span> <span class="op"><</span><span class="op">=</span> <span class="ident">eps</span> <span class="op">*</span> <span class="ident">eps</span> {
<span class="self">Self</span>::<span class="ident">identity</span>()
} <span class="kw">else</span> {
<span class="kw">let</span> <span class="ident">w_exp</span> <span class="op">=</span> <span class="self">self</span>.<span class="ident">scalar</span>().<span class="ident">exp</span>();
<span class="kw">let</span> <span class="ident">n</span> <span class="op">=</span> <span class="ident">nn</span>.<span class="ident">sqrt</span>();
<span class="kw">let</span> <span class="ident">nv</span> <span class="op">=</span> <span class="ident">v</span> <span class="op">*</span> (<span class="ident">w_exp</span> <span class="op">*</span> <span class="ident">n</span>.<span class="ident">sin</span>() <span class="op">/</span> <span class="ident">n</span>);
<span class="self">Self</span>::<span class="ident">from_parts</span>(<span class="ident">w_exp</span> <span class="op">*</span> <span class="ident">n</span>.<span class="ident">cos</span>(), <span class="ident">nv</span>)
}
}
<span class="doccomment">/// Raise the quaternion to a given floating power.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// assert_relative_eq!(q.powf(1.5), Quaternion::new( -6.2576659, 4.1549037, 6.2323556, 8.3098075), epsilon = 1.0e-6);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">powf</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">n</span>: <span class="ident">N</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
(<span class="self">self</span>.<span class="ident">ln</span>() <span class="op">*</span> <span class="ident">n</span>).<span class="ident">exp</span>()
}
<span class="doccomment">/// Transforms this quaternion into its 4D vector form (Vector part, Scalar part).</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # use nalgebra::{Quaternion, Vector4};</span>
<span class="doccomment">/// let mut q = Quaternion::identity();</span>
<span class="doccomment">/// *q.as_vector_mut() = Vector4::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// assert!(q.i == 1.0 && q.j == 2.0 && q.k == 3.0 && q.w == 4.0);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">as_vector_mut</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">Vector4</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="self">self</span>.<span class="ident">coords</span>
}
<span class="doccomment">/// The mutable vector part `(i, j, k)` of this quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # use nalgebra::{Quaternion, Vector4};</span>
<span class="doccomment">/// let mut q = Quaternion::identity();</span>
<span class="doccomment">/// {</span>
<span class="doccomment">/// let mut v = q.vector_mut();</span>
<span class="doccomment">/// v[0] = 2.0;</span>
<span class="doccomment">/// v[1] = 3.0;</span>
<span class="doccomment">/// v[2] = 4.0;</span>
<span class="doccomment">/// }</span>
<span class="doccomment">/// assert!(q.i == 2.0 && q.j == 3.0 && q.k == 4.0 && q.w == 1.0);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">vector_mut</span>(
<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="self">self</span>,
) <span class="op">-</span><span class="op">></span> <span class="ident">MatrixSliceMut</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">U3</span>, <span class="ident">U1</span>, <span class="ident">RStride</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">U4</span>, <span class="ident">U1</span><span class="op">></span>, <span class="ident">CStride</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">U4</span>, <span class="ident">U1</span><span class="op">></span><span class="op">></span> {
<span class="self">self</span>.<span class="ident">coords</span>.<span class="ident">fixed_rows_mut</span>::<span class="op"><</span><span class="ident">U3</span><span class="op">></span>(<span class="number">0</span>)
}
<span class="doccomment">/// Replaces this quaternion by its conjugate.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let mut q = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// q.conjugate_mut();</span>
<span class="doccomment">/// assert!(q.i == -2.0 && q.j == -3.0 && q.k == -4.0 && q.w == 1.0);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">conjugate_mut</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="self">self</span>) {
<span class="self">self</span>.<span class="ident">coords</span>[<span class="number">0</span>] <span class="op">=</span> <span class="op">-</span><span class="self">self</span>.<span class="ident">coords</span>[<span class="number">0</span>];
<span class="self">self</span>.<span class="ident">coords</span>[<span class="number">1</span>] <span class="op">=</span> <span class="op">-</span><span class="self">self</span>.<span class="ident">coords</span>[<span class="number">1</span>];
<span class="self">self</span>.<span class="ident">coords</span>[<span class="number">2</span>] <span class="op">=</span> <span class="op">-</span><span class="self">self</span>.<span class="ident">coords</span>[<span class="number">2</span>];
}
<span class="doccomment">/// Inverts this quaternion in-place if it is not zero.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let mut q = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">///</span>
<span class="doccomment">/// assert!(q.try_inverse_mut());</span>
<span class="doccomment">/// assert_relative_eq!(q * Quaternion::new(1.0, 2.0, 3.0, 4.0), Quaternion::identity());</span>
<span class="doccomment">///</span>
<span class="doccomment">/// //Non-invertible case</span>
<span class="doccomment">/// let mut q = Quaternion::new(0.0, 0.0, 0.0, 0.0);</span>
<span class="doccomment">/// assert!(!q.try_inverse_mut());</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">try_inverse_mut</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">bool</span> {
<span class="kw">let</span> <span class="ident">norm_squared</span> <span class="op">=</span> <span class="self">self</span>.<span class="ident">norm_squared</span>();
<span class="kw">if</span> <span class="macro">relative_eq</span><span class="macro">!</span>(<span class="kw-2">&</span><span class="ident">norm_squared</span>, <span class="kw-2">&</span><span class="ident">N</span>::<span class="ident">zero</span>()) {
<span class="bool-val">false</span>
} <span class="kw">else</span> {
<span class="self">self</span>.<span class="ident">conjugate_mut</span>();
<span class="self">self</span>.<span class="ident">coords</span> <span class="op">/</span><span class="op">=</span> <span class="ident">norm_squared</span>;
<span class="bool-val">true</span>
}
}
<span class="doccomment">/// Normalizes this quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let mut q = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// q.normalize_mut();</span>
<span class="doccomment">/// assert_relative_eq!(q.norm(), 1.0);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">normalize_mut</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">N</span> {
<span class="self">self</span>.<span class="ident">coords</span>.<span class="ident">normalize_mut</span>()
}
<span class="doccomment">/// Calculates square of a quaternion.</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">squared</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="self">self</span> <span class="op">*</span> <span class="self">self</span>
}
<span class="doccomment">/// Divides quaternion into two.</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">half</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="self">self</span> <span class="op">/</span> <span class="kw">crate</span>::<span class="ident">convert</span>(<span class="number">2.0f64</span>)
}
<span class="doccomment">/// Calculates square root.</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">sqrt</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="self">self</span>.<span class="ident">powf</span>(<span class="kw">crate</span>::<span class="ident">convert</span>(<span class="number">0.5</span>))
}
<span class="doccomment">/// Check if the quaternion is pure.</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">is_pure</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">bool</span> {
<span class="self">self</span>.<span class="ident">w</span>.<span class="ident">is_zero</span>()
}
<span class="doccomment">/// Convert quaternion to pure quaternion.</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">pure</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="self">Self</span>::<span class="ident">from_imag</span>(<span class="self">self</span>.<span class="ident">imag</span>())
}
<span class="doccomment">/// Left quaternionic division.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// Calculates B<sup>-1</sup> * A where A = self, B = other.</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">left_div</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">other</span>: <span class="kw-2">&</span><span class="self">Self</span>) <span class="op">-</span><span class="op">></span> <span class="prelude-ty">Option</span><span class="op"><</span><span class="self">Self</span><span class="op">></span> {
