<!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 of the Rust file `/Users/erlendbasso/.cargo/registry/src/github.com-1ecc6299db9ec823/nalgebra-0.32.1/src/geometry/rotation_specialization.rs`."><meta name="keywords" content="rust, rustlang, rust-lang"><title>rotation_specialization.rs - source</title><link rel="preload" as="font" type="font/woff2" crossorigin href="../../../static.files/SourceSerif4-Regular-1f7d512b176f0f72.ttf.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../../static.files/FiraSans-Regular-018c141bf0843ffd.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../../static.files/FiraSans-Medium-8f9a781e4970d388.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../../static.files/SourceCodePro-Regular-562dcc5011b6de7d.ttf.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../../static.files/SourceSerif4-Bold-124a1ca42af929b6.ttf.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../../static.files/SourceCodePro-Semibold-d899c5a5c4aeb14a.ttf.woff2"><link rel="stylesheet" href="../../../static.files/normalize-76eba96aa4d2e634.css"><link rel="stylesheet" href="../../../static.files/rustdoc-6827029ac823cab7.css" id="mainThemeStyle"><link rel="stylesheet" id="themeStyle" href="../../../static.files/light-ebce58d0a40c3431.css"><link rel="stylesheet" disabled href="../../../static.files/dark-f23faae4a2daf9a6.css"><link rel="stylesheet" disabled href="../../../static.files/ayu-8af5e100b21cd173.css"><script id="default-settings" ></script><script src="../../../static.files/storage-d43fa987303ecbbb.js"></script><script defer src="../../../static.files/source-script-5cf2e01a42cc9858.js"></script><script defer src="../../../source-files.js"></script><script defer src="../../../static.files/main-c55e1eb52e1886b4.js"></script><noscript><link rel="stylesheet" href="../../../static.files/noscript-13285aec31fa243e.css"></noscript><link rel="icon" href="https://nalgebra.org/img/favicon.ico"></head><body class="rustdoc source"><!--[if lte IE 11]><div class="warning">This old browser is unsupported and will most likely display funky things.</div><![endif]--><nav class="sidebar"></nav><main><div class="width-limiter"><nav class="sub"><a class="sub-logo-container" href="../../../nalgebra/index.html"><img class="rust-logo" src="../../../static.files/rust-logo-151179464ae7ed46.svg" alt="logo"></a><form class="search-form"><span></span><input class="search-input" name="search" aria-label="Run search in the documentation" autocomplete="off" spellcheck="false" placeholder="Click or press ‘S’ to search, ‘?’ for more options…" type="search"><div id="help-button" title="help" tabindex="-1"><a href="../../../help.html">?</a></div><div id="settings-menu" tabindex="-1"><a href="../../../settings.html" title="settings"><img width="22" height="22" alt="Change settings" src="../../../static.files/wheel-5ec35bf9ca753509.svg"></a></div></form></nav><section id="main-content" class="content"><div class="example-wrap"><pre class="src-line-numbers"><a href="#1" id="1">1</a>
<a href="#2" id="2">2</a>
<a href="#3" id="3">3</a>
<a href="#4" id="4">4</a>
<a href="#5" id="5">5</a>
<a href="#6" id="6">6</a>
<a href="#7" id="7">7</a>
<a href="#8" id="8">8</a>
<a href="#9" id="9">9</a>
<a href="#10" id="10">10</a>
<a href="#11" id="11">11</a>
<a href="#12" id="12">12</a>
<a href="#13" id="13">13</a>
<a href="#14" id="14">14</a>
<a href="#15" id="15">15</a>
<a href="#16" id="16">16</a>
<a href="#17" id="17">17</a>
<a href="#18" id="18">18</a>
<a href="#19" id="19">19</a>
<a href="#20" id="20">20</a>
<a href="#21" id="21">21</a>
<a href="#22" id="22">22</a>
<a href="#23" id="23">23</a>
<a href="#24" id="24">24</a>
<a href="#25" id="25">25</a>
<a href="#26" id="26">26</a>
<a href="#27" id="27">27</a>
<a href="#28" id="28">28</a>
<a href="#29" id="29">29</a>
<a href="#30" id="30">30</a>
<a href="#31" id="31">31</a>
<a href="#32" id="32">32</a>
<a href="#33" id="33">33</a>
<a href="#34" id="34">34</a>
<a href="#35" id="35">35</a>
<a href="#36" id="36">36</a>
<a href="#37" id="37">37</a>
<a href="#38" id="38">38</a>
<a href="#39" id="39">39</a>
<a href="#40" id="40">40</a>
<a href="#41" id="41">41</a>
<a href="#42" id="42">42</a>
<a href="#43" id="43">43</a>
<a href="#44" id="44">44</a>
<a href="#45" id="45">45</a>
<a href="#46" id="46">46</a>
<a href="#47" id="47">47</a>
<a href="#48" id="48">48</a>
<a href="#49" id="49">49</a>
<a href="#50" id="50">50</a>
<a href="#51" id="51">51</a>
<a href="#52" id="52">52</a>
<a href="#53" id="53">53</a>
<a href="#54" id="54">54</a>
<a href="#55" id="55">55</a>
<a href="#56" id="56">56</a>
<a href="#57" id="57">57</a>
<a href="#58" id="58">58</a>
<a href="#59" id="59">59</a>
<a href="#60" id="60">60</a>
<a href="#61" id="61">61</a>
<a href="#62" id="62">62</a>
<a href="#63" id="63">63</a>
<a href="#64" id="64">64</a>
<a href="#65" id="65">65</a>
<a href="#66" id="66">66</a>
<a href="#67" id="67">67</a>
<a href="#68" id="68">68</a>
<a href="#69" id="69">69</a>
<a href="#70" id="70">70</a>
<a href="#71" id="71">71</a>
<a href="#72" id="72">72</a>
<a href="#73" id="73">73</a>
<a href="#74" id="74">74</a>
<a href="#75" id="75">75</a>
<a href="#76" id="76">76</a>
<a href="#77" id="77">77</a>
<a href="#78" id="78">78</a>
<a href="#79" id="79">79</a>
<a href="#80" id="80">80</a>
<a href="#81" id="81">81</a>
<a href="#82" id="82">82</a>
<a href="#83" id="83">83</a>
<a href="#84" id="84">84</a>
<a href="#85" id="85">85</a>
<a href="#86" id="86">86</a>
<a href="#87" id="87">87</a>
<a href="#88" id="88">88</a>
<a href="#89" id="89">89</a>
<a href="#90" id="90">90</a>
<a href="#91" id="91">91</a>
<a href="#92" id="92">92</a>
<a href="#93" id="93">93</a>
<a href="#94" id="94">94</a>
<a href="#95" id="95">95</a>
<a href="#96" id="96">96</a>
<a href="#97" id="97">97</a>
<a href="#98" id="98">98</a>
<a href="#99" id="99">99</a>
<a href="#100" id="100">100</a>
<a href="#101" id="101">101</a>
<a href="#102" id="102">102</a>
<a href="#103" id="103">103</a>
<a href="#104" id="104">104</a>
<a href="#105" id="105">105</a>
<a href="#106" id="106">106</a>
<a href="#107" id="107">107</a>
<a href="#108" id="108">108</a>
<a href="#109" id="109">109</a>
<a href="#110" id="110">110</a>
<a href="#111" id="111">111</a>
<a href="#112" id="112">112</a>
<a href="#113" id="113">113</a>
<a href="#114" id="114">114</a>
<a href="#115" id="115">115</a>
<a href="#116" id="116">116</a>
<a href="#117" id="117">117</a>
<a href="#118" id="118">118</a>
<a href="#119" id="119">119</a>
<a href="#120" id="120">120</a>
<a href="#121" id="121">121</a>
<a href="#122" id="122">122</a>
<a href="#123" id="123">123</a>
<a href="#124" id="124">124</a>
<a href="#125" id="125">125</a>
<a href="#126" id="126">126</a>
<a href="#127" id="127">127</a>
<a href="#128" id="128">128</a>
<a href="#129" id="129">129</a>
<a href="#130" id="130">130</a>
<a href="#131" id="131">131</a>
<a href="#132" id="132">132</a>
<a href="#133" id="133">133</a>
<a href="#134" id="134">134</a>
<a href="#135" id="135">135</a>
<a href="#136" id="136">136</a>
<a href="#137" id="137">137</a>
<a href="#138" id="138">138</a>
<a href="#139" id="139">139</a>
<a href="#140" id="140">140</a>
<a href="#141" id="141">141</a>
<a href="#142" id="142">142</a>
<a href="#143" id="143">143</a>
<a href="#144" id="144">144</a>
<a href="#145" id="145">145</a>
<a href="#146" id="146">146</a>
<a href="#147" id="147">147</a>
<a href="#148" id="148">148</a>
<a href="#149" id="149">149</a>
<a href="#150" id="150">150</a>
<a href="#151" id="151">151</a>
<a href="#152" id="152">152</a>
<a href="#153" id="153">153</a>
<a href="#154" id="154">154</a>
<a href="#155" id="155">155</a>
<a href="#156" id="156">156</a>
<a href="#157" id="157">157</a>
<a href="#158" id="158">158</a>
<a href="#159" id="159">159</a>
<a href="#160" id="160">160</a>
<a href="#161" id="161">161</a>
<a href="#162" id="162">162</a>
<a href="#163" id="163">163</a>
<a href="#164" id="164">164</a>
<a href="#165" id="165">165</a>
