build based on 6d90f17

JuliaGeometry · Feb 5, 2024 · e986692 · e986692
1 parent d7fa633
commit e986692
Show file tree

Hide file tree

Showing 8 changed files with 58 additions and 36 deletions.
diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json
@@ -1 +1 @@
-{"documenter":{"julia_version":"1.9.4","generation_timestamp":"2023-12-23T03:53:19","documenter_version":"1.2.1"}}
+{"documenter":{"julia_version":"1.10.0","generation_timestamp":"2024-02-05T03:38:28","documenter_version":"1.2.1"}}
diff --git a/dev/api/index.html b/dev/api/index.html
diff --git a/dev/examples/basics/index.html b/dev/examples/basics/index.html
diff --git a/dev/examples/dual_quaternions/index.html b/dev/examples/dual_quaternions/index.html
diff --git a/dev/examples/rotations/index.html b/dev/examples/rotations/index.html
diff --git a/dev/examples/type_parameter/index.html b/dev/examples/type_parameter/index.html
@@ -0,0 +1,2 @@
+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>The type parameter T in Quaternion{T} · Quaternions.jl</title><meta name="title" content="The type parameter T in Quaternion{T} · Quaternions.jl"/><meta property="og:title" content="The type parameter T in Quaternion{T} · Quaternions.jl"/><meta property="twitter:title" content="The type parameter T in Quaternion{T} · Quaternions.jl"/><meta name="description" content="Documentation for Quaternions.jl."/><meta property="og:description" content="Documentation for Quaternions.jl."/><meta property="twitter:description" content="Documentation for Quaternions.jl."/><meta property="og:url" content="https://JuliaGeometry.github.io/Quaternions.jl/examples/type_parameter/"/><meta property="twitter:url" content="https://JuliaGeometry.github.io/Quaternions.jl/examples/type_parameter/"/><link rel="canonical" href="https://JuliaGeometry.github.io/Quaternions.jl/examples/type_parameter/"/><script data-outdated-warner src="../../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="../.."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../../assets/documenter.js"></script><script src="../../search_index.js"></script><script src="../../siteinfo.js"></script><script src="../../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../../assets/themeswap.js"></script><link href="../../assets/custom.css" rel="stylesheet" type="text/css"/><link href="../../assets/favicon.ico" rel="icon" type="image/x-icon"/></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../../"><img src="../../assets/logo.svg" alt="Quaternions.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href="../../">Quaternions.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../../">Home</a></li><li><a class="tocitem" href="../../api/">APIs</a></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="../basics/">Basics</a></li><li class="is-active"><a class="tocitem" href>The type parameter <code>T</code> in <code>Quaternion{T}</code></a><ul class="internal"><li><a class="tocitem" href="#Lipschitz-quaternions"><span>Lipschitz quaternions</span></a></li><li><a class="tocitem" href="#Hurwitz-quaternions"><span>Hurwitz quaternions</span></a></li><li><a class="tocitem" href="#Biquaternions"><span>Biquaternions</span></a></li></ul></li><li><a class="tocitem" href="../rotations/">Rotations with quaternions</a></li><li><a class="tocitem" href="../dual_quaternions/">Dual quaternions</a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Examples</a></li><li class="is-active"><a href>The type parameter <code>T</code> in <code>Quaternion{T}</code></a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>The type parameter <code>T</code> in <code>Quaternion{T}</code></a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/JuliaGeometry/Quaternions.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/JuliaGeometry/Quaternions.jl/blob/main/docs/src/examples/type_parameter.md#" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="The-type-parameter-T-in-Quaternion{T}"><a class="docs-heading-anchor" href="#The-type-parameter-T-in-Quaternion{T}">The type parameter <code>T</code> in <code>Quaternion{T}</code></a><a id="The-type-parameter-T-in-Quaternion{T}-1"></a><a class="docs-heading-anchor-permalink" href="#The-type-parameter-T-in-Quaternion{T}" title="Permalink"></a></h1><p>The type parameter <code>T &lt;: Real</code> in <code>Quaternion{T}</code> represents the type of real and imaginary parts of a quaternion.