<span class="ident">other</span>.<span class="ident">try_inverse</span>().<span class="ident">map</span>(<span class="op">|</span><span class="ident">inv</span><span class="op">|</span> <span class="ident">inv</span> <span class="op">*</span> <span class="self">self</span>)
}
<span class="doccomment">/// Right quaternionic division.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// Calculates A * B<sup>-1</sup> where A = self, B = other.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let a = Quaternion::new(0.0, 1.0, 2.0, 3.0);</span>
<span class="doccomment">/// let b = Quaternion::new(0.0, 5.0, 2.0, 1.0);</span>
<span class="doccomment">/// let result = a.right_div(&b).unwrap();</span>
<span class="doccomment">/// let expected = Quaternion::new(0.4, 0.13333333333333336, -0.4666666666666667, 0.26666666666666666);</span>
<span class="doccomment">/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">right_div</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">other</span>: <span class="kw-2">&</span><span class="self">Self</span>) <span class="op">-</span><span class="op">></span> <span class="prelude-ty">Option</span><span class="op"><</span><span class="self">Self</span><span class="op">></span> {
<span class="ident">other</span>.<span class="ident">try_inverse</span>().<span class="ident">map</span>(<span class="op">|</span><span class="ident">inv</span><span class="op">|</span> <span class="self">self</span> <span class="op">*</span> <span class="ident">inv</span>)
}
<span class="doccomment">/// Calculates the quaternionic cosinus.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// let expected = Quaternion::new(58.93364616794395, -34.086183690465596, -51.1292755356984, -68.17236738093119);</span>
<span class="doccomment">/// let result = input.cos();</span>
<span class="doccomment">/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">cos</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="kw">let</span> <span class="ident">z</span> <span class="op">=</span> <span class="self">self</span>.<span class="ident">imag</span>().<span class="ident">magnitude</span>();
<span class="kw">let</span> <span class="ident">w</span> <span class="op">=</span> <span class="op">-</span><span class="self">self</span>.<span class="ident">w</span>.<span class="ident">sin</span>() <span class="op">*</span> <span class="ident">z</span>.<span class="ident">sinhc</span>();
<span class="self">Self</span>::<span class="ident">from_parts</span>(<span class="self">self</span>.<span class="ident">w</span>.<span class="ident">cos</span>() <span class="op">*</span> <span class="ident">z</span>.<span class="ident">cosh</span>(), <span class="self">self</span>.<span class="ident">imag</span>() <span class="op">*</span> <span class="ident">w</span>)
}
<span class="doccomment">/// Calculates the quaternionic arccosinus.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// let result = input.cos().acos();</span>
<span class="doccomment">/// assert_relative_eq!(input, result, epsilon = 1.0e-7);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">acos</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="kw">let</span> <span class="ident">u</span> <span class="op">=</span> <span class="self">Self</span>::<span class="ident">from_imag</span>(<span class="self">self</span>.<span class="ident">imag</span>().<span class="ident">normalize</span>());
<span class="kw">let</span> <span class="ident">identity</span> <span class="op">=</span> <span class="self">Self</span>::<span class="ident">identity</span>();
<span class="kw">let</span> <span class="ident">z</span> <span class="op">=</span> (<span class="self">self</span> <span class="op">+</span> (<span class="self">self</span>.<span class="ident">squared</span>() <span class="op">-</span> <span class="ident">identity</span>).<span class="ident">sqrt</span>()).<span class="ident">ln</span>();
<span class="op">-</span>(<span class="ident">u</span> <span class="op">*</span> <span class="ident">z</span>)
}
<span class="doccomment">/// Calculates the quaternionic sinus.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// let expected = Quaternion::new(91.78371578403467, 21.886486853029176, 32.82973027954377, 43.77297370605835);</span>
<span class="doccomment">/// let result = input.sin();</span>
<span class="doccomment">/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">sin</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="kw">let</span> <span class="ident">z</span> <span class="op">=</span> <span class="self">self</span>.<span class="ident">imag</span>().<span class="ident">magnitude</span>();
<span class="kw">let</span> <span class="ident">w</span> <span class="op">=</span> <span class="self">self</span>.<span class="ident">w</span>.<span class="ident">cos</span>() <span class="op">*</span> <span class="ident">z</span>.<span class="ident">sinhc</span>();
<span class="self">Self</span>::<span class="ident">from_parts</span>(<span class="self">self</span>.<span class="ident">w</span>.<span class="ident">sin</span>() <span class="op">*</span> <span class="ident">z</span>.<span class="ident">cosh</span>(), <span class="self">self</span>.<span class="ident">imag</span>() <span class="op">*</span> <span class="ident">w</span>)
}
<span class="doccomment">/// Calculates the quaternionic arcsinus.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// let result = input.sin().asin();</span>
<span class="doccomment">/// assert_relative_eq!(input, result, epsilon = 1.0e-7);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">asin</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="kw">let</span> <span class="ident">u</span> <span class="op">=</span> <span class="self">Self</span>::<span class="ident">from_imag</span>(<span class="self">self</span>.<span class="ident">imag</span>().<span class="ident">normalize</span>());
<span class="kw">let</span> <span class="ident">identity</span> <span class="op">=</span> <span class="self">Self</span>::<span class="ident">identity</span>();
<span class="kw">let</span> <span class="ident">z</span> <span class="op">=</span> ((<span class="ident">u</span> <span class="op">*</span> <span class="self">self</span>) <span class="op">+</span> (<span class="ident">identity</span> <span class="op">-</span> <span class="self">self</span>.<span class="ident">squared</span>()).<span class="ident">sqrt</span>()).<span class="ident">ln</span>();
<span class="op">-</span>(<span class="ident">u</span> <span class="op">*</span> <span class="ident">z</span>)
}
<span class="doccomment">/// Calculates the quaternionic tangent.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// let expected = Quaternion::new(0.00003821631725009489, 0.3713971716439371, 0.5570957574659058, 0.7427943432878743);</span>
<span class="doccomment">/// let result = input.tan();</span>
<span class="doccomment">/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">tan</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="self">self</span>.<span class="ident">sin</span>().<span class="ident">right_div</span>(<span class="kw-2">&</span><span class="self">self</span>.<span class="ident">cos</span>()).<span class="ident">unwrap</span>()
}
<span class="doccomment">/// Calculates the quaternionic arctangent.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// let result = input.tan().atan();</span>
<span class="doccomment">/// assert_relative_eq!(input, result, epsilon = 1.0e-7);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">atan</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="kw">let</span> <span class="ident">u</span> <span class="op">=</span> <span class="self">Self</span>::<span class="ident">from_imag</span>(<span class="self">self</span>.<span class="ident">imag</span>().<span class="ident">normalize</span>());
<span class="kw">let</span> <span class="ident">num</span> <span class="op">=</span> <span class="ident">u</span> <span class="op">+</span> <span class="self">self</span>;
<span class="kw">let</span> <span class="ident">den</span> <span class="op">=</span> <span class="ident">u</span> <span class="op">-</span> <span class="self">self</span>;
<span class="kw">let</span> <span class="ident">fr</span> <span class="op">=</span> <span class="ident">num</span>.<span class="ident">right_div</span>(<span class="kw-2">&</span><span class="ident">den</span>).<span class="ident">unwrap</span>();
<span class="kw">let</span> <span class="ident">ln</span> <span class="op">=</span> <span class="ident">fr</span>.<span class="ident">ln</span>();
(<span class="ident">u</span>.<span class="ident">half</span>()) <span class="op">*</span> <span class="ident">ln</span>
}
<span class="doccomment">/// Calculates the hyperbolic quaternionic sinus.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// let expected = Quaternion::new(0.7323376060463428, -0.4482074499805421, -0.6723111749708133, -0.8964148999610843);</span>
<span class="doccomment">/// let result = input.sinh();</span>
<span class="doccomment">/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">sinh</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
(<span class="self">self</span>.<span class="ident">exp</span>() <span class="op">-</span> (<span class="op">-</span><span class="self">self</span>).<span class="ident">exp</span>()).<span class="ident">half</span>()