<a href="#166" id="166">166</a>
<a href="#167" id="167">167</a>
<a href="#168" id="168">168</a>
<a href="#169" id="169">169</a>
<a href="#170" id="170">170</a>
<a href="#171" id="171">171</a>
<a href="#172" id="172">172</a>
<a href="#173" id="173">173</a>
<a href="#174" id="174">174</a>
<a href="#175" id="175">175</a>
<a href="#176" id="176">176</a>
<a href="#177" id="177">177</a>
<a href="#178" id="178">178</a>
<a href="#179" id="179">179</a>
<a href="#180" id="180">180</a>
<a href="#181" id="181">181</a>
<a href="#182" id="182">182</a>
<a href="#183" id="183">183</a>
<a href="#184" id="184">184</a>
<a href="#185" id="185">185</a>
<a href="#186" id="186">186</a>
<a href="#187" id="187">187</a>
<a href="#188" id="188">188</a>
<a href="#189" id="189">189</a>
<a href="#190" id="190">190</a>
<a href="#191" id="191">191</a>
<a href="#192" id="192">192</a>
<a href="#193" id="193">193</a>
<a href="#194" id="194">194</a>
<a href="#195" id="195">195</a>
<a href="#196" id="196">196</a>
<a href="#197" id="197">197</a>
<a href="#198" id="198">198</a>
<a href="#199" id="199">199</a>
<a href="#200" id="200">200</a>
<a href="#201" id="201">201</a>
<a href="#202" id="202">202</a>
<a href="#203" id="203">203</a>
<a href="#204" id="204">204</a>
<a href="#205" id="205">205</a>
<a href="#206" id="206">206</a>
<a href="#207" id="207">207</a>
<a href="#208" id="208">208</a>
<a href="#209" id="209">209</a>
<a href="#210" id="210">210</a>
<a href="#211" id="211">211</a>
<a href="#212" id="212">212</a>
<a href="#213" id="213">213</a>
<a href="#214" id="214">214</a>
<a href="#215" id="215">215</a>
<a href="#216" id="216">216</a>
<a href="#217" id="217">217</a>
<a href="#218" id="218">218</a>
<a href="#219" id="219">219</a>
<a href="#220" id="220">220</a>
<a href="#221" id="221">221</a>
<a href="#222" id="222">222</a>
<a href="#223" id="223">223</a>
<a href="#224" id="224">224</a>
<a href="#225" id="225">225</a>
<a href="#226" id="226">226</a>
<a href="#227" id="227">227</a>
<a href="#228" id="228">228</a>
<a href="#229" id="229">229</a>
<a href="#230" id="230">230</a>
<a href="#231" id="231">231</a>
<a href="#232" id="232">232</a>
<a href="#233" id="233">233</a>
<a href="#234" id="234">234</a>
<a href="#235" id="235">235</a>
<a href="#236" id="236">236</a>
<a href="#237" id="237">237</a>
<a href="#238" id="238">238</a>
<a href="#239" id="239">239</a>
<a href="#240" id="240">240</a>
<a href="#241" id="241">241</a>
<a href="#242" id="242">242</a>
<a href="#243" id="243">243</a>
<a href="#244" id="244">244</a>
<a href="#245" id="245">245</a>
<a href="#246" id="246">246</a>
<a href="#247" id="247">247</a>
<a href="#248" id="248">248</a>
<a href="#249" id="249">249</a>
<a href="#250" id="250">250</a>
<a href="#251" id="251">251</a>
<a href="#252" id="252">252</a>
<a href="#253" id="253">253</a>
<a href="#254" id="254">254</a>
<a href="#255" id="255">255</a>
<a href="#256" id="256">256</a>
<a href="#257" id="257">257</a>
<a href="#258" id="258">258</a>
<a href="#259" id="259">259</a>
<a href="#260" id="260">260</a>
<a href="#261" id="261">261</a>
<a href="#262" id="262">262</a>
<a href="#263" id="263">263</a>
<a href="#264" id="264">264</a>
<a href="#265" id="265">265</a>
<a href="#266" id="266">266</a>
<a href="#267" id="267">267</a>
<a href="#268" id="268">268</a>
<a href="#269" id="269">269</a>
<a href="#270" id="270">270</a>
<a href="#271" id="271">271</a>
<a href="#272" id="272">272</a>
<a href="#273" id="273">273</a>
<a href="#274" id="274">274</a>
<a href="#275" id="275">275</a>
<a href="#276" id="276">276</a>
<a href="#277" id="277">277</a>
<a href="#278" id="278">278</a>
<a href="#279" id="279">279</a>
<a href="#280" id="280">280</a>
<a href="#281" id="281">281</a>
<a href="#282" id="282">282</a>
<a href="#283" id="283">283</a>
<a href="#284" id="284">284</a>
<a href="#285" id="285">285</a>
<a href="#286" id="286">286</a>
<a href="#287" id="287">287</a>
<a href="#288" id="288">288</a>
<a href="#289" id="289">289</a>
<a href="#290" id="290">290</a>
<a href="#291" id="291">291</a>
<a href="#292" id="292">292</a>
<a href="#293" id="293">293</a>
<a href="#294" id="294">294</a>
<a href="#295" id="295">295</a>
<a href="#296" id="296">296</a>
<a href="#297" id="297">297</a>
<a href="#298" id="298">298</a>
<a href="#299" id="299">299</a>
<a href="#300" id="300">300</a>
<a href="#301" id="301">301</a>
<a href="#302" id="302">302</a>
<a href="#303" id="303">303</a>
<a href="#304" id="304">304</a>
<a href="#305" id="305">305</a>
<a href="#306" id="306">306</a>
<a href="#307" id="307">307</a>
<a href="#308" id="308">308</a>
<a href="#309" id="309">309</a>
<a href="#310" id="310">310</a>
<a href="#311" id="311">311</a>
<a href="#312" id="312">312</a>
<a href="#313" id="313">313</a>
<a href="#314" id="314">314</a>
<a href="#315" id="315">315</a>
<a href="#316" id="316">316</a>
<a href="#317" id="317">317</a>
<a href="#318" id="318">318</a>
<a href="#319" id="319">319</a>
<a href="#320" id="320">320</a>
<a href="#321" id="321">321</a>
<a href="#322" id="322">322</a>
<a href="#323" id="323">323</a>
<a href="#324" id="324">324</a>
<a href="#325" id="325">325</a>
<a href="#326" id="326">326</a>
<a href="#327" id="327">327</a>
<a href="#328" id="328">328</a>
<a href="#329" id="329">329</a>
<a href="#330" id="330">330</a>
<a href="#331" id="331">331</a>
<a href="#332" id="332">332</a>
<a href="#333" id="333">333</a>
<a href="#334" id="334">334</a>
<a href="#335" id="335">335</a>
<a href="#336" id="336">336</a>
<a href="#337" id="337">337</a>
<a href="#338" id="338">338</a>
<a href="#339" id="339">339</a>
<a href="#340" id="340">340</a>
<a href="#341" id="341">341</a>
<a href="#342" id="342">342</a>
<a href="#343" id="343">343</a>
<a href="#344" id="344">344</a>
<a href="#345" id="345">345</a>
<a href="#346" id="346">346</a>
<a href="#347" id="347">347</a>
<a href="#348" id="348">348</a>
<a href="#349" id="349">349</a>
<a href="#350" id="350">350</a>
<a href="#351" id="351">351</a>
<a href="#352" id="352">352</a>
<a href="#353" id="353">353</a>
<a href="#354" id="354">354</a>
<a href="#355" id="355">355</a>
<a href="#356" id="356">356</a>
<a href="#357" id="357">357</a>
<a href="#358" id="358">358</a>
<a href="#359" id="359">359</a>
<a href="#360" id="360">360</a>
<a href="#361" id="361">361</a>
<a href="#362" id="362">362</a>
<a href="#363" id="363">363</a>
<a href="#364" id="364">364</a>
<a href="#365" id="365">365</a>
<a href="#366" id="366">366</a>
<a href="#367" id="367">367</a>
<a href="#368" id="368">368</a>
<a href="#369" id="369">369</a>
<a href="#370" id="370">370</a>
<a href="#371" id="371">371</a>
<a href="#372" id="372">372</a>
<a href="#373" id="373">373</a>
<a href="#374" id="374">374</a>
<a href="#375" id="375">375</a>
<a href="#376" id="376">376</a>
<a href="#377" id="377">377</a>
<a href="#378" id="378">378</a>
<a href="#379" id="379">379</a>
<a href="#380" id="380">380</a>
<a href="#381" id="381">381</a>
<a href="#382" id="382">382</a>
<a href="#383" id="383">383</a>
<a href="#384" id="384">384</a>
<a href="#385" id="385">385</a>
<a href="#386" id="386">386</a>
<a href="#387" id="387">387</a>
<a href="#388" id="388">388</a>
<a href="#389" id="389">389</a>
<a href="#390" id="390">390</a>
<a href="#391" id="391">391</a>
<a href="#392" id="392">392</a>
<a href="#393" id="393">393</a>
<a href="#394" id="394">394</a>
<a href="#395" id="395">395</a>
<a href="#396" id="396">396</a>
<a href="#397" id="397">397</a>
<a href="#398" id="398">398</a>
<a href="#399" id="399">399</a>
<a href="#400" id="400">400</a>
<a href="#401" id="401">401</a>
<a href="#402" id="402">402</a>
<a href="#403" id="403">403</a>
<a href="#404" id="404">404</a>
<a href="#405" id="405">405</a>
<a href="#406" id="406">406</a>
<a href="#407" id="407">407</a>
<a href="#408" id="408">408</a>
<a href="#409" id="409">409</a>
<a href="#410" id="410">410</a>
<a href="#411" id="411">411</a>
<a href="#412" id="412">412</a>
<a href="#413" id="413">413</a>