</p><h2 id="Lipschitz-quaternions"><a class="docs-heading-anchor" href="#Lipschitz-quaternions">Lipschitz quaternions</a><a id="Lipschitz-quaternions-1"></a><a class="docs-heading-anchor-permalink" href="#Lipschitz-quaternions" title="Permalink"></a></h2><p>By using this type parameter, some special quaternions such as <a href="https://en.wikipedia.org/wiki/Hurwitz_quaternion"><strong>Lipschitz quaternions</strong></a> <span>$L$</span> can be represented.</p><p class="math-container">\[L = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z}\right\}\]</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; q1 = Quaternion{Int}(1,2,3,4)</code><code class="nohighlight hljs ansi" style="display:block;">Quaternion{Int64}(1, 2, 3, 4)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; q2 = Quaternion{Int}(5,6,7,8)</code><code class="nohighlight hljs ansi" style="display:block;">Quaternion{Int64}(5, 6, 7, 8)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; islipschitz(q::Quaternion) = isinteger(q.s) &amp; isinteger(q.v1) &amp; isinteger(q.v2) &amp; isinteger(q.v3)</code><code class="nohighlight hljs ansi" style="display:block;">islipschitz (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; islipschitz(q1)</code><code class="nohighlight hljs ansi" style="display:block;">true</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; islipschitz(q2)</code><code class="nohighlight hljs ansi" style="display:block;">true</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; islipschitz(q1 + q2)</code><code class="nohighlight hljs ansi" style="display:block;">true</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; islipschitz(q1 * q2)</code><code class="nohighlight hljs ansi" style="display:block;">true</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; islipschitz(q1 / q2)  # Division is not defined on L.</code><code class="nohighlight hljs ansi" style="display:block;">false</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; q1 * q2 == q2 * q1  # non-commutative</code><code class="nohighlight hljs ansi" style="display:block;">false</code></pre><h2 id="Hurwitz-quaternions"><a class="docs-heading-anchor" href="#Hurwitz-quaternions">Hurwitz quaternions</a><a id="Hurwitz-quaternions-1"></a><a class="docs-heading-anchor-permalink" href="#Hurwitz-quaternions" title="Permalink"></a></h2><p>If all coefficients of a quaternion are integers or half-integers, the quaternion is called a <a href="https://en.wikipedia.org/wiki/Hurwitz_quaternion"><strong>Hurwitz quaternion</strong></a>. The set of Hurwitz quaternions is defined by</p><p class="math-container">\[H = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z} \ \text{or} \ a,b,c,d \in \mathbb{Z} + \tfrac{1}{2}\right\}.\]</p><p>Hurwitz quaternions can be implemented with <a href="https://github.com/sostock/HalfIntegers.jl">HalfIntegers.jl</a> package.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; using HalfIntegers</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; q1 = Quaternion{HalfInt}(1, 2, 3, 4)</code><code class="nohighlight hljs ansi" style="display:block;">Quaternion{Half{Int64}}(1, 2, 3, 4)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; q2 = Quaternion{HalfInt}(5.5, 6.5, 7.5, 8.5)</code><code class="nohighlight hljs ansi" style="display:block;">Quaternion{Half{Int64}}(11/2, 13/2, 15/2, 17/2)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; q3 = Quaternion{HalfInt}(1, 2, 3, 4.5)  # not Hurwitz quaternion</code><code class="nohighlight hljs ansi" style="display:block;">Quaternion{Half{Int64}}(1, 2, 3, 9/2)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; ishalfodd(x::Number) = isodd(twice(x))  # Should be defined in HalfIntegers.jl (HalfIntegers.jl#59)</code><code class="nohighlight hljs ansi" style="display:block;">ishalfodd (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; ishurwitz(q::Quaternion) = (isinteger(q.s) &amp; isinteger(q.v1) &amp; isinteger(q.v2) &amp; isinteger(q.v3)) | (ishalfodd(q.s) &amp; ishalfodd(q.v1) &amp; ishalfodd(q.v2) &amp; ishalfodd(q.v3))</code><code class="nohighlight hljs ansi" style="display:block;">ishurwitz (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; ishurwitz(q1)</code><code class="nohighlight hljs ansi" style="display:block;">true</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; ishurwitz(q2)</code><code class="nohighlight hljs ansi" style="display:block;">true</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; ishurwitz(q3)</code><code class="nohighlight hljs ansi" style="display:block;">false</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; ishurwitz(q1 + q2)</code><code class="nohighlight hljs ansi" style="display:block;">true</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; ishurwitz(q1 * q2)</code><code class="nohighlight hljs ansi" style="display:block;">true</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; ishurwitz(q1 / q2)  # Division is not defined on H.