}
<span class="doccomment">/// Calculates the hyperbolic quaternionic arcsinus.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// let expected = Quaternion::new(2.385889902585242, 0.514052600662788, 0.7710789009941821, 1.028105201325576);</span>
<span class="doccomment">/// let result = input.asinh();</span>
<span class="doccomment">/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">asinh</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="kw">let</span> <span class="ident">identity</span> <span class="op">=</span> <span class="self">Self</span>::<span class="ident">identity</span>();
(<span class="self">self</span> <span class="op">+</span> (<span class="ident">identity</span> <span class="op">+</span> <span class="self">self</span>.<span class="ident">squared</span>()).<span class="ident">sqrt</span>()).<span class="ident">ln</span>()
}
<span class="doccomment">/// Calculates the hyperbolic quaternionic cosinus.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// let expected = Quaternion::new(0.9615851176369566, -0.3413521745610167, -0.5120282618415251, -0.6827043491220334);</span>
<span class="doccomment">/// let result = input.cosh();</span>
<span class="doccomment">/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">cosh</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
(<span class="self">self</span>.<span class="ident">exp</span>() <span class="op">+</span> (<span class="op">-</span><span class="self">self</span>).<span class="ident">exp</span>()).<span class="ident">half</span>()
}
<span class="doccomment">/// Calculates the hyperbolic quaternionic arccosinus.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// let expected = Quaternion::new(2.4014472020074007, 0.5162761016176176, 0.7744141524264264, 1.0325522032352352);</span>
<span class="doccomment">/// let result = input.acosh();</span>
<span class="doccomment">/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">acosh</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="kw">let</span> <span class="ident">identity</span> <span class="op">=</span> <span class="self">Self</span>::<span class="ident">identity</span>();
(<span class="self">self</span> <span class="op">+</span> (<span class="self">self</span> <span class="op">+</span> <span class="ident">identity</span>).<span class="ident">sqrt</span>() <span class="op">*</span> (<span class="self">self</span> <span class="op">-</span> <span class="ident">identity</span>).<span class="ident">sqrt</span>()).<span class="ident">ln</span>()
}
<span class="doccomment">/// Calculates the hyperbolic quaternionic tangent.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// let expected = Quaternion::new(1.0248695360556623, -0.10229568178876419, -0.1534435226831464, -0.20459136357752844);</span>
<span class="doccomment">/// let result = input.tanh();</span>
<span class="doccomment">/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">tanh</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="self">self</span>.<span class="ident">sinh</span>().<span class="ident">right_div</span>(<span class="kw-2">&</span><span class="self">self</span>.<span class="ident">cosh</span>()).<span class="ident">unwrap</span>()
}
<span class="doccomment">/// Calculates the hyperbolic quaternionic arctangent.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::Quaternion;</span>
<span class="doccomment">/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);</span>
<span class="doccomment">/// let expected = Quaternion::new(0.03230293287000163, 0.5173453683196951, 0.7760180524795426, 1.0346907366393903);</span>
<span class="doccomment">/// let result = input.atanh();</span>
<span class="doccomment">/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">atanh</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="kw">let</span> <span class="ident">identity</span> <span class="op">=</span> <span class="self">Self</span>::<span class="ident">identity</span>();
((<span class="ident">identity</span> <span class="op">+</span> <span class="self">self</span>).<span class="ident">ln</span>() <span class="op">-</span> (<span class="ident">identity</span> <span class="op">-</span> <span class="self">self</span>).<span class="ident">ln</span>()).<span class="ident">half</span>()
}
}
<span class="kw">impl</span><span class="op"><</span><span class="ident">N</span>: <span class="ident">RealField</span> <span class="op">+</span> <span class="ident">AbsDiffEq</span><span class="op"><</span><span class="ident">Epsilon</span> <span class="op">=</span> <span class="ident">N</span><span class="op">></span><span class="op">></span> <span class="ident">AbsDiffEq</span> <span class="kw">for</span> <span class="ident">Quaternion</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="kw">type</span> <span class="ident">Epsilon</span> <span class="op">=</span> <span class="ident">N</span>;
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">fn</span> <span class="ident">default_epsilon</span>() <span class="op">-</span><span class="op">></span> <span class="self">Self</span>::<span class="ident">Epsilon</span> {
<span class="ident">N</span>::<span class="ident">default_epsilon</span>()
}
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">fn</span> <span class="ident">abs_diff_eq</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">other</span>: <span class="kw-2">&</span><span class="self">Self</span>, <span class="ident">epsilon</span>: <span class="self">Self</span>::<span class="ident">Epsilon</span>) <span class="op">-</span><span class="op">></span> <span class="ident">bool</span> {
<span class="self">self</span>.<span class="ident">as_vector</span>().<span class="ident">abs_diff_eq</span>(<span class="ident">other</span>.<span class="ident">as_vector</span>(), <span class="ident">epsilon</span>) <span class="op">|</span><span class="op">|</span>
<span class="comment">// Account for the double-covering of S², i.e. q = -q</span>
<span class="self">self</span>.<span class="ident">as_vector</span>().<span class="ident">iter</span>().<span class="ident">zip</span>(<span class="ident">other</span>.<span class="ident">as_vector</span>().<span class="ident">iter</span>()).<span class="ident">all</span>(<span class="op">|</span>(<span class="ident">a</span>, <span class="ident">b</span>)<span class="op">|</span> <span class="ident">a</span>.<span class="ident">abs_diff_eq</span>(<span class="kw-2">&</span><span class="op">-</span><span class="kw-2">*</span><span class="ident">b</span>, <span class="ident">epsilon</span>))
}
}
<span class="kw">impl</span><span class="op"><</span><span class="ident">N</span>: <span class="ident">RealField</span> <span class="op">+</span> <span class="ident">RelativeEq</span><span class="op"><</span><span class="ident">Epsilon</span> <span class="op">=</span> <span class="ident">N</span><span class="op">></span><span class="op">></span> <span class="ident">RelativeEq</span> <span class="kw">for</span> <span class="ident">Quaternion</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">fn</span> <span class="ident">default_max_relative</span>() <span class="op">-</span><span class="op">></span> <span class="self">Self</span>::<span class="ident">Epsilon</span> {
<span class="ident">N</span>::<span class="ident">default_max_relative</span>()
}
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">fn</span> <span class="ident">relative_eq</span>(
<span class="kw-2">&</span><span class="self">self</span>,
<span class="ident">other</span>: <span class="kw-2">&</span><span class="self">Self</span>,
<span class="ident">epsilon</span>: <span class="self">Self</span>::<span class="ident">Epsilon</span>,
<span class="ident">max_relative</span>: <span class="self">Self</span>::<span class="ident">Epsilon</span>,
) <span class="op">-</span><span class="op">></span> <span class="ident">bool</span>
{
<span class="self">self</span>.<span class="ident">as_vector</span>().<span class="ident">relative_eq</span>(<span class="ident">other</span>.<span class="ident">as_vector</span>(), <span class="ident">epsilon</span>, <span class="ident">max_relative</span>) <span class="op">|</span><span class="op">|</span>
<span class="comment">// Account for the double-covering of S², i.e. q = -q</span>
<span class="self">self</span>.<span class="ident">as_vector</span>().<span class="ident">iter</span>().<span class="ident">zip</span>(<span class="ident">other</span>.<span class="ident">as_vector</span>().<span class="ident">iter</span>()).<span class="ident">all</span>(<span class="op">|</span>(<span class="ident">a</span>, <span class="ident">b</span>)<span class="op">|</span> <span class="ident">a</span>.<span class="ident">relative_eq</span>(<span class="kw-2">&</span><span class="op">-</span><span class="kw-2">*</span><span class="ident">b</span>, <span class="ident">epsilon</span>, <span class="ident">max_relative</span>))
}
}
<span class="kw">impl</span><span class="op"><</span><span class="ident">N</span>: <span class="ident">RealField</span> <span class="op">+</span> <span class="ident">UlpsEq</span><span class="op"><</span><span class="ident">Epsilon</span> <span class="op">=</span> <span class="ident">N</span><span class="op">></span><span class="op">></span> <span class="ident">UlpsEq</span> <span class="kw">for</span> <span class="ident">Quaternion</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">fn</span> <span class="ident">default_max_ulps</span>() <span class="op">-</span><span class="op">></span> <span class="ident">u32</span> {