<a href="#414" id="414">414</a>
<a href="#415" id="415">415</a>
<a href="#416" id="416">416</a>
<a href="#417" id="417">417</a>
<a href="#418" id="418">418</a>
<a href="#419" id="419">419</a>
<a href="#420" id="420">420</a>
<a href="#421" id="421">421</a>
<a href="#422" id="422">422</a>
<a href="#423" id="423">423</a>
<a href="#424" id="424">424</a>
<a href="#425" id="425">425</a>
<a href="#426" id="426">426</a>
<a href="#427" id="427">427</a>
<a href="#428" id="428">428</a>
<a href="#429" id="429">429</a>
<a href="#430" id="430">430</a>
<a href="#431" id="431">431</a>
<a href="#432" id="432">432</a>
<a href="#433" id="433">433</a>
<a href="#434" id="434">434</a>
<a href="#435" id="435">435</a>
<a href="#436" id="436">436</a>
<a href="#437" id="437">437</a>
<a href="#438" id="438">438</a>
<a href="#439" id="439">439</a>
<a href="#440" id="440">440</a>
<a href="#441" id="441">441</a>
<a href="#442" id="442">442</a>
<a href="#443" id="443">443</a>
<a href="#444" id="444">444</a>
<a href="#445" id="445">445</a>
<a href="#446" id="446">446</a>
<a href="#447" id="447">447</a>
<a href="#448" id="448">448</a>
<a href="#449" id="449">449</a>
<a href="#450" id="450">450</a>
<a href="#451" id="451">451</a>
<a href="#452" id="452">452</a>
<a href="#453" id="453">453</a>
<a href="#454" id="454">454</a>
<a href="#455" id="455">455</a>
<a href="#456" id="456">456</a>
<a href="#457" id="457">457</a>
<a href="#458" id="458">458</a>
<a href="#459" id="459">459</a>
<a href="#460" id="460">460</a>
<a href="#461" id="461">461</a>
<a href="#462" id="462">462</a>
<a href="#463" id="463">463</a>
<a href="#464" id="464">464</a>
<a href="#465" id="465">465</a>
<a href="#466" id="466">466</a>
<a href="#467" id="467">467</a>
<a href="#468" id="468">468</a>
<a href="#469" id="469">469</a>
<a href="#470" id="470">470</a>
<a href="#471" id="471">471</a>
<a href="#472" id="472">472</a>
<a href="#473" id="473">473</a>
<a href="#474" id="474">474</a>
<a href="#475" id="475">475</a>
<a href="#476" id="476">476</a>
<a href="#477" id="477">477</a>
<a href="#478" id="478">478</a>
<a href="#479" id="479">479</a>
<a href="#480" id="480">480</a>
<a href="#481" id="481">481</a>
<a href="#482" id="482">482</a>
<a href="#483" id="483">483</a>
<a href="#484" id="484">484</a>
<a href="#485" id="485">485</a>
<a href="#486" id="486">486</a>
<a href="#487" id="487">487</a>
<a href="#488" id="488">488</a>
<a href="#489" id="489">489</a>
<a href="#490" id="490">490</a>
<a href="#491" id="491">491</a>
<a href="#492" id="492">492</a>
<a href="#493" id="493">493</a>
<a href="#494" id="494">494</a>
<a href="#495" id="495">495</a>
<a href="#496" id="496">496</a>
<a href="#497" id="497">497</a>
<a href="#498" id="498">498</a>
<a href="#499" id="499">499</a>
<a href="#500" id="500">500</a>
<a href="#501" id="501">501</a>
<a href="#502" id="502">502</a>
<a href="#503" id="503">503</a>
<a href="#504" id="504">504</a>
<a href="#505" id="505">505</a>
<a href="#506" id="506">506</a>
<a href="#507" id="507">507</a>
<a href="#508" id="508">508</a>
<a href="#509" id="509">509</a>
<a href="#510" id="510">510</a>
<a href="#511" id="511">511</a>
<a href="#512" id="512">512</a>
<a href="#513" id="513">513</a>
<a href="#514" id="514">514</a>
<a href="#515" id="515">515</a>
<a href="#516" id="516">516</a>
<a href="#517" id="517">517</a>
<a href="#518" id="518">518</a>
<a href="#519" id="519">519</a>
<a href="#520" id="520">520</a>
<a href="#521" id="521">521</a>
<a href="#522" id="522">522</a>
<a href="#523" id="523">523</a>
<a href="#524" id="524">524</a>
<a href="#525" id="525">525</a>
<a href="#526" id="526">526</a>
<a href="#527" id="527">527</a>
<a href="#528" id="528">528</a>
<a href="#529" id="529">529</a>
<a href="#530" id="530">530</a>
<a href="#531" id="531">531</a>
<a href="#532" id="532">532</a>
<a href="#533" id="533">533</a>
<a href="#534" id="534">534</a>
<a href="#535" id="535">535</a>
<a href="#536" id="536">536</a>
<a href="#537" id="537">537</a>
<a href="#538" id="538">538</a>
<a href="#539" id="539">539</a>
<a href="#540" id="540">540</a>
<a href="#541" id="541">541</a>
<a href="#542" id="542">542</a>
<a href="#543" id="543">543</a>
<a href="#544" id="544">544</a>
<a href="#545" id="545">545</a>
<a href="#546" id="546">546</a>
<a href="#547" id="547">547</a>
<a href="#548" id="548">548</a>
<a href="#549" id="549">549</a>
<a href="#550" id="550">550</a>
<a href="#551" id="551">551</a>
<a href="#552" id="552">552</a>
<a href="#553" id="553">553</a>
<a href="#554" id="554">554</a>
<a href="#555" id="555">555</a>
<a href="#556" id="556">556</a>
<a href="#557" id="557">557</a>
<a href="#558" id="558">558</a>
<a href="#559" id="559">559</a>
<a href="#560" id="560">560</a>
<a href="#561" id="561">561</a>
<a href="#562" id="562">562</a>
<a href="#563" id="563">563</a>
<a href="#564" id="564">564</a>
<a href="#565" id="565">565</a>
<a href="#566" id="566">566</a>
<a href="#567" id="567">567</a>
<a href="#568" id="568">568</a>
<a href="#569" id="569">569</a>
<a href="#570" id="570">570</a>
<a href="#571" id="571">571</a>
<a href="#572" id="572">572</a>
<a href="#573" id="573">573</a>
<a href="#574" id="574">574</a>
<a href="#575" id="575">575</a>
<a href="#576" id="576">576</a>
<a href="#577" id="577">577</a>
<a href="#578" id="578">578</a>
<a href="#579" id="579">579</a>
<a href="#580" id="580">580</a>
<a href="#581" id="581">581</a>
<a href="#582" id="582">582</a>
<a href="#583" id="583">583</a>
<a href="#584" id="584">584</a>
<a href="#585" id="585">585</a>
<a href="#586" id="586">586</a>
<a href="#587" id="587">587</a>
<a href="#588" id="588">588</a>
<a href="#589" id="589">589</a>
<a href="#590" id="590">590</a>
<a href="#591" id="591">591</a>
<a href="#592" id="592">592</a>
<a href="#593" id="593">593</a>
<a href="#594" id="594">594</a>
<a href="#595" id="595">595</a>
<a href="#596" id="596">596</a>
<a href="#597" id="597">597</a>
<a href="#598" id="598">598</a>
<a href="#599" id="599">599</a>
<a href="#600" id="600">600</a>
<a href="#601" id="601">601</a>
<a href="#602" id="602">602</a>
<a href="#603" id="603">603</a>
<a href="#604" id="604">604</a>
<a href="#605" id="605">605</a>
<a href="#606" id="606">606</a>
<a href="#607" id="607">607</a>
<a href="#608" id="608">608</a>
<a href="#609" id="609">609</a>
<a href="#610" id="610">610</a>
<a href="#611" id="611">611</a>
<a href="#612" id="612">612</a>
<a href="#613" id="613">613</a>
<a href="#614" id="614">614</a>
<a href="#615" id="615">615</a>
<a href="#616" id="616">616</a>
<a href="#617" id="617">617</a>
<a href="#618" id="618">618</a>
<a href="#619" id="619">619</a>
<a href="#620" id="620">620</a>
<a href="#621" id="621">621</a>
<a href="#622" id="622">622</a>
<a href="#623" id="623">623</a>
<a href="#624" id="624">624</a>
<a href="#625" id="625">625</a>
<a href="#626" id="626">626</a>
<a href="#627" id="627">627</a>
<a href="#628" id="628">628</a>
<a href="#629" id="629">629</a>
<a href="#630" id="630">630</a>
<a href="#631" id="631">631</a>
<a href="#632" id="632">632</a>
<a href="#633" id="633">633</a>
<a href="#634" id="634">634</a>
<a href="#635" id="635">635</a>
<a href="#636" id="636">636</a>
<a href="#637" id="637">637</a>
<a href="#638" id="638">638</a>
<a href="#639" id="639">639</a>
<a href="#640" id="640">640</a>
<a href="#641" id="641">641</a>
<a href="#642" id="642">642</a>
<a href="#643" id="643">643</a>
<a href="#644" id="644">644</a>
<a href="#645" id="645">645</a>
<a href="#646" id="646">646</a>
<a href="#647" id="647">647</a>
<a href="#648" id="648">648</a>
<a href="#649" id="649">649</a>
<a href="#650" id="650">650</a>
<a href="#651" id="651">651</a>
<a href="#652" id="652">652</a>
<a href="#653" id="653">653</a>
<a href="#654" id="654">654</a>
<a href="#655" id="655">655</a>
<a href="#656" id="656">656</a>
<a href="#657" id="657">657</a>
<a href="#658" id="658">658</a>
<a href="#659" id="659">659</a>
<a href="#660" id="660">660</a>
<a href="#661" id="661">661</a>
<a href="#662" id="662">662</a>