</code><code class="nohighlight hljs ansi" style="display:block;">false</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; q1 * q2 == q2 * q1  # non-commucative</code><code class="nohighlight hljs ansi" style="display:block;">false</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; abs2(q1)  # Squared norm is always an integer.</code><code class="nohighlight hljs ansi" style="display:block;">30.0</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; abs2(q2)  # Squared norm is always an integer.</code><code class="nohighlight hljs ansi" style="display:block;">201.0</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; abs2(q3)  # Squared norm is not an integer because `q3` is not Hurwitz quaternion.</code><code class="nohighlight hljs ansi" style="display:block;">34.25</code></pre><h2 id="Biquaternions"><a class="docs-heading-anchor" href="#Biquaternions">Biquaternions</a><a id="Biquaternions-1"></a><a class="docs-heading-anchor-permalink" href="#Biquaternions" title="Permalink"></a></h2><p>If all coefficients of a quaternion are complex numbers, the quaternion is called a <a href="https://en.wikipedia.org/wiki/Biquaternion"><strong>Biquaternion</strong></a>. However, the type parameter <code>T</code> is restricted to <code>&lt;:Real</code>, so biquaternions are not supported in this package. Note that <code>Base.Complex</code> has the same type parameter restriction, and <a href="https://en.wikipedia.org/wiki/Bicomplex_number">bicomplex numbers</a> are not supported in Base. See <a href="https://github.com/JuliaGeometry/Quaternions.jl/issues/79">issue#79</a> for more discussion.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../basics/">« Basics</a><a class="docs-footer-nextpage" href="../rotations/">Rotations with quaternions »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.2.1 on <span class="colophon-date" title="Monday 5 February 2024 03:38">Monday 5 February 2024</span>. Using Julia version 1.10.0.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
Original file line number	Diff line number	Diff line change
		@@ -1 +1 @@
		{"documenter":{"julia_version":"1.9.4","generation_timestamp":"2023-12-23T03:53:19","documenter_version":"1.2.1"}}
		{"documenter":{"julia_version":"1.10.0","generation_timestamp":"2024-02-05T03:38:28","documenter_version":"1.2.1"}}
Original file line number	Diff line number	Diff line change
		@@ -0,0 +1,2 @@
		<!DOCTYPE html>
		<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>The type parameter T in Quaternion{T} · Quaternions.jl</title><meta name="title" content="The type parameter T in Quaternion{T} · Quaternions.jl"/><meta property="og:title" content="The type parameter T in Quaternion{T} · Quaternions.jl"/><meta property="twitter:title" content="The type parameter T in Quaternion{T} · Quaternions.jl"/><meta name="description" content="Documentation for Quaternions.jl."/><meta property="og:description" content="Documentation for Quaternions.jl."/><meta property="twitter:description" content="Documentation for Quaternions.jl."/><meta property="og:url" content="https://JuliaGeometry.github.io/Quaternions.jl/examples/type_parameter/"/><meta property="twitter:url" content="https://JuliaGeometry.github.io/Quaternions.jl/examples/type_parameter/"/><link rel="canonical" href="https://JuliaGeometry.github.io/Quaternions.jl/examples/type_parameter/"/><script data-outdated-warner src="../../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="../.."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../../assets/documenter.js"></script><script src="../../search_index.js"></script><script src="../../siteinfo.js"></script><script src="../../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../../assets/themeswap.js"></script><link href="../../assets/custom.css" rel="stylesheet" type="text/css"/><link href="../../assets/favicon.ico" rel="icon" type="image/x-icon"/></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../../"><img src="../../assets/logo.svg" alt="Quaternions.