<span class="ident">N</span>::<span class="ident">default_max_ulps</span>()
}
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">fn</span> <span class="ident">ulps_eq</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">other</span>: <span class="kw-2">&</span><span class="self">Self</span>, <span class="ident">epsilon</span>: <span class="self">Self</span>::<span class="ident">Epsilon</span>, <span class="ident">max_ulps</span>: <span class="ident">u32</span>) <span class="op">-</span><span class="op">></span> <span class="ident">bool</span> {
<span class="self">self</span>.<span class="ident">as_vector</span>().<span class="ident">ulps_eq</span>(<span class="ident">other</span>.<span class="ident">as_vector</span>(), <span class="ident">epsilon</span>, <span class="ident">max_ulps</span>) <span class="op">|</span><span class="op">|</span>
<span class="comment">// Account for the double-covering of S², i.e. q = -q.</span>
<span class="self">self</span>.<span class="ident">as_vector</span>().<span class="ident">iter</span>().<span class="ident">zip</span>(<span class="ident">other</span>.<span class="ident">as_vector</span>().<span class="ident">iter</span>()).<span class="ident">all</span>(<span class="op">|</span>(<span class="ident">a</span>, <span class="ident">b</span>)<span class="op">|</span> <span class="ident">a</span>.<span class="ident">ulps_eq</span>(<span class="kw-2">&</span><span class="op">-</span><span class="kw-2">*</span><span class="ident">b</span>, <span class="ident">epsilon</span>, <span class="ident">max_ulps</span>))
}
}
<span class="kw">impl</span><span class="op"><</span><span class="ident">N</span>: <span class="ident">RealField</span> <span class="op">+</span> <span class="ident">fmt</span>::<span class="ident">Display</span><span class="op">></span> <span class="ident">fmt</span>::<span class="ident">Display</span> <span class="kw">for</span> <span class="ident">Quaternion</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="kw">fn</span> <span class="ident">fmt</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">f</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">fmt</span>::<span class="ident">Formatter</span>) <span class="op">-</span><span class="op">></span> <span class="ident">fmt</span>::<span class="prelude-ty">Result</span> {
<span class="macro">write</span><span class="macro">!</span>(
<span class="ident">f</span>,
<span class="string">"Quaternion {} − ({}, {}, {})"</span>,
<span class="self">self</span>[<span class="number">3</span>], <span class="self">self</span>[<span class="number">0</span>], <span class="self">self</span>[<span class="number">1</span>], <span class="self">self</span>[<span class="number">2</span>]
)
}
}
<span class="doccomment">/// A unit quaternions. May be used to represent a rotation.</span>
<span class="kw">pub</span> <span class="kw">type</span> <span class="ident">UnitQuaternion</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> <span class="op">=</span> <span class="ident">Unit</span><span class="op"><</span><span class="ident">Quaternion</span><span class="op"><</span><span class="ident">N</span><span class="op">></span><span class="op">></span>;
<span class="kw">impl</span><span class="op"><</span><span class="ident">N</span>: <span class="ident">RealField</span><span class="op">></span> <span class="ident">UnitQuaternion</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="doccomment">/// Moves this unit quaternion into one that owns its data.</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="attribute">#[<span class="ident">deprecated</span>(
<span class="ident">note</span> <span class="op">=</span> <span class="string">"This method is unnecessary and will be removed in a future release. Use `.clone()` instead."</span>
)]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">into_owned</span>(<span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="self">self</span>
}
<span class="doccomment">/// Clones this unit quaternion into one that owns its data.</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="attribute">#[<span class="ident">deprecated</span>(
<span class="ident">note</span> <span class="op">=</span> <span class="string">"This method is unnecessary and will be removed in a future release. Use `.clone()` instead."</span>
)]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">clone_owned</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="kw-2">*</span><span class="self">self</span>
}
<span class="doccomment">/// The rotation angle in [0; pi] of this unit quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # use nalgebra::{Unit, UnitQuaternion, Vector3};</span>
<span class="doccomment">/// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));</span>
<span class="doccomment">/// let rot = UnitQuaternion::from_axis_angle(&axis, 1.78);</span>
<span class="doccomment">/// assert_eq!(rot.angle(), 1.78);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">angle</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">N</span> {
<span class="kw">let</span> <span class="ident">w</span> <span class="op">=</span> <span class="self">self</span>.<span class="ident">quaternion</span>().<span class="ident">scalar</span>().<span class="ident">abs</span>();
<span class="self">self</span>.<span class="ident">quaternion</span>().<span class="ident">imag</span>().<span class="ident">norm</span>().<span class="ident">atan2</span>(<span class="ident">w</span>) <span class="op">*</span> <span class="kw">crate</span>::<span class="ident">convert</span>(<span class="number">2.0f64</span>)
}
<span class="doccomment">/// The underlying quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// Same as `self.as_ref()`.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # use nalgebra::{UnitQuaternion, Quaternion};</span>
<span class="doccomment">/// let axis = UnitQuaternion::identity();</span>
<span class="doccomment">/// assert_eq!(*axis.quaternion(), Quaternion::new(1.0, 0.0, 0.0, 0.0));</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">quaternion</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="kw-2">&</span><span class="ident">Quaternion</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="self">self</span>.<span class="ident">as_ref</span>()
}
<span class="doccomment">/// Compute the conjugate of this unit quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # use nalgebra::{Unit, UnitQuaternion, Vector3};</span>
<span class="doccomment">/// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));</span>
<span class="doccomment">/// let rot = UnitQuaternion::from_axis_angle(&axis, 1.78);</span>
<span class="doccomment">/// let conj = rot.conjugate();</span>
<span class="doccomment">/// assert_eq!(conj, UnitQuaternion::from_axis_angle(&-axis, 1.78));</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">conjugate</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="self">Self</span>::<span class="ident">new_unchecked</span>(<span class="self">self</span>.<span class="ident">as_ref</span>().<span class="ident">conjugate</span>())
}
<span class="doccomment">/// Inverts this quaternion if it is not zero.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # use nalgebra::{Unit, UnitQuaternion, Vector3};</span>
<span class="doccomment">/// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));</span>
<span class="doccomment">/// let rot = UnitQuaternion::from_axis_angle(&axis, 1.78);</span>
<span class="doccomment">/// let inv = rot.inverse();</span>
<span class="doccomment">/// assert_eq!(rot * inv, UnitQuaternion::identity());</span>
<span class="doccomment">/// assert_eq!(inv * rot, UnitQuaternion::identity());</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">inverse</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="self">self</span>.<span class="ident">conjugate</span>()
}
<span class="doccomment">/// The rotation angle needed to make `self` and `other` coincide.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::{UnitQuaternion, Vector3};</span>
<span class="doccomment">/// let rot1 = UnitQuaternion::from_axis_angle(&Vector3::y_axis(), 1.0);</span>
<span class="doccomment">/// let rot2 = UnitQuaternion::from_axis_angle(&Vector3::x_axis(), 0.1);</span>
<span class="doccomment">/// assert_relative_eq!(rot1.angle_to(&rot2), 1.0045657, epsilon = 1.0e-6);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">angle_to</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">other</span>: <span class="kw-2">&</span><span class="self">Self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">N</span> {
<span class="kw">let</span> <span class="ident">delta</span> <span class="op">=</span> <span class="self">self</span>.<span class="ident">rotation_to</span>(<span class="ident">other</span>);
<span class="ident">delta</span>.<span class="ident">angle</span>()
}