<a href="#663" id="663">663</a>
<a href="#664" id="664">664</a>
<a href="#665" id="665">665</a>
<a href="#666" id="666">666</a>
<a href="#667" id="667">667</a>
<a href="#668" id="668">668</a>
<a href="#669" id="669">669</a>
<a href="#670" id="670">670</a>
<a href="#671" id="671">671</a>
<a href="#672" id="672">672</a>
<a href="#673" id="673">673</a>
<a href="#674" id="674">674</a>
<a href="#675" id="675">675</a>
<a href="#676" id="676">676</a>
<a href="#677" id="677">677</a>
<a href="#678" id="678">678</a>
<a href="#679" id="679">679</a>
<a href="#680" id="680">680</a>
<a href="#681" id="681">681</a>
<a href="#682" id="682">682</a>
<a href="#683" id="683">683</a>
<a href="#684" id="684">684</a>
<a href="#685" id="685">685</a>
<a href="#686" id="686">686</a>
<a href="#687" id="687">687</a>
<a href="#688" id="688">688</a>
<a href="#689" id="689">689</a>
<a href="#690" id="690">690</a>
<a href="#691" id="691">691</a>
<a href="#692" id="692">692</a>
<a href="#693" id="693">693</a>
<a href="#694" id="694">694</a>
<a href="#695" id="695">695</a>
<a href="#696" id="696">696</a>
<a href="#697" id="697">697</a>
<a href="#698" id="698">698</a>
<a href="#699" id="699">699</a>
<a href="#700" id="700">700</a>
<a href="#701" id="701">701</a>
<a href="#702" id="702">702</a>
<a href="#703" id="703">703</a>
<a href="#704" id="704">704</a>
<a href="#705" id="705">705</a>
<a href="#706" id="706">706</a>
<a href="#707" id="707">707</a>
<a href="#708" id="708">708</a>
<a href="#709" id="709">709</a>
<a href="#710" id="710">710</a>
<a href="#711" id="711">711</a>
<a href="#712" id="712">712</a>
<a href="#713" id="713">713</a>
<a href="#714" id="714">714</a>
<a href="#715" id="715">715</a>
<a href="#716" id="716">716</a>
<a href="#717" id="717">717</a>
<a href="#718" id="718">718</a>
<a href="#719" id="719">719</a>
<a href="#720" id="720">720</a>
<a href="#721" id="721">721</a>
<a href="#722" id="722">722</a>
<a href="#723" id="723">723</a>
<a href="#724" id="724">724</a>
<a href="#725" id="725">725</a>
<a href="#726" id="726">726</a>
<a href="#727" id="727">727</a>
<a href="#728" id="728">728</a>
<a href="#729" id="729">729</a>
<a href="#730" id="730">730</a>
<a href="#731" id="731">731</a>
<a href="#732" id="732">732</a>
<a href="#733" id="733">733</a>
<a href="#734" id="734">734</a>
<a href="#735" id="735">735</a>
<a href="#736" id="736">736</a>
<a href="#737" id="737">737</a>
<a href="#738" id="738">738</a>
<a href="#739" id="739">739</a>
<a href="#740" id="740">740</a>
<a href="#741" id="741">741</a>
<a href="#742" id="742">742</a>
<a href="#743" id="743">743</a>
<a href="#744" id="744">744</a>
<a href="#745" id="745">745</a>
<a href="#746" id="746">746</a>
<a href="#747" id="747">747</a>
<a href="#748" id="748">748</a>
<a href="#749" id="749">749</a>
<a href="#750" id="750">750</a>
<a href="#751" id="751">751</a>
<a href="#752" id="752">752</a>
<a href="#753" id="753">753</a>
<a href="#754" id="754">754</a>
<a href="#755" id="755">755</a>
<a href="#756" id="756">756</a>
<a href="#757" id="757">757</a>
<a href="#758" id="758">758</a>
<a href="#759" id="759">759</a>
<a href="#760" id="760">760</a>
<a href="#761" id="761">761</a>
<a href="#762" id="762">762</a>
<a href="#763" id="763">763</a>
<a href="#764" id="764">764</a>
<a href="#765" id="765">765</a>
<a href="#766" id="766">766</a>
<a href="#767" id="767">767</a>
<a href="#768" id="768">768</a>
<a href="#769" id="769">769</a>
<a href="#770" id="770">770</a>
<a href="#771" id="771">771</a>
<a href="#772" id="772">772</a>
<a href="#773" id="773">773</a>
<a href="#774" id="774">774</a>
<a href="#775" id="775">775</a>
<a href="#776" id="776">776</a>
<a href="#777" id="777">777</a>
<a href="#778" id="778">778</a>
<a href="#779" id="779">779</a>
<a href="#780" id="780">780</a>
<a href="#781" id="781">781</a>
<a href="#782" id="782">782</a>
<a href="#783" id="783">783</a>
<a href="#784" id="784">784</a>
<a href="#785" id="785">785</a>
<a href="#786" id="786">786</a>
<a href="#787" id="787">787</a>
<a href="#788" id="788">788</a>
<a href="#789" id="789">789</a>
<a href="#790" id="790">790</a>
<a href="#791" id="791">791</a>
<a href="#792" id="792">792</a>
<a href="#793" id="793">793</a>
<a href="#794" id="794">794</a>
<a href="#795" id="795">795</a>
<a href="#796" id="796">796</a>
<a href="#797" id="797">797</a>
<a href="#798" id="798">798</a>
<a href="#799" id="799">799</a>
<a href="#800" id="800">800</a>
<a href="#801" id="801">801</a>
<a href="#802" id="802">802</a>
<a href="#803" id="803">803</a>
<a href="#804" id="804">804</a>
<a href="#805" id="805">805</a>
<a href="#806" id="806">806</a>
<a href="#807" id="807">807</a>
<a href="#808" id="808">808</a>
<a href="#809" id="809">809</a>
<a href="#810" id="810">810</a>
<a href="#811" id="811">811</a>
<a href="#812" id="812">812</a>
<a href="#813" id="813">813</a>
<a href="#814" id="814">814</a>
<a href="#815" id="815">815</a>
<a href="#816" id="816">816</a>
<a href="#817" id="817">817</a>
<a href="#818" id="818">818</a>
<a href="#819" id="819">819</a>
<a href="#820" id="820">820</a>
<a href="#821" id="821">821</a>
<a href="#822" id="822">822</a>
<a href="#823" id="823">823</a>
<a href="#824" id="824">824</a>
<a href="#825" id="825">825</a>
<a href="#826" id="826">826</a>
<a href="#827" id="827">827</a>
<a href="#828" id="828">828</a>
<a href="#829" id="829">829</a>
<a href="#830" id="830">830</a>
<a href="#831" id="831">831</a>
<a href="#832" id="832">832</a>
<a href="#833" id="833">833</a>
<a href="#834" id="834">834</a>
<a href="#835" id="835">835</a>
<a href="#836" id="836">836</a>
<a href="#837" id="837">837</a>
<a href="#838" id="838">838</a>
<a href="#839" id="839">839</a>
<a href="#840" id="840">840</a>
<a href="#841" id="841">841</a>
<a href="#842" id="842">842</a>
<a href="#843" id="843">843</a>
<a href="#844" id="844">844</a>
<a href="#845" id="845">845</a>
<a href="#846" id="846">846</a>
<a href="#847" id="847">847</a>
<a href="#848" id="848">848</a>
<a href="#849" id="849">849</a>
<a href="#850" id="850">850</a>
<a href="#851" id="851">851</a>
<a href="#852" id="852">852</a>
<a href="#853" id="853">853</a>
<a href="#854" id="854">854</a>
<a href="#855" id="855">855</a>
<a href="#856" id="856">856</a>
<a href="#857" id="857">857</a>
<a href="#858" id="858">858</a>
<a href="#859" id="859">859</a>
<a href="#860" id="860">860</a>
<a href="#861" id="861">861</a>
<a href="#862" id="862">862</a>
<a href="#863" id="863">863</a>
<a href="#864" id="864">864</a>
<a href="#865" id="865">865</a>
<a href="#866" id="866">866</a>
<a href="#867" id="867">867</a>
<a href="#868" id="868">868</a>
<a href="#869" id="869">869</a>
<a href="#870" id="870">870</a>
<a href="#871" id="871">871</a>
<a href="#872" id="872">872</a>
<a href="#873" id="873">873</a>
<a href="#874" id="874">874</a>
<a href="#875" id="875">875</a>
<a href="#876" id="876">876</a>
<a href="#877" id="877">877</a>
<a href="#878" id="878">878</a>
<a href="#879" id="879">879</a>
<a href="#880" id="880">880</a>
<a href="#881" id="881">881</a>
<a href="#882" id="882">882</a>
<a href="#883" id="883">883</a>
<a href="#884" id="884">884</a>
<a href="#885" id="885">885</a>
<a href="#886" id="886">886</a>
<a href="#887" id="887">887</a>
<a href="#888" id="888">888</a>
<a href="#889" id="889">889</a>
<a href="#890" id="890">890</a>
<a href="#891" id="891">891</a>
<a href="#892" id="892">892</a>
<a href="#893" id="893">893</a>
<a href="#894" id="894">894</a>
<a href="#895" id="895">895</a>
<a href="#896" id="896">896</a>
<a href="#897" id="897">897</a>
<a href="#898" id="898">898</a>
<a href="#899" id="899">899</a>
<a href="#900" id="900">900</a>
<a href="#901" id="901">901</a>
<a href="#902" id="902">902</a>
<a href="#903" id="903">903</a>
<a href="#904" id="904">904</a>
<a href="#905" id="905">905</a>
<a href="#906" id="906">906</a>
<a href="#907" id="907">907</a>
<a href="#908" id="908">908</a>
<a href="#909" id="909">909</a>
<a href="#910" id="910">910</a>