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href="../../">Quaternions.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../../">Home</a></li><li><a class="tocitem" href="../../api/">APIs</a></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="../basics/">Basics</a></li><li class="is-active"><a class="tocitem" href>The type parameter <code>T</code> in <code>Quaternion{T}</code></a><ul class="internal"><li><a class="tocitem" href="#Lipschitz-quaternions"><span>Lipschitz quaternions</span></a></li><li><a class="tocitem" href="#Hurwitz-quaternions"><span>Hurwitz quaternions</span></a></li><li><a class="tocitem" href="#Biquaternions"><span>Biquaternions</span></a></li></ul></li><li><a class="tocitem" href="../rotations/">Rotations with quaternions</a></li><li><a class="tocitem" href="../dual_quaternions/">Dual quaternions</a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Examples</a></li><li class="is-active"><a href>The type parameter <code>T</code> in <code>Quaternion{T}</code></a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>The type parameter <code>T</code> in <code>Quaternion{T}</code></a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/JuliaGeometry/Quaternions.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/JuliaGeometry/Quaternions.jl/blob/main/docs/src/examples/type_parameter.md#" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="The-type-parameter-T-in-Quaternion{T}"><a class="docs-heading-anchor" href="#The-type-parameter-T-in-Quaternion{T}">The type parameter <code>T</code> in <code>Quaternion{T}</code></a><a id="The-type-parameter-T-in-Quaternion{T}-1"></a><a class="docs-heading-anchor-permalink" href="#The-type-parameter-T-in-Quaternion{T}" title="Permalink"></a></h1><p>The type parameter <code>T <: Real</code> in <code>Quaternion{T}</code> represents the type of real and imaginary parts of a quaternion.</p><h2 id="Lipschitz-quaternions"><a class="docs-heading-anchor" href="#Lipschitz-quaternions">Lipschitz quaternions</a><a id="Lipschitz-quaternions-1"></a><a class="docs-heading-anchor-permalink" href="#Lipschitz-quaternions" title="Permalink"></a></h2><p>By using this type parameter, some special quaternions such as <a href="https://en.wikipedia.org/wiki/Hurwitz_quaternion"><strong>Lipschitz quaternions</strong></a> <span>$L$</span> can be represented.</p><p class="math-container">\[L = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z}\right\}\]</p><pre><code class="language-julia-repl hljs" style="display:block;">julia> q1 = Quaternion{Int}(1,2,3,4)</code><code class="nohighlight hljs ansi" style="display:block;">Quaternion{Int64}(1, 2, 3, 4)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> q2 = Quaternion{Int}(5,6,7,8)</code><code class="nohighlight hljs ansi" style="display:block;">Quaternion{Int64}(5, 6, 7, 8)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> islipschitz(q::Quaternion) = isinteger(q.s) & isinteger(q.v1) & isinteger(q.v2) & isinteger(q.v3)</code><code class="nohighlight hljs ansi" style="display:block;">islipschitz (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> islipschitz(q1)</code><code class="nohighlight hljs ansi" style="display:block;">true</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> islipschitz(q2)</code><code class="nohighlight hljs ansi" style="display:block;">true</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> islipschitz(q1 + q2)</code><code class="nohighlight hljs ansi" style="display:block;">true</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> islipschitz(q1 * q2)</code><code class="nohighlight hljs ansi" style="display:block;">true</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> islipschitz(q1 / q2) # Division is not defined on L.</code><code class="nohighlight hljs ansi" style="display:block;">false</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> q1 * q2 == q2 * q1 # non-commutative</code><code class="nohighlight hljs ansi" style="display:block;">false</code></pre><h2 id="Hurwitz-quaternions"><a class="docs-heading-anchor" href="#Hurwitz-quaternions">Hurwitz quaternions</a><a id="Hurwitz-quaternions-1"></a><a class="docs-heading-anchor-permalink" href="#Hurwitz-quaternions" title="Permalink"></a></h2><p>If all coefficients of a quaternion are integers or half-integers, the quaternion is called a <a href="https://en.wikipedia.org/wiki/Hurwitz_quaternion"><strong>Hurwitz quaternion</strong></a>. The set of Hurwitz quaternions is defined by</p><p class="math-container">\[H = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z} \ \text{or} \ a,b,c,d \in \mathbb{Z} + \tfrac{1}{2}\right\}.