<span class="doccomment">/// The unit quaternion needed to make `self` and `other` coincide.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// The result is such that: `self.rotation_to(other) * self == other`.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::{UnitQuaternion, Vector3};</span>
<span class="doccomment">/// let rot1 = UnitQuaternion::from_axis_angle(&Vector3::y_axis(), 1.0);</span>
<span class="doccomment">/// let rot2 = UnitQuaternion::from_axis_angle(&Vector3::x_axis(), 0.1);</span>
<span class="doccomment">/// let rot_to = rot1.rotation_to(&rot2);</span>
<span class="doccomment">/// assert_relative_eq!(rot_to * rot1, rot2, epsilon = 1.0e-6);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">rotation_to</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">other</span>: <span class="kw-2">&</span><span class="self">Self</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span>{
<span class="ident">other</span> <span class="op">/</span> <span class="self">self</span>
}
<span class="doccomment">/// Linear interpolation between two unit quaternions.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// The result is not normalized.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # use nalgebra::{UnitQuaternion, Quaternion};</span>
<span class="doccomment">/// let q1 = UnitQuaternion::new_normalize(Quaternion::new(1.0, 0.0, 0.0, 0.0));</span>
<span class="doccomment">/// let q2 = UnitQuaternion::new_normalize(Quaternion::new(0.0, 1.0, 0.0, 0.0));</span>
<span class="doccomment">/// assert_eq!(q1.lerp(&q2, 0.1), Quaternion::new(0.9, 0.1, 0.0, 0.0));</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">lerp</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">other</span>: <span class="kw-2">&</span><span class="self">Self</span>, <span class="ident">t</span>: <span class="ident">N</span>) <span class="op">-</span><span class="op">></span> <span class="ident">Quaternion</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="self">self</span>.<span class="ident">as_ref</span>().<span class="ident">lerp</span>(<span class="ident">other</span>.<span class="ident">as_ref</span>(), <span class="ident">t</span>)
}
<span class="doccomment">/// Normalized linear interpolation between two unit quaternions.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// This is the same as `self.lerp` except that the result is normalized.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # use nalgebra::{UnitQuaternion, Quaternion};</span>
<span class="doccomment">/// let q1 = UnitQuaternion::new_normalize(Quaternion::new(1.0, 0.0, 0.0, 0.0));</span>
<span class="doccomment">/// let q2 = UnitQuaternion::new_normalize(Quaternion::new(0.0, 1.0, 0.0, 0.0));</span>
<span class="doccomment">/// assert_eq!(q1.nlerp(&q2, 0.1), UnitQuaternion::new_normalize(Quaternion::new(0.9, 0.1, 0.0, 0.0)));</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">nlerp</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">other</span>: <span class="kw-2">&</span><span class="self">Self</span>, <span class="ident">t</span>: <span class="ident">N</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">res</span> <span class="op">=</span> <span class="self">self</span>.<span class="ident">lerp</span>(<span class="ident">other</span>, <span class="ident">t</span>);
<span class="kw">let</span> <span class="kw">_</span> <span class="op">=</span> <span class="ident">res</span>.<span class="ident">normalize_mut</span>();
<span class="self">Self</span>::<span class="ident">new_unchecked</span>(<span class="ident">res</span>)
}
<span class="doccomment">/// Spherical linear interpolation between two unit quaternions.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// Panics if the angle between both quaternion is 180 degrees (in which case the interpolation</span>
<span class="doccomment">/// is not well-defined). Use `.try_slerp` instead to avoid the panic.</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">slerp</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">other</span>: <span class="kw-2">&</span><span class="self">Self</span>, <span class="ident">t</span>: <span class="ident">N</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="ident">Unit</span>::<span class="ident">new_unchecked</span>(<span class="ident">Quaternion</span>::<span class="ident">from</span>(
<span class="ident">Unit</span>::<span class="ident">new_unchecked</span>(<span class="self">self</span>.<span class="ident">coords</span>)
.<span class="ident">slerp</span>(<span class="kw-2">&</span><span class="ident">Unit</span>::<span class="ident">new_unchecked</span>(<span class="ident">other</span>.<span class="ident">coords</span>), <span class="ident">t</span>)
.<span class="ident">into_inner</span>(),
))
}
<span class="doccomment">/// Computes the spherical linear interpolation between two unit quaternions or returns `None`</span>
<span class="doccomment">/// if both quaternions are approximately 180 degrees apart (in which case the interpolation is</span>
<span class="doccomment">/// not well-defined).</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Arguments</span>
<span class="doccomment">/// * `self`: the first quaternion to interpolate from.</span>
<span class="doccomment">/// * `other`: the second quaternion to interpolate toward.</span>
<span class="doccomment">/// * `t`: the interpolation parameter. Should be between 0 and 1.</span>
<span class="doccomment">/// * `epsilon`: the value below which the sinus of the angle separating both quaternion</span>
<span class="doccomment">/// must be to return `None`.</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">try_slerp</span>(
<span class="kw-2">&</span><span class="self">self</span>,
<span class="ident">other</span>: <span class="kw-2">&</span><span class="self">Self</span>,
<span class="ident">t</span>: <span class="ident">N</span>,
<span class="ident">epsilon</span>: <span class="ident">N</span>,
) <span class="op">-</span><span class="op">></span> <span class="prelude-ty">Option</span><span class="op"><</span><span class="self">Self</span><span class="op">></span>
{
<span class="ident">Unit</span>::<span class="ident">new_unchecked</span>(<span class="self">self</span>.<span class="ident">coords</span>)
.<span class="ident">try_slerp</span>(<span class="kw-2">&</span><span class="ident">Unit</span>::<span class="ident">new_unchecked</span>(<span class="ident">other</span>.<span class="ident">coords</span>), <span class="ident">t</span>, <span class="ident">epsilon</span>)
.<span class="ident">map</span>(<span class="op">|</span><span class="ident">q</span><span class="op">|</span> <span class="ident">Unit</span>::<span class="ident">new_unchecked</span>(<span class="ident">Quaternion</span>::<span class="ident">from</span>(<span class="ident">q</span>.<span class="ident">into_inner</span>())))
}
<span class="doccomment">/// Compute the conjugate of this unit quaternion in-place.</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">conjugate_mut</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="self">self</span>) {
<span class="self">self</span>.<span class="ident">as_mut_unchecked</span>().<span class="ident">conjugate_mut</span>()
}
<span class="doccomment">/// Inverts this quaternion if it is not zero.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::{UnitQuaternion, Vector3, Unit};</span>
<span class="doccomment">/// let axisangle = Vector3::new(0.1, 0.2, 0.3);</span>
<span class="doccomment">/// let mut rot = UnitQuaternion::new(axisangle);</span>
<span class="doccomment">/// rot.inverse_mut();</span>
<span class="doccomment">/// assert_relative_eq!(rot * UnitQuaternion::new(axisangle), UnitQuaternion::identity());</span>
<span class="doccomment">/// assert_relative_eq!(UnitQuaternion::new(axisangle) * rot, UnitQuaternion::identity());</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">inverse_mut</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="self">self</span>) {
<span class="self">self</span>.<span class="ident">as_mut_unchecked</span>().<span class="ident">conjugate_mut</span>()
}
<span class="doccomment">/// The rotation axis of this unit quaternion or `None` if the rotation is zero.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # use nalgebra::{UnitQuaternion, Vector3, Unit};</span>
<span class="doccomment">/// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));</span>
<span class="doccomment">/// let angle = 1.2;</span>
<span class="doccomment">/// let rot = UnitQuaternion::from_axis_angle(&axis, angle);</span>
<span class="doccomment">/// assert_eq!(rot.axis(), Some(axis));</span>
<span class="doccomment">///</span>
<span class="doccomment">/// // Case with a zero angle.</span>
<span class="doccomment">/// let rot = UnitQuaternion::from_axis_angle(&axis, 0.0);</span>
<span class="doccomment">/// assert!(rot.axis().is_none());</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">axis</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="prelude-ty">Option</span><span class="op"><</span><span class="ident">Unit</span><span class="op"><</span><span class="ident">Vector3</span><span class="op"><</span><span class="ident">N</span><span class="op">></span><span class="op">></span><span class="op">></span> {