<a href="#911" id="911">911</a>
<a href="#912" id="912">912</a>
<a href="#913" id="913">913</a>
<a href="#914" id="914">914</a>
<a href="#915" id="915">915</a>
<a href="#916" id="916">916</a>
<a href="#917" id="917">917</a>
<a href="#918" id="918">918</a>
<a href="#919" id="919">919</a>
<a href="#920" id="920">920</a>
<a href="#921" id="921">921</a>
<a href="#922" id="922">922</a>
<a href="#923" id="923">923</a>
<a href="#924" id="924">924</a>
<a href="#925" id="925">925</a>
<a href="#926" id="926">926</a>
<a href="#927" id="927">927</a>
<a href="#928" id="928">928</a>
<a href="#929" id="929">929</a>
<a href="#930" id="930">930</a>
<a href="#931" id="931">931</a>
<a href="#932" id="932">932</a>
<a href="#933" id="933">933</a>
<a href="#934" id="934">934</a>
<a href="#935" id="935">935</a>
<a href="#936" id="936">936</a>
<a href="#937" id="937">937</a>
<a href="#938" id="938">938</a>
<a href="#939" id="939">939</a>
<a href="#940" id="940">940</a>
<a href="#941" id="941">941</a>
<a href="#942" id="942">942</a>
<a href="#943" id="943">943</a>
<a href="#944" id="944">944</a>
<a href="#945" id="945">945</a>
<a href="#946" id="946">946</a>
<a href="#947" id="947">947</a>
<a href="#948" id="948">948</a>
<a href="#949" id="949">949</a>
<a href="#950" id="950">950</a>
<a href="#951" id="951">951</a>
<a href="#952" id="952">952</a>
<a href="#953" id="953">953</a>
<a href="#954" id="954">954</a>
<a href="#955" id="955">955</a>
<a href="#956" id="956">956</a>
<a href="#957" id="957">957</a>
<a href="#958" id="958">958</a>
<a href="#959" id="959">959</a>
<a href="#960" id="960">960</a>
<a href="#961" id="961">961</a>
<a href="#962" id="962">962</a>
<a href="#963" id="963">963</a>
<a href="#964" id="964">964</a>
<a href="#965" id="965">965</a>
<a href="#966" id="966">966</a>
<a href="#967" id="967">967</a>
<a href="#968" id="968">968</a>
<a href="#969" id="969">969</a>
<a href="#970" id="970">970</a>
<a href="#971" id="971">971</a>
<a href="#972" id="972">972</a>
<a href="#973" id="973">973</a>
<a href="#974" id="974">974</a>
<a href="#975" id="975">975</a>
<a href="#976" id="976">976</a>
<a href="#977" id="977">977</a>
<a href="#978" id="978">978</a>
<a href="#979" id="979">979</a>
<a href="#980" id="980">980</a>
<a href="#981" id="981">981</a>
<a href="#982" id="982">982</a>
<a href="#983" id="983">983</a>
<a href="#984" id="984">984</a>
<a href="#985" id="985">985</a>
<a href="#986" id="986">986</a>
<a href="#987" id="987">987</a>
<a href="#988" id="988">988</a>
<a href="#989" id="989">989</a>
<a href="#990" id="990">990</a>
<a href="#991" id="991">991</a>
<a href="#992" id="992">992</a>
<a href="#993" id="993">993</a>
<a href="#994" id="994">994</a>
<a href="#995" id="995">995</a>
<a href="#996" id="996">996</a>
<a href="#997" id="997">997</a>
<a href="#998" id="998">998</a>
<a href="#999" id="999">999</a>
<a href="#1000" id="1000">1000</a>
<a href="#1001" id="1001">1001</a>
<a href="#1002" id="1002">1002</a>
<a href="#1003" id="1003">1003</a>
<a href="#1004" id="1004">1004</a>
<a href="#1005" id="1005">1005</a>
<a href="#1006" id="1006">1006</a>
<a href="#1007" id="1007">1007</a>
<a href="#1008" id="1008">1008</a>
<a href="#1009" id="1009">1009</a>
<a href="#1010" id="1010">1010</a>
<a href="#1011" id="1011">1011</a>
<a href="#1012" id="1012">1012</a>
<a href="#1013" id="1013">1013</a>
<a href="#1014" id="1014">1014</a>
<a href="#1015" id="1015">1015</a>
<a href="#1016" id="1016">1016</a>
<a href="#1017" id="1017">1017</a>
<a href="#1018" id="1018">1018</a>
<a href="#1019" id="1019">1019</a>
<a href="#1020" id="1020">1020</a>
<a href="#1021" id="1021">1021</a>
<a href="#1022" id="1022">1022</a>
<a href="#1023" id="1023">1023</a>
<a href="#1024" id="1024">1024</a>
<a href="#1025" id="1025">1025</a>
<a href="#1026" id="1026">1026</a>
<a href="#1027" id="1027">1027</a>
<a href="#1028" id="1028">1028</a>
<a href="#1029" id="1029">1029</a>
<a href="#1030" id="1030">1030</a>
<a href="#1031" id="1031">1031</a>
<a href="#1032" id="1032">1032</a>
<a href="#1033" id="1033">1033</a>
<a href="#1034" id="1034">1034</a>
<a href="#1035" id="1035">1035</a>
<a href="#1036" id="1036">1036</a>
<a href="#1037" id="1037">1037</a>
<a href="#1038" id="1038">1038</a>
<a href="#1039" id="1039">1039</a>
</pre><pre class="rust"><code><span class="attr">#[cfg(feature = <span class="string">"arbitrary"</span>)]
</span><span class="kw">use </span><span class="kw">crate</span>::base::storage::Owned;
<span class="attr">#[cfg(feature = <span class="string">"arbitrary"</span>)]
</span><span class="kw">use </span>quickcheck::{Arbitrary, Gen};
<span class="kw">use </span>num::Zero;
<span class="attr">#[cfg(feature = <span class="string">"rand-no-std"</span>)]
</span><span class="kw">use </span>rand::{
distributions::{uniform::SampleUniform, Distribution, OpenClosed01, Standard, Uniform},
Rng,
};
<span class="kw">use </span>simba::scalar::RealField;
<span class="kw">use </span>simba::simd::{SimdBool, SimdRealField};
<span class="kw">use </span>std::ops::Neg;
<span class="kw">use </span><span class="kw">crate</span>::base::dimension::{U1, U2, U3};
<span class="kw">use </span><span class="kw">crate</span>::base::storage::Storage;
<span class="kw">use </span><span class="kw">crate</span>::base::{
Matrix2, Matrix3, SMatrix, SVector, Unit, UnitVector3, Vector, Vector1, Vector2, Vector3,
};
<span class="kw">use </span><span class="kw">crate</span>::geometry::{Rotation2, Rotation3, UnitComplex, UnitQuaternion};
<span class="comment">/*
*
* 2D Rotation matrix.
*
*/
</span><span class="doccomment">/// # Construction from a 2D rotation angle
</span><span class="kw">impl</span><T: SimdRealField> Rotation2<T> {
<span class="doccomment">/// Builds a 2 dimensional rotation matrix from an angle in radian.
///
/// # Example
///
/// ```
/// # #[macro_use] extern crate approx;
/// # use std::f32;
/// # use nalgebra::{Rotation2, Point2};
/// let rot = Rotation2::new(f32::consts::FRAC_PI_2);
///
/// assert_relative_eq!(rot * Point2::new(3.0, 4.0), Point2::new(-4.0, 3.0));
/// ```
</span><span class="kw">pub fn </span>new(angle: T) -> <span class="self">Self </span>{
<span class="kw">let </span>(sia, coa) = angle.simd_sin_cos();
<span class="self">Self</span>::from_matrix_unchecked(Matrix2::new(coa.clone(), -sia.clone(), sia, coa))
}
<span class="doccomment">/// Builds a 2 dimensional rotation matrix from an angle in radian wrapped in a 1-dimensional vector.
///
///
/// This is generally used in the context of generic programming. Using
/// the `::new(angle)` method instead is more common.
</span><span class="attr">#[inline]
</span><span class="kw">pub fn </span>from_scaled_axis<SB: Storage<T, U1>>(axisangle: Vector<T, U1, SB>) -> <span class="self">Self </span>{
<span class="self">Self</span>::new(axisangle[<span class="number">0</span>].clone())
}
}
<span class="doccomment">/// # Construction from an existing 2D matrix or rotations
</span><span class="kw">impl</span><T: SimdRealField> Rotation2<T> {
<span class="doccomment">/// Builds a rotation from a basis assumed to be orthonormal.
///
/// In order to get a valid rotation matrix, the input must be an
/// orthonormal basis, i.e., all vectors are normalized, and the are
/// all orthogonal to each other. These invariants are not checked
/// by this method.
</span><span class="kw">pub fn </span>from_basis_unchecked(basis: <span class="kw-2">&</span>[Vector2<T>; <span class="number">2</span>]) -> <span class="self">Self </span>{
<span class="kw">let </span>mat = Matrix2::from_columns(<span class="kw-2">&</span>basis[..]);
<span class="self">Self</span>::from_matrix_unchecked(mat)
}
<span class="doccomment">/// Builds a rotation matrix by extracting the rotation part of the given transformation `m`.
///
/// This is an iterative method. See `.from_matrix_eps` to provide mover
/// convergence parameters and starting solution.
/// This implements "A Robust Method to Extract the Rotational Part of Deformations" by Müller et al.