\]</p><p>Hurwitz quaternions can be implemented with <a href="https://github.com/sostock/HalfIntegers.jl">HalfIntegers.jl</a> package.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia> using HalfIntegers</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> q1 = Quaternion{HalfInt}(1, 2, 3, 4)</code><code class="nohighlight hljs ansi" style="display:block;">Quaternion{Half{Int64}}(1, 2, 3, 4)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> q2 = Quaternion{HalfInt}(5.5, 6.5, 7.5, 8.5)</code><code class="nohighlight hljs ansi" style="display:block;">Quaternion{Half{Int64}}(11/2, 13/2, 15/2, 17/2)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> q3 = Quaternion{HalfInt}(1, 2, 3, 4.5) # not Hurwitz quaternion</code><code class="nohighlight hljs ansi" style="display:block;">Quaternion{Half{Int64}}(1, 2, 3, 9/2)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> ishalfodd(x::Number) = isodd(twice(x)) # Should be defined in HalfIntegers.jl (HalfIntegers.jl#59)</code><code class="nohighlight hljs ansi" style="display:block;">ishalfodd (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> ishurwitz(q::Quaternion) = (isinteger(q.s) & isinteger(q.v1) & isinteger(q.v2) & isinteger(q.v3)) \| (ishalfodd(q.s) & ishalfodd(q.v1) & ishalfodd(q.v2) & ishalfodd(q.v3))</code><code class="nohighlight hljs ansi" style="display:block;">ishurwitz (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> ishurwitz(q1)</code><code class="nohighlight hljs ansi" style="display:block;">true</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> ishurwitz(q2)</code><code class="nohighlight hljs ansi" style="display:block;">true</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> ishurwitz(q3)</code><code class="nohighlight hljs ansi" style="display:block;">false</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> ishurwitz(q1 + q2)</code><code class="nohighlight hljs ansi" style="display:block;">true</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> ishurwitz(q1 * q2)</code><code class="nohighlight hljs ansi" style="display:block;">true</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> ishurwitz(q1 / q2) # Division is not defined on H.</code><code class="nohighlight hljs ansi" style="display:block;">false</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> q1 * q2 == q2 * q1 # non-commucative</code><code class="nohighlight hljs ansi" style="display:block;">false</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> abs2(q1) # Squared norm is always an integer.</code><code class="nohighlight hljs ansi" style="display:block;">30.0</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> abs2(q2) # Squared norm is always an integer.</code><code class="nohighlight hljs ansi" style="display:block;">201.0</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> abs2(q3) # Squared norm is not an integer because `q3` is not Hurwitz quaternion.</code><code class="nohighlight hljs ansi" style="display:block;">34.25</code></pre><h2 id="Biquaternions"><a class="docs-heading-anchor" href="#Biquaternions">Biquaternions</a><a id="Biquaternions-1"></a><a class="docs-heading-anchor-permalink" href="#Biquaternions" title="Permalink"></a></h2><p>If all coefficients of a quaternion are complex numbers, the quaternion is called a <a href="https://en.wikipedia.org/wiki/Biquaternion"><strong>Biquaternion</strong></a>. However, the type parameter <code>T</code> is restricted to <code><:Real</code>, so biquaternions are not supported in this package. Note that <code>Base.Complex</code> has the same type parameter restriction, and <a href="https://en.wikipedia.org/wiki/Bicomplex_number">bicomplex numbers</a> are not supported in Base. See <a href="https://github.com/JuliaGeometry/Quaternions.jl/issues/79">issue#79</a> for more discussion.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../basics/">« Basics</a><a class="docs-footer-nextpage" href="../rotations/">Rotations with quaternions »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.2.1 on <span class="colophon-date" title="Monday 5 February 2024 03:38">Monday 5 February 2024</span>. Using Julia version 1.10.0.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>