<span class="kw">let</span> <span class="ident">v</span> <span class="op">=</span> <span class="kw">if</span> <span class="self">self</span>.<span class="ident">quaternion</span>().<span class="ident">scalar</span>() <span class="op">></span><span class="op">=</span> <span class="ident">N</span>::<span class="ident">zero</span>() {
<span class="self">self</span>.<span class="ident">as_ref</span>().<span class="ident">vector</span>().<span class="ident">clone_owned</span>()
} <span class="kw">else</span> {
<span class="op">-</span><span class="self">self</span>.<span class="ident">as_ref</span>().<span class="ident">vector</span>()
};
<span class="ident">Unit</span>::<span class="ident">try_new</span>(<span class="ident">v</span>, <span class="ident">N</span>::<span class="ident">zero</span>())
}
<span class="doccomment">/// The rotation axis of this unit quaternion multiplied by the rotation angle.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::{UnitQuaternion, Vector3, Unit};</span>
<span class="doccomment">/// let axisangle = Vector3::new(0.1, 0.2, 0.3);</span>
<span class="doccomment">/// let rot = UnitQuaternion::new(axisangle);</span>
<span class="doccomment">/// assert_relative_eq!(rot.scaled_axis(), axisangle, epsilon = 1.0e-6);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">scaled_axis</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">Vector3</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="kw">if</span> <span class="kw">let</span> <span class="prelude-val">Some</span>(<span class="ident">axis</span>) <span class="op">=</span> <span class="self">self</span>.<span class="ident">axis</span>() {
<span class="ident">axis</span>.<span class="ident">into_inner</span>() <span class="op">*</span> <span class="self">self</span>.<span class="ident">angle</span>()
} <span class="kw">else</span> {
<span class="ident">Vector3</span>::<span class="ident">zero</span>()
}
}
<span class="doccomment">/// The rotation axis and angle in ]0, pi] of this unit quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// Returns `None` if the angle is zero.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # use nalgebra::{UnitQuaternion, Vector3, Unit};</span>
<span class="doccomment">/// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));</span>
<span class="doccomment">/// let angle = 1.2;</span>
<span class="doccomment">/// let rot = UnitQuaternion::from_axis_angle(&axis, angle);</span>
<span class="doccomment">/// assert_eq!(rot.axis_angle(), Some((axis, angle)));</span>
<span class="doccomment">///</span>
<span class="doccomment">/// // Case with a zero angle.</span>
<span class="doccomment">/// let rot = UnitQuaternion::from_axis_angle(&axis, 0.0);</span>
<span class="doccomment">/// assert!(rot.axis_angle().is_none());</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">axis_angle</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="prelude-ty">Option</span><span class="op"><</span>(<span class="ident">Unit</span><span class="op"><</span><span class="ident">Vector3</span><span class="op"><</span><span class="ident">N</span><span class="op">></span><span class="op">></span>, <span class="ident">N</span>)<span class="op">></span> {
<span class="self">self</span>.<span class="ident">axis</span>().<span class="ident">map</span>(<span class="op">|</span><span class="ident">axis</span><span class="op">|</span> (<span class="ident">axis</span>, <span class="self">self</span>.<span class="ident">angle</span>()))
}
<span class="doccomment">/// Compute the exponential of a quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// Note that this function yields a `Quaternion<N>` because it loses the unit property.</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">exp</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">Quaternion</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="self">self</span>.<span class="ident">as_ref</span>().<span class="ident">exp</span>()
}
<span class="doccomment">/// Compute the natural logarithm of a quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// Note that this function yields a `Quaternion<N>` because it loses the unit property.</span>
<span class="doccomment">/// The vector part of the return value corresponds to the axis-angle representation (divided</span>
<span class="doccomment">/// by 2.0) of this unit quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::{Vector3, UnitQuaternion};</span>
<span class="doccomment">/// let axisangle = Vector3::new(0.1, 0.2, 0.3);</span>
<span class="doccomment">/// let q = UnitQuaternion::new(axisangle);</span>
<span class="doccomment">/// assert_relative_eq!(q.ln().vector().into_owned(), axisangle, epsilon = 1.0e-6);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">ln</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">Quaternion</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="kw">if</span> <span class="kw">let</span> <span class="prelude-val">Some</span>(<span class="ident">v</span>) <span class="op">=</span> <span class="self">self</span>.<span class="ident">axis</span>() {
<span class="ident">Quaternion</span>::<span class="ident">from_imag</span>(<span class="ident">v</span>.<span class="ident">into_inner</span>() <span class="op">*</span> <span class="self">self</span>.<span class="ident">angle</span>())
} <span class="kw">else</span> {
<span class="ident">Quaternion</span>::<span class="ident">zero</span>()
}
}
<span class="doccomment">/// Raise the quaternion to a given floating power.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// This returns the unit quaternion that identifies a rotation with axis `self.axis()` and</span>
<span class="doccomment">/// angle `self.angle() × n`.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::{UnitQuaternion, Vector3, Unit};</span>
<span class="doccomment">/// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));</span>
<span class="doccomment">/// let angle = 1.2;</span>
<span class="doccomment">/// let rot = UnitQuaternion::from_axis_angle(&axis, angle);</span>
<span class="doccomment">/// let pow = rot.powf(2.0);</span>
<span class="doccomment">/// assert_relative_eq!(pow.axis().unwrap(), axis, epsilon = 1.0e-6);</span>
<span class="doccomment">/// assert_eq!(pow.angle(), 2.4);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">powf</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">n</span>: <span class="ident">N</span>) <span class="op">-</span><span class="op">></span> <span class="self">Self</span> {
<span class="kw">if</span> <span class="kw">let</span> <span class="prelude-val">Some</span>(<span class="ident">v</span>) <span class="op">=</span> <span class="self">self</span>.<span class="ident">axis</span>() {
<span class="self">Self</span>::<span class="ident">from_axis_angle</span>(<span class="kw-2">&</span><span class="ident">v</span>, <span class="self">self</span>.<span class="ident">angle</span>() <span class="op">*</span> <span class="ident">n</span>)
} <span class="kw">else</span> {
<span class="self">Self</span>::<span class="ident">identity</span>()
}
}
<span class="doccomment">/// Builds a rotation matrix from this unit quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">///</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use std::f32;</span>
<span class="doccomment">/// # use nalgebra::{UnitQuaternion, Vector3, Matrix3};</span>
<span class="doccomment">/// let q = UnitQuaternion::from_axis_angle(&Vector3::z_axis(), f32::consts::FRAC_PI_6);</span>
<span class="doccomment">/// let rot = q.to_rotation_matrix();</span>
<span class="doccomment">/// let expected = Matrix3::new(0.8660254, -0.5, 0.0,</span>
<span class="doccomment">/// 0.5, 0.8660254, 0.0,</span>
<span class="doccomment">/// 0.0, 0.0, 1.0);</span>
<span class="doccomment">///</span>
<span class="doccomment">/// assert_relative_eq!(*rot.matrix(), expected, epsilon = 1.0e-6);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">to_rotation_matrix</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">Rotation</span><span class="op"><</span><span class="ident">N</span>, <span class="ident">U3</span><span class="op">></span> {
<span class="kw">let</span> <span class="ident">i</span> <span class="op">=</span> <span class="self">self</span>.<span class="ident">as_ref</span>()[<span class="number">0</span>];
<span class="kw">let</span> <span class="ident">j</span> <span class="op">=</span> <span class="self">self</span>.<span class="ident">as_ref</span>()[<span class="number">1</span>];
<span class="kw">let</span> <span class="ident">k</span> <span class="op">=</span> <span class="self">self</span>.<span class="ident">as_ref</span>()[<span class="number">2</span>];
<span class="kw">let</span> <span class="ident">w</span> <span class="op">=</span> <span class="self">self</span>.<span class="ident">as_ref</span>()[<span class="number">3</span>];
<span class="kw">let</span> <span class="ident">ww</span> <span class="op">=</span> <span class="ident">w</span> <span class="op">*</span> <span class="ident">w</span>;
<span class="kw">let</span> <span class="ident">ii</span> <span class="op">=</span> <span class="ident">i</span> <span class="op">*</span> <span class="ident">i</span>;
<span class="kw">let</span> <span class="ident">jj</span> <span class="op">=</span> <span class="ident">j</span> <span class="op">*</span> <span class="ident">j</span>;