</span><span class="kw">pub fn </span>from_matrix(m: <span class="kw-2">&</span>Matrix2<T>) -> <span class="self">Self
</span><span class="kw">where
</span>T: RealField,
{
<span class="self">Self</span>::from_matrix_eps(m, T::default_epsilon(), <span class="number">0</span>, <span class="self">Self</span>::identity())
}
<span class="doccomment">/// Builds a rotation matrix by extracting the rotation part of the given transformation `m`.
///
/// This implements "A Robust Method to Extract the Rotational Part of Deformations" by Müller et al.
///
/// # Parameters
///
/// * `m`: the matrix from which the rotational part is to be extracted.
/// * `eps`: the angular errors tolerated between the current rotation and the optimal one.
/// * `max_iter`: the maximum number of iterations. Loops indefinitely until convergence if set to `0`.
/// * `guess`: an estimate of the solution. Convergence will be significantly faster if an initial solution close
/// to the actual solution is provided. Can be set to `Rotation2::identity()` if no other
/// guesses come to mind.
</span><span class="kw">pub fn </span>from_matrix_eps(m: <span class="kw-2">&</span>Matrix2<T>, eps: T, <span class="kw-2">mut </span>max_iter: usize, guess: <span class="self">Self</span>) -> <span class="self">Self
</span><span class="kw">where
</span>T: RealField,
{
<span class="kw">if </span>max_iter == <span class="number">0 </span>{
max_iter = usize::max_value();
}
<span class="kw">let </span><span class="kw-2">mut </span>rot = guess.into_inner();
<span class="kw">for _ in </span><span class="number">0</span>..max_iter {
<span class="kw">let </span>axis = rot.column(<span class="number">0</span>).perp(<span class="kw-2">&</span>m.column(<span class="number">0</span>)) + rot.column(<span class="number">1</span>).perp(<span class="kw-2">&</span>m.column(<span class="number">1</span>));
<span class="kw">let </span>denom = rot.column(<span class="number">0</span>).dot(<span class="kw-2">&</span>m.column(<span class="number">0</span>)) + rot.column(<span class="number">1</span>).dot(<span class="kw-2">&</span>m.column(<span class="number">1</span>));
<span class="kw">let </span>angle = axis / (denom.abs() + T::default_epsilon());
<span class="kw">if </span>angle.clone().abs() > eps {
rot = <span class="self">Self</span>::new(angle) * rot;
} <span class="kw">else </span>{
<span class="kw">break</span>;
}
}
<span class="self">Self</span>::from_matrix_unchecked(rot)
}
<span class="doccomment">/// The rotation matrix required to align `a` and `b` but with its angle.
///
/// This is the rotation `R` such that `(R * a).angle(b) == 0 && (R * a).dot(b).is_positive()`.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::{Vector2, Rotation2};
/// let a = Vector2::new(1.0, 2.0);
/// let b = Vector2::new(2.0, 1.0);
/// let rot = Rotation2::rotation_between(&a, &b);
/// assert_relative_eq!(rot * a, b);
/// assert_relative_eq!(rot.inverse() * b, a);
/// ```
</span><span class="attr">#[inline]
</span><span class="kw">pub fn </span>rotation_between<SB, SC>(a: <span class="kw-2">&</span>Vector<T, U2, SB>, b: <span class="kw-2">&</span>Vector<T, U2, SC>) -> <span class="self">Self
</span><span class="kw">where
</span>T: RealField,
SB: Storage<T, U2>,
SC: Storage<T, U2>,
{
<span class="kw">crate</span>::convert(UnitComplex::rotation_between(a, b).to_rotation_matrix())
}
<span class="doccomment">/// The smallest rotation needed to make `a` and `b` collinear and point toward the same
/// direction, raised to the power `s`.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::{Vector2, Rotation2};
/// let a = Vector2::new(1.0, 2.0);
/// let b = Vector2::new(2.0, 1.0);
/// let rot2 = Rotation2::scaled_rotation_between(&a, &b, 0.2);
/// let rot5 = Rotation2::scaled_rotation_between(&a, &b, 0.5);
/// assert_relative_eq!(rot2 * rot2 * rot2 * rot2 * rot2 * a, b, epsilon = 1.0e-6);
/// assert_relative_eq!(rot5 * rot5 * a, b, epsilon = 1.0e-6);
/// ```
</span><span class="attr">#[inline]
</span><span class="kw">pub fn </span>scaled_rotation_between<SB, SC>(
a: <span class="kw-2">&</span>Vector<T, U2, SB>,
b: <span class="kw-2">&</span>Vector<T, U2, SC>,
s: T,
) -> <span class="self">Self
</span><span class="kw">where
</span>T: RealField,
SB: Storage<T, U2>,
SC: Storage<T, U2>,
{
<span class="kw">crate</span>::convert(UnitComplex::scaled_rotation_between(a, b, s).to_rotation_matrix())
}
<span class="doccomment">/// The rotation matrix needed to make `self` and `other` coincide.
///
/// The result is such that: `self.rotation_to(other) * self == other`.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::Rotation2;
/// let rot1 = Rotation2::new(0.1);
/// let rot2 = Rotation2::new(1.7);
/// let rot_to = rot1.rotation_to(&rot2);
///
/// assert_relative_eq!(rot_to * rot1, rot2);
/// assert_relative_eq!(rot_to.inverse() * rot2, rot1);
/// ```
</span><span class="attr">#[inline]
#[must_use]
</span><span class="kw">pub fn </span>rotation_to(<span class="kw-2">&</span><span class="self">self</span>, other: <span class="kw-2">&</span><span class="self">Self</span>) -> <span class="self">Self </span>{
other * <span class="self">self</span>.inverse()
}
<span class="doccomment">/// Ensure this rotation is an orthonormal rotation matrix. This is useful when repeated
/// computations might cause the matrix from progressively not being orthonormal anymore.
</span><span class="attr">#[inline]
</span><span class="kw">pub fn </span>renormalize(<span class="kw-2">&mut </span><span class="self">self</span>)
<span class="kw">where
</span>T: RealField,
{
<span class="kw">let </span><span class="kw-2">mut </span>c = UnitComplex::from(<span class="self">self</span>.clone());
<span class="kw">let _ </span>= c.renormalize();
<span class="kw-2">*</span><span class="self">self </span>= <span class="self">Self</span>::from_matrix_eps(<span class="self">self</span>.matrix(), T::default_epsilon(), <span class="number">0</span>, c.into())
}
<span class="doccomment">/// Raise the rotation to a given floating power, i.e., returns the rotation with the angle
/// of `self` multiplied by `n`.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::Rotation2;
/// let rot = Rotation2::new(0.78);
/// let pow = rot.powf(2.0);
/// assert_relative_eq!(pow.angle(), 2.0 * 0.78);
/// ```
</span><span class="attr">#[inline]
#[must_use]
</span><span class="kw">pub fn </span>powf(<span class="kw-2">&</span><span class="self">self</span>, n: T) -> <span class="self">Self </span>{
<span class="self">Self</span>::new(<span class="self">self</span>.angle() * n)
}
}
<span class="doccomment">/// # 2D angle extraction
</span><span class="kw">impl</span><T: SimdRealField> Rotation2<T> {
<span class="doccomment">/// The rotation angle.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::Rotation2;
/// let rot = Rotation2::new(1.78);
/// assert_relative_eq!(rot.angle(), 1.78);
/// ```
</span><span class="attr">#[inline]
#[must_use]
</span><span class="kw">pub fn </span>angle(<span class="kw-2">&</span><span class="self">self</span>) -> T {
<span class="self">self</span>.matrix()[(<span class="number">1</span>, <span class="number">0</span>)]
.clone()
.simd_atan2(<span class="self">self</span>.matrix()[(<span class="number">0</span>, <span class="number">0</span>)].clone())
}
<span class="doccomment">/// The rotation angle needed to make `self` and `other` coincide.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::Rotation2;
/// let rot1 = Rotation2::new(0.1);
/// let rot2 = Rotation2::new(1.7);
/// assert_relative_eq!(rot1.angle_to(&rot2), 1.6);
/// ```
</span><span class="attr">#[inline]
#[must_use]
</span><span class="kw">pub fn </span>angle_to(<span class="kw-2">&</span><span class="self">self</span>, other: <span class="kw-2">&</span><span class="self">Self</span>) -> T {
<span class="self">self</span>.rotation_to(other).angle()
}
<span class="doccomment">/// The rotation angle returned as a 1-dimensional vector.
///
/// This is generally used in the context of generic programming. Using
/// the `.angle()` method instead is more common.
</span><span class="attr">#[inline]
#[must_use]
</span><span class="kw">pub fn </span>scaled_axis(<span class="kw-2">&</span><span class="self">self</span>) -> SVector<T, <span class="number">1</span>> {
Vector1::new(<span class="self">self</span>.angle())
}
}
<span class="attr">#[cfg(feature = <span class="string">"rand-no-std"</span>)]
</span><span class="kw">impl</span><T: SimdRealField> Distribution<Rotation2<T>> <span class="kw">for </span>Standard
<span class="kw">where
</span>T::Element: SimdRealField,
T: SampleUniform,
{
<span class="doccomment">/// Generate a uniformly distributed random rotation.