<span class="kw">let</span> <span class="ident">kk</span> <span class="op">=</span> <span class="ident">k</span> <span class="op">*</span> <span class="ident">k</span>;
<span class="kw">let</span> <span class="ident">ij</span> <span class="op">=</span> <span class="ident">i</span> <span class="op">*</span> <span class="ident">j</span> <span class="op">*</span> <span class="kw">crate</span>::<span class="ident">convert</span>(<span class="number">2.0f64</span>);
<span class="kw">let</span> <span class="ident">wk</span> <span class="op">=</span> <span class="ident">w</span> <span class="op">*</span> <span class="ident">k</span> <span class="op">*</span> <span class="kw">crate</span>::<span class="ident">convert</span>(<span class="number">2.0f64</span>);
<span class="kw">let</span> <span class="ident">wj</span> <span class="op">=</span> <span class="ident">w</span> <span class="op">*</span> <span class="ident">j</span> <span class="op">*</span> <span class="kw">crate</span>::<span class="ident">convert</span>(<span class="number">2.0f64</span>);
<span class="kw">let</span> <span class="ident">ik</span> <span class="op">=</span> <span class="ident">i</span> <span class="op">*</span> <span class="ident">k</span> <span class="op">*</span> <span class="kw">crate</span>::<span class="ident">convert</span>(<span class="number">2.0f64</span>);
<span class="kw">let</span> <span class="ident">jk</span> <span class="op">=</span> <span class="ident">j</span> <span class="op">*</span> <span class="ident">k</span> <span class="op">*</span> <span class="kw">crate</span>::<span class="ident">convert</span>(<span class="number">2.0f64</span>);
<span class="kw">let</span> <span class="ident">wi</span> <span class="op">=</span> <span class="ident">w</span> <span class="op">*</span> <span class="ident">i</span> <span class="op">*</span> <span class="kw">crate</span>::<span class="ident">convert</span>(<span class="number">2.0f64</span>);
<span class="ident">Rotation</span>::<span class="ident">from_matrix_unchecked</span>(<span class="ident">Matrix3</span>::<span class="ident">new</span>(
<span class="ident">ww</span> <span class="op">+</span> <span class="ident">ii</span> <span class="op">-</span> <span class="ident">jj</span> <span class="op">-</span> <span class="ident">kk</span>,
<span class="ident">ij</span> <span class="op">-</span> <span class="ident">wk</span>,
<span class="ident">wj</span> <span class="op">+</span> <span class="ident">ik</span>,
<span class="ident">wk</span> <span class="op">+</span> <span class="ident">ij</span>,
<span class="ident">ww</span> <span class="op">-</span> <span class="ident">ii</span> <span class="op">+</span> <span class="ident">jj</span> <span class="op">-</span> <span class="ident">kk</span>,
<span class="ident">jk</span> <span class="op">-</span> <span class="ident">wi</span>,
<span class="ident">ik</span> <span class="op">-</span> <span class="ident">wj</span>,
<span class="ident">wi</span> <span class="op">+</span> <span class="ident">jk</span>,
<span class="ident">ww</span> <span class="op">-</span> <span class="ident">ii</span> <span class="op">-</span> <span class="ident">jj</span> <span class="op">+</span> <span class="ident">kk</span>,
))
}
<span class="doccomment">/// Converts this unit quaternion into its equivalent Euler angles.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// The angles are produced in the form (roll, pitch, yaw).</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="attribute">#[<span class="ident">deprecated</span>(<span class="ident">note</span> <span class="op">=</span> <span class="string">"This is renamed to use `.euler_angles()`."</span>)]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">to_euler_angles</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> (<span class="ident">N</span>, <span class="ident">N</span>, <span class="ident">N</span>) {
<span class="self">self</span>.<span class="ident">euler_angles</span>()
}
<span class="doccomment">/// Retrieves the euler angles corresponding to this unit quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// The angles are produced in the form (roll, pitch, yaw).</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use nalgebra::UnitQuaternion;</span>
<span class="doccomment">/// let rot = UnitQuaternion::from_euler_angles(0.1, 0.2, 0.3);</span>
<span class="doccomment">/// let euler = rot.euler_angles();</span>
<span class="doccomment">/// assert_relative_eq!(euler.0, 0.1, epsilon = 1.0e-6);</span>
<span class="doccomment">/// assert_relative_eq!(euler.1, 0.2, epsilon = 1.0e-6);</span>
<span class="doccomment">/// assert_relative_eq!(euler.2, 0.3, epsilon = 1.0e-6);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">euler_angles</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> (<span class="ident">N</span>, <span class="ident">N</span>, <span class="ident">N</span>) {
<span class="self">self</span>.<span class="ident">to_rotation_matrix</span>().<span class="ident">euler_angles</span>()
}
<span class="doccomment">/// Converts this unit quaternion into its equivalent homogeneous transformation matrix.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">///</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use std::f32;</span>
<span class="doccomment">/// # use nalgebra::{UnitQuaternion, Vector3, Matrix4};</span>
<span class="doccomment">/// let rot = UnitQuaternion::from_axis_angle(&Vector3::z_axis(), f32::consts::FRAC_PI_6);</span>
<span class="doccomment">/// let expected = Matrix4::new(0.8660254, -0.5, 0.0, 0.0,</span>
<span class="doccomment">/// 0.5, 0.8660254, 0.0, 0.0,</span>
<span class="doccomment">/// 0.0, 0.0, 1.0, 0.0,</span>
<span class="doccomment">/// 0.0, 0.0, 0.0, 1.0);</span>
<span class="doccomment">///</span>
<span class="doccomment">/// assert_relative_eq!(rot.to_homogeneous(), expected, epsilon = 1.0e-6);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">to_homogeneous</span>(<span class="kw-2">&</span><span class="self">self</span>) <span class="op">-</span><span class="op">></span> <span class="ident">Matrix4</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="self">self</span>.<span class="ident">to_rotation_matrix</span>().<span class="ident">to_homogeneous</span>()
}
<span class="doccomment">/// Rotate a point by this unit quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// This is the same as the multiplication `self * pt`.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">///</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use std::f32;</span>
<span class="doccomment">/// # use nalgebra::{UnitQuaternion, Vector3, Point3};</span>
<span class="doccomment">/// let rot = UnitQuaternion::from_axis_angle(&Vector3::y_axis(), f32::consts::FRAC_PI_2);</span>
<span class="doccomment">/// let transformed_point = rot.transform_point(&Point3::new(1.0, 2.0, 3.0));</span>
<span class="doccomment">///</span>
<span class="doccomment">/// assert_relative_eq!(transformed_point, Point3::new(3.0, 2.0, -1.0), epsilon = 1.0e-6);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">transform_point</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">pt</span>: <span class="kw-2">&</span><span class="ident">Point3</span><span class="op"><</span><span class="ident">N</span><span class="op">></span>) <span class="op">-</span><span class="op">></span> <span class="ident">Point3</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="self">self</span> <span class="op">*</span> <span class="ident">pt</span>
}
<span class="doccomment">/// Rotate a vector by this unit quaternion.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// This is the same as the multiplication `self * v`.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">///</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use std::f32;</span>
<span class="doccomment">/// # use nalgebra::{UnitQuaternion, Vector3};</span>
<span class="doccomment">/// let rot = UnitQuaternion::from_axis_angle(&Vector3::y_axis(), f32::consts::FRAC_PI_2);</span>
<span class="doccomment">/// let transformed_vector = rot.transform_vector(&Vector3::new(1.0, 2.0, 3.0));</span>
<span class="doccomment">///</span>
<span class="doccomment">/// assert_relative_eq!(transformed_vector, Vector3::new(3.0, 2.0, -1.0), epsilon = 1.0e-6);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">transform_vector</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">v</span>: <span class="kw-2">&</span><span class="ident">Vector3</span><span class="op"><</span><span class="ident">N</span><span class="op">></span>) <span class="op">-</span><span class="op">></span> <span class="ident">Vector3</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="self">self</span> <span class="op">*</span> <span class="ident">v</span>
}
<span class="doccomment">/// Rotate a point by the inverse of this unit quaternion. This may be</span>
<span class="doccomment">/// cheaper than inverting the unit quaternion and transforming the</span>
<span class="doccomment">/// point.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">///</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use std::f32;</span>
<span class="doccomment">/// # use nalgebra::{UnitQuaternion, Vector3, Point3};</span>