</span><span class="attr">#[inline]
</span><span class="kw">fn </span>sample<R: Rng + <span class="question-mark">?</span>Sized>(<span class="kw-2">&</span><span class="self">self</span>, rng: <span class="kw-2">&mut </span>R) -> Rotation2<T> {
<span class="kw">let </span>twopi = Uniform::new(T::zero(), T::simd_two_pi());
Rotation2::new(rng.sample(twopi))
}
}
<span class="attr">#[cfg(feature = <span class="string">"arbitrary"</span>)]
</span><span class="kw">impl</span><T: SimdRealField + Arbitrary> Arbitrary <span class="kw">for </span>Rotation2<T>
<span class="kw">where
</span>T::Element: SimdRealField,
Owned<T, U2, U2>: Send,
{
<span class="attr">#[inline]
</span><span class="kw">fn </span>arbitrary(g: <span class="kw-2">&mut </span>Gen) -> <span class="self">Self </span>{
<span class="self">Self</span>::new(T::arbitrary(g))
}
}
<span class="comment">/*
*
* 3D Rotation matrix.
*
*/
</span><span class="doccomment">/// # Construction from a 3D axis and/or angles
</span><span class="kw">impl</span><T: SimdRealField> Rotation3<T>
<span class="kw">where
</span>T::Element: SimdRealField,
{
<span class="doccomment">/// Builds a 3 dimensional rotation matrix from an axis and an angle.
///
/// # Arguments
/// * `axisangle` - A vector representing the rotation. Its magnitude is the amount of rotation
/// in radian. Its direction is the axis of rotation.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use std::f32;
/// # use nalgebra::{Rotation3, Point3, Vector3};
/// let axisangle = Vector3::y() * f32::consts::FRAC_PI_2;
/// // Point and vector being transformed in the tests.
/// let pt = Point3::new(4.0, 5.0, 6.0);
/// let vec = Vector3::new(4.0, 5.0, 6.0);
/// let rot = Rotation3::new(axisangle);
///
/// assert_relative_eq!(rot * pt, Point3::new(6.0, 5.0, -4.0), epsilon = 1.0e-6);
/// assert_relative_eq!(rot * vec, Vector3::new(6.0, 5.0, -4.0), epsilon = 1.0e-6);
///
/// // A zero vector yields an identity.
/// assert_eq!(Rotation3::new(Vector3::<f32>::zeros()), Rotation3::identity());
/// ```
</span><span class="kw">pub fn </span>new<SB: Storage<T, U3>>(axisangle: Vector<T, U3, SB>) -> <span class="self">Self </span>{
<span class="kw">let </span>axisangle = axisangle.into_owned();
<span class="kw">let </span>(axis, angle) = Unit::new_and_get(axisangle);
<span class="self">Self</span>::from_axis_angle(<span class="kw-2">&</span>axis, angle)
}
<span class="doccomment">/// Builds a 3D rotation matrix from an axis scaled by the rotation angle.
///
/// This is the same as `Self::new(axisangle)`.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use std::f32;
/// # use nalgebra::{Rotation3, Point3, Vector3};
/// let axisangle = Vector3::y() * f32::consts::FRAC_PI_2;
/// // Point and vector being transformed in the tests.
/// let pt = Point3::new(4.0, 5.0, 6.0);
/// let vec = Vector3::new(4.0, 5.0, 6.0);
/// let rot = Rotation3::new(axisangle);
///
/// assert_relative_eq!(rot * pt, Point3::new(6.0, 5.0, -4.0), epsilon = 1.0e-6);
/// assert_relative_eq!(rot * vec, Vector3::new(6.0, 5.0, -4.0), epsilon = 1.0e-6);
///
/// // A zero vector yields an identity.
/// assert_eq!(Rotation3::from_scaled_axis(Vector3::<f32>::zeros()), Rotation3::identity());
/// ```
</span><span class="kw">pub fn </span>from_scaled_axis<SB: Storage<T, U3>>(axisangle: Vector<T, U3, SB>) -> <span class="self">Self </span>{
<span class="self">Self</span>::new(axisangle)
}
<span class="doccomment">/// Builds a 3D rotation matrix from an axis and a rotation angle.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use std::f32;
/// # use nalgebra::{Rotation3, Point3, Vector3};
/// let axis = Vector3::y_axis();
/// let angle = f32::consts::FRAC_PI_2;
/// // Point and vector being transformed in the tests.
/// let pt = Point3::new(4.0, 5.0, 6.0);
/// let vec = Vector3::new(4.0, 5.0, 6.0);
/// let rot = Rotation3::from_axis_angle(&axis, angle);
///
/// assert_eq!(rot.axis().unwrap(), axis);
/// assert_eq!(rot.angle(), angle);
/// assert_relative_eq!(rot * pt, Point3::new(6.0, 5.0, -4.0), epsilon = 1.0e-6);
/// assert_relative_eq!(rot * vec, Vector3::new(6.0, 5.0, -4.0), epsilon = 1.0e-6);
///
/// // A zero vector yields an identity.
/// assert_eq!(Rotation3::from_scaled_axis(Vector3::<f32>::zeros()), Rotation3::identity());
/// ```
</span><span class="kw">pub fn </span>from_axis_angle<SB>(axis: <span class="kw-2">&</span>Unit<Vector<T, U3, SB>>, angle: T) -> <span class="self">Self
</span><span class="kw">where
</span>SB: Storage<T, U3>,
{
angle.clone().simd_ne(T::zero()).if_else(
|| {
<span class="kw">let </span>ux = axis.as_ref()[<span class="number">0</span>].clone();
<span class="kw">let </span>uy = axis.as_ref()[<span class="number">1</span>].clone();
<span class="kw">let </span>uz = axis.as_ref()[<span class="number">2</span>].clone();
<span class="kw">let </span>sqx = ux.clone() * ux.clone();
<span class="kw">let </span>sqy = uy.clone() * uy.clone();
<span class="kw">let </span>sqz = uz.clone() * uz.clone();
<span class="kw">let </span>(sin, cos) = angle.simd_sin_cos();
<span class="kw">let </span>one_m_cos = T::one() - cos.clone();
<span class="self">Self</span>::from_matrix_unchecked(SMatrix::<T, <span class="number">3</span>, <span class="number">3</span>>::new(
sqx.clone() + (T::one() - sqx) * cos.clone(),
ux.clone() * uy.clone() * one_m_cos.clone() - uz.clone() * sin.clone(),
ux.clone() * uz.clone() * one_m_cos.clone() + uy.clone() * sin.clone(),
ux.clone() * uy.clone() * one_m_cos.clone() + uz.clone() * sin.clone(),
sqy.clone() + (T::one() - sqy) * cos.clone(),
uy.clone() * uz.clone() * one_m_cos.clone() - ux.clone() * sin.clone(),
ux.clone() * uz.clone() * one_m_cos.clone() - uy.clone() * sin.clone(),
uy * uz * one_m_cos + ux * sin,
sqz.clone() + (T::one() - sqz) * cos,
))
},
<span class="self">Self</span>::identity,
)
}
<span class="doccomment">/// Creates a new rotation from Euler angles.
///
/// The primitive rotations are applied in order: 1 roll − 2 pitch − 3 yaw.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::Rotation3;
/// let rot = Rotation3::from_euler_angles(0.1, 0.2, 0.3);
/// let euler = rot.euler_angles();
/// assert_relative_eq!(euler.0, 0.1, epsilon = 1.0e-6);
/// assert_relative_eq!(euler.1, 0.2, epsilon = 1.0e-6);
/// assert_relative_eq!(euler.2, 0.3, epsilon = 1.0e-6);
/// ```
</span><span class="kw">pub fn </span>from_euler_angles(roll: T, pitch: T, yaw: T) -> <span class="self">Self </span>{
<span class="kw">let </span>(sr, cr) = roll.simd_sin_cos();
<span class="kw">let </span>(sp, cp) = pitch.simd_sin_cos();
<span class="kw">let </span>(sy, cy) = yaw.simd_sin_cos();
<span class="self">Self</span>::from_matrix_unchecked(SMatrix::<T, <span class="number">3</span>, <span class="number">3</span>>::new(
cy.clone() * cp.clone(),
cy.clone() * sp.clone() * sr.clone() - sy.clone() * cr.clone(),
cy.clone() * sp.clone() * cr.clone() + sy.clone() * sr.clone(),
sy.clone() * cp.clone(),
sy.clone() * sp.clone() * sr.clone() + cy.clone() * cr.clone(),
sy * sp.clone() * cr.clone() - cy * sr.clone(),
-sp,
cp.clone() * sr,
cp * cr,
))
}
}
<span class="doccomment">/// # Construction from a 3D eye position and target point
</span><span class="kw">impl</span><T: SimdRealField> Rotation3<T>
<span class="kw">where
</span>T::Element: SimdRealField,
{
<span class="doccomment">/// Creates a rotation that corresponds to the local frame of an observer standing at the
/// origin and looking toward `dir`.
///
/// It maps the `z` axis to the direction `dir`.
///
/// # Arguments
/// * dir - The look direction, that is, direction the matrix `z` axis will be aligned with.
/// * up - The vertical direction. The only requirement of this parameter is to not be
/// collinear to `dir`. Non-collinearity is not checked.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use std::f32;
/// # use nalgebra::{Rotation3, Vector3};
/// let dir = Vector3::new(1.0, 2.0, 3.0);
/// let up = Vector3::y();
///
/// let rot = Rotation3::face_towards(&dir, &up);
/// assert_relative_eq!(rot * Vector3::z(), dir.normalize());
/// ```
</span><span class="attr">#[inline]
</span><span class="kw">pub fn </span>face_towards<SB, SC>(dir: <span class="kw-2">&</span>Vector<T, U3, SB>, up: <span class="kw-2">&</span>Vector<T, U3, SC>) -> <span class="self">Self
</span><span class="kw">where
</span>SB: Storage<T, U3>,
SC: Storage<T, U3>,
{
<span class="kw">let </span>zaxis = dir.normalize();
<span class="kw">let </span>xaxis = up.cross(<span class="kw-2">&</span>zaxis).normalize();
<span class="kw">let </span>yaxis = zaxis.cross(<span class="kw-2">&</span>xaxis).normalize();
<span class="self">Self</span>::from_matrix_unchecked(SMatrix::<T, <span class="number">3</span>, <span class="number">3</span>>::new(
xaxis.x.clone(),
yaxis.x.clone(),
zaxis.x.clone(),
xaxis.y.clone(),
yaxis.y.clone(),
zaxis.y.clone(),
xaxis.z.clone(),
yaxis.z.clone(),
zaxis.z.clone(),
))
}
<span class="doccomment">/// Deprecated: Use [`Rotation3::face_towards`] instead.