<span class="doccomment">/// let rot = UnitQuaternion::from_axis_angle(&Vector3::y_axis(), f32::consts::FRAC_PI_2);</span>
<span class="doccomment">/// let transformed_point = rot.inverse_transform_point(&Point3::new(1.0, 2.0, 3.0));</span>
<span class="doccomment">///</span>
<span class="doccomment">/// assert_relative_eq!(transformed_point, Point3::new(-3.0, 2.0, 1.0), epsilon = 1.0e-6);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">inverse_transform_point</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">pt</span>: <span class="kw-2">&</span><span class="ident">Point3</span><span class="op"><</span><span class="ident">N</span><span class="op">></span>) <span class="op">-</span><span class="op">></span> <span class="ident">Point3</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="comment">// FIXME: would it be useful performancewise not to call inverse explicitly (i-e. implement</span>
<span class="comment">// the inverse transformation explicitly here) ?</span>
<span class="self">self</span>.<span class="ident">inverse</span>() <span class="op">*</span> <span class="ident">pt</span>
}
<span class="doccomment">/// Rotate a vector by the inverse of this unit quaternion. This may be</span>
<span class="doccomment">/// cheaper than inverting the unit quaternion and transforming the</span>
<span class="doccomment">/// vector.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// # Example</span>
<span class="doccomment">///</span>
<span class="doccomment">/// ```</span>
<span class="doccomment">/// # #[macro_use] extern crate approx;</span>
<span class="doccomment">/// # use std::f32;</span>
<span class="doccomment">/// # use nalgebra::{UnitQuaternion, Vector3};</span>
<span class="doccomment">/// let rot = UnitQuaternion::from_axis_angle(&Vector3::y_axis(), f32::consts::FRAC_PI_2);</span>
<span class="doccomment">/// let transformed_vector = rot.inverse_transform_vector(&Vector3::new(1.0, 2.0, 3.0));</span>
<span class="doccomment">///</span>
<span class="doccomment">/// assert_relative_eq!(transformed_vector, Vector3::new(-3.0, 2.0, 1.0), epsilon = 1.0e-6);</span>
<span class="doccomment">/// ```</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">inverse_transform_vector</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">v</span>: <span class="kw-2">&</span><span class="ident">Vector3</span><span class="op"><</span><span class="ident">N</span><span class="op">></span>) <span class="op">-</span><span class="op">></span> <span class="ident">Vector3</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="self">self</span>.<span class="ident">inverse</span>() <span class="op">*</span> <span class="ident">v</span>
}
}
<span class="kw">impl</span><span class="op"><</span><span class="ident">N</span>: <span class="ident">RealField</span> <span class="op">+</span> <span class="ident">fmt</span>::<span class="ident">Display</span><span class="op">></span> <span class="ident">fmt</span>::<span class="ident">Display</span> <span class="kw">for</span> <span class="ident">UnitQuaternion</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="kw">fn</span> <span class="ident">fmt</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">f</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">fmt</span>::<span class="ident">Formatter</span>) <span class="op">-</span><span class="op">></span> <span class="ident">fmt</span>::<span class="prelude-ty">Result</span> {
<span class="kw">if</span> <span class="kw">let</span> <span class="prelude-val">Some</span>(<span class="ident">axis</span>) <span class="op">=</span> <span class="self">self</span>.<span class="ident">axis</span>() {
<span class="kw">let</span> <span class="ident">axis</span> <span class="op">=</span> <span class="ident">axis</span>.<span class="ident">into_inner</span>();
<span class="macro">write</span><span class="macro">!</span>(
<span class="ident">f</span>,
<span class="string">"UnitQuaternion angle: {} − axis: ({}, {}, {})"</span>,
<span class="self">self</span>.<span class="ident">angle</span>(),
<span class="ident">axis</span>[<span class="number">0</span>],
<span class="ident">axis</span>[<span class="number">1</span>],
<span class="ident">axis</span>[<span class="number">2</span>]
)
} <span class="kw">else</span> {
<span class="macro">write</span><span class="macro">!</span>(
<span class="ident">f</span>,
<span class="string">"UnitQuaternion angle: {} − axis: (undefined)"</span>,
<span class="self">self</span>.<span class="ident">angle</span>()
)
}
}
}
<span class="kw">impl</span><span class="op"><</span><span class="ident">N</span>: <span class="ident">RealField</span> <span class="op">+</span> <span class="ident">AbsDiffEq</span><span class="op"><</span><span class="ident">Epsilon</span> <span class="op">=</span> <span class="ident">N</span><span class="op">></span><span class="op">></span> <span class="ident">AbsDiffEq</span> <span class="kw">for</span> <span class="ident">UnitQuaternion</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="kw">type</span> <span class="ident">Epsilon</span> <span class="op">=</span> <span class="ident">N</span>;
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">fn</span> <span class="ident">default_epsilon</span>() <span class="op">-</span><span class="op">></span> <span class="self">Self</span>::<span class="ident">Epsilon</span> {
<span class="ident">N</span>::<span class="ident">default_epsilon</span>()
}
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">fn</span> <span class="ident">abs_diff_eq</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">other</span>: <span class="kw-2">&</span><span class="self">Self</span>, <span class="ident">epsilon</span>: <span class="self">Self</span>::<span class="ident">Epsilon</span>) <span class="op">-</span><span class="op">></span> <span class="ident">bool</span> {
<span class="self">self</span>.<span class="ident">as_ref</span>().<span class="ident">abs_diff_eq</span>(<span class="ident">other</span>.<span class="ident">as_ref</span>(), <span class="ident">epsilon</span>)
}
}
<span class="kw">impl</span><span class="op"><</span><span class="ident">N</span>: <span class="ident">RealField</span> <span class="op">+</span> <span class="ident">RelativeEq</span><span class="op"><</span><span class="ident">Epsilon</span> <span class="op">=</span> <span class="ident">N</span><span class="op">></span><span class="op">></span> <span class="ident">RelativeEq</span> <span class="kw">for</span> <span class="ident">UnitQuaternion</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">fn</span> <span class="ident">default_max_relative</span>() <span class="op">-</span><span class="op">></span> <span class="self">Self</span>::<span class="ident">Epsilon</span> {
<span class="ident">N</span>::<span class="ident">default_max_relative</span>()
}
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">fn</span> <span class="ident">relative_eq</span>(
<span class="kw-2">&</span><span class="self">self</span>,
<span class="ident">other</span>: <span class="kw-2">&</span><span class="self">Self</span>,
<span class="ident">epsilon</span>: <span class="self">Self</span>::<span class="ident">Epsilon</span>,
<span class="ident">max_relative</span>: <span class="self">Self</span>::<span class="ident">Epsilon</span>,
) <span class="op">-</span><span class="op">></span> <span class="ident">bool</span>
{
<span class="self">self</span>.<span class="ident">as_ref</span>()
.<span class="ident">relative_eq</span>(<span class="ident">other</span>.<span class="ident">as_ref</span>(), <span class="ident">epsilon</span>, <span class="ident">max_relative</span>)
}
}
<span class="kw">impl</span><span class="op"><</span><span class="ident">N</span>: <span class="ident">RealField</span> <span class="op">+</span> <span class="ident">UlpsEq</span><span class="op"><</span><span class="ident">Epsilon</span> <span class="op">=</span> <span class="ident">N</span><span class="op">></span><span class="op">></span> <span class="ident">UlpsEq</span> <span class="kw">for</span> <span class="ident">UnitQuaternion</span><span class="op"><</span><span class="ident">N</span><span class="op">></span> {
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">fn</span> <span class="ident">default_max_ulps</span>() <span class="op">-</span><span class="op">></span> <span class="ident">u32</span> {
<span class="ident">N</span>::<span class="ident">default_max_ulps</span>()
}
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">fn</span> <span class="ident">ulps_eq</span>(<span class="kw-2">&</span><span class="self">self</span>, <span class="ident">other</span>: <span class="kw-2">&</span><span class="self">Self</span>, <span class="ident">epsilon</span>: <span class="self">Self</span>::<span class="ident">Epsilon</span>, <span class="ident">max_ulps</span>: <span class="ident">u32</span>) <span class="op">-</span><span class="op">></span> <span class="ident">bool</span> {
<span class="self">self</span>.<span class="ident">as_ref</span>().<span class="ident">ulps_eq</span>(<span class="ident">other</span>.<span class="ident">as_ref</span>(), <span class="ident">epsilon</span>, <span class="ident">max_ulps</span>)
}
}
</pre></div>
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