</span><span class="attr">#[deprecated(note = <span class="string">"renamed to `face_towards`"</span>)]
</span><span class="kw">pub fn </span>new_observer_frames<SB, SC>(dir: <span class="kw-2">&</span>Vector<T, U3, SB>, up: <span class="kw-2">&</span>Vector<T, U3, SC>) -> <span class="self">Self
</span><span class="kw">where
</span>SB: Storage<T, U3>,
SC: Storage<T, U3>,
{
<span class="self">Self</span>::face_towards(dir, up)
}
<span class="doccomment">/// Builds a right-handed look-at view matrix without translation.
///
/// It maps the view direction `dir` to the **negative** `z` axis.
/// This conforms to the common notion of right handed look-at matrix from the computer
/// graphics community.
///
/// # Arguments
/// * dir - The direction toward which the camera looks.
/// * up - A vector approximately aligned with required the vertical axis. The only
/// requirement of this parameter is to not be collinear to `dir`.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use std::f32;
/// # use nalgebra::{Rotation3, Vector3};
/// let dir = Vector3::new(1.0, 2.0, 3.0);
/// let up = Vector3::y();
///
/// let rot = Rotation3::look_at_rh(&dir, &up);
/// assert_relative_eq!(rot * dir.normalize(), -Vector3::z());
/// ```
</span><span class="attr">#[inline]
</span><span class="kw">pub fn </span>look_at_rh<SB, SC>(dir: <span class="kw-2">&</span>Vector<T, U3, SB>, up: <span class="kw-2">&</span>Vector<T, U3, SC>) -> <span class="self">Self
</span><span class="kw">where
</span>SB: Storage<T, U3>,
SC: Storage<T, U3>,
{
<span class="self">Self</span>::face_towards(<span class="kw-2">&</span>dir.neg(), up).inverse()
}
<span class="doccomment">/// Builds a left-handed look-at view matrix without translation.
///
/// It maps the view direction `dir` to the **positive** `z` axis.
/// This conforms to the common notion of left handed look-at matrix from the computer
/// graphics community.
///
/// # Arguments
/// * dir - The direction toward which the camera looks.
/// * up - A vector approximately aligned with required the vertical axis. The only
/// requirement of this parameter is to not be collinear to `dir`.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use std::f32;
/// # use nalgebra::{Rotation3, Vector3};
/// let dir = Vector3::new(1.0, 2.0, 3.0);
/// let up = Vector3::y();
///
/// let rot = Rotation3::look_at_lh(&dir, &up);
/// assert_relative_eq!(rot * dir.normalize(), Vector3::z());
/// ```
</span><span class="attr">#[inline]
</span><span class="kw">pub fn </span>look_at_lh<SB, SC>(dir: <span class="kw-2">&</span>Vector<T, U3, SB>, up: <span class="kw-2">&</span>Vector<T, U3, SC>) -> <span class="self">Self
</span><span class="kw">where
</span>SB: Storage<T, U3>,
SC: Storage<T, U3>,
{
<span class="self">Self</span>::face_towards(dir, up).inverse()
}
}
<span class="doccomment">/// # Construction from an existing 3D matrix or rotations
</span><span class="kw">impl</span><T: SimdRealField> Rotation3<T>
<span class="kw">where
</span>T::Element: SimdRealField,
{
<span class="doccomment">/// The rotation matrix required to align `a` and `b` but with its angle.
///
/// This is the rotation `R` such that `(R * a).angle(b) == 0 && (R * a).dot(b).is_positive()`.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::{Vector3, Rotation3};
/// let a = Vector3::new(1.0, 2.0, 3.0);
/// let b = Vector3::new(3.0, 1.0, 2.0);
/// let rot = Rotation3::rotation_between(&a, &b).unwrap();
/// assert_relative_eq!(rot * a, b, epsilon = 1.0e-6);
/// assert_relative_eq!(rot.inverse() * b, a, epsilon = 1.0e-6);
/// ```
</span><span class="attr">#[inline]
</span><span class="kw">pub fn </span>rotation_between<SB, SC>(a: <span class="kw-2">&</span>Vector<T, U3, SB>, b: <span class="kw-2">&</span>Vector<T, U3, SC>) -> <span class="prelude-ty">Option</span><<span class="self">Self</span>>
<span class="kw">where
</span>T: RealField,
SB: Storage<T, U3>,
SC: Storage<T, U3>,
{
<span class="self">Self</span>::scaled_rotation_between(a, b, T::one())
}
<span class="doccomment">/// The smallest rotation needed to make `a` and `b` collinear and point toward the same
/// direction, raised to the power `s`.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::{Vector3, Rotation3};
/// let a = Vector3::new(1.0, 2.0, 3.0);
/// let b = Vector3::new(3.0, 1.0, 2.0);
/// let rot2 = Rotation3::scaled_rotation_between(&a, &b, 0.2).unwrap();
/// let rot5 = Rotation3::scaled_rotation_between(&a, &b, 0.5).unwrap();
/// assert_relative_eq!(rot2 * rot2 * rot2 * rot2 * rot2 * a, b, epsilon = 1.0e-6);
/// assert_relative_eq!(rot5 * rot5 * a, b, epsilon = 1.0e-6);
/// ```
</span><span class="attr">#[inline]
</span><span class="kw">pub fn </span>scaled_rotation_between<SB, SC>(
a: <span class="kw-2">&</span>Vector<T, U3, SB>,
b: <span class="kw-2">&</span>Vector<T, U3, SC>,
n: T,
) -> <span class="prelude-ty">Option</span><<span class="self">Self</span>>
<span class="kw">where
</span>T: RealField,
SB: Storage<T, U3>,
SC: Storage<T, U3>,
{
<span class="comment">// TODO: code duplication with Rotation.
</span><span class="kw">if let </span>(<span class="prelude-val">Some</span>(na), <span class="prelude-val">Some</span>(nb)) = (a.try_normalize(T::zero()), b.try_normalize(T::zero())) {
<span class="kw">let </span>c = na.cross(<span class="kw-2">&</span>nb);
<span class="kw">if let </span><span class="prelude-val">Some</span>(axis) = Unit::try_new(c, T::default_epsilon()) {
<span class="kw">return </span><span class="prelude-val">Some</span>(<span class="self">Self</span>::from_axis_angle(<span class="kw-2">&</span>axis, na.dot(<span class="kw-2">&</span>nb).acos() * n));
}
<span class="comment">// Zero or PI.
</span><span class="kw">if </span>na.dot(<span class="kw-2">&</span>nb) < T::zero() {
<span class="comment">// PI
//
// The rotation axis is undefined but the angle not zero. This is not a
// simple rotation.
</span><span class="kw">return </span><span class="prelude-val">None</span>;
}
}
<span class="prelude-val">Some</span>(<span class="self">Self</span>::identity())
}
<span class="doccomment">/// The rotation matrix needed to make `self` and `other` coincide.
///
/// The result is such that: `self.rotation_to(other) * self == other`.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::{Rotation3, Vector3};
/// let rot1 = Rotation3::from_axis_angle(&Vector3::y_axis(), 1.0);
/// let rot2 = Rotation3::from_axis_angle(&Vector3::x_axis(), 0.1);
/// let rot_to = rot1.rotation_to(&rot2);
/// assert_relative_eq!(rot_to * rot1, rot2, epsilon = 1.0e-6);
/// ```
</span><span class="attr">#[inline]
#[must_use]
</span><span class="kw">pub fn </span>rotation_to(<span class="kw-2">&</span><span class="self">self</span>, other: <span class="kw-2">&</span><span class="self">Self</span>) -> <span class="self">Self </span>{
other * <span class="self">self</span>.inverse()
}
<span class="doccomment">/// Raise the rotation to a given floating power, i.e., returns the rotation with the same
/// axis as `self` and an angle equal to `self.angle()` multiplied by `n`.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::{Rotation3, Vector3, Unit};
/// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
/// let angle = 1.2;
/// let rot = Rotation3::from_axis_angle(&axis, angle);
/// let pow = rot.powf(2.0);
/// assert_relative_eq!(pow.axis().unwrap(), axis, epsilon = 1.0e-6);
/// assert_eq!(pow.angle(), 2.4);
/// ```
</span><span class="attr">#[inline]
#[must_use]
</span><span class="kw">pub fn </span>powf(<span class="kw-2">&</span><span class="self">self</span>, n: T) -> <span class="self">Self
</span><span class="kw">where
</span>T: RealField,
{
<span class="kw">if let </span><span class="prelude-val">Some</span>(axis) = <span class="self">self</span>.axis() {
<span class="self">Self</span>::from_axis_angle(<span class="kw-2">&</span>axis, <span class="self">self</span>.angle() * n)
} <span class="kw">else if </span><span class="self">self</span>.matrix()[(<span class="number">0</span>, <span class="number">0</span>)] < T::zero() {
<span class="kw">let </span>minus_id = SMatrix::<T, <span class="number">3</span>, <span class="number">3</span>>::from_diagonal_element(-T::one());
<span class="self">Self</span>::from_matrix_unchecked(minus_id)
} <span class="kw">else </span>{
<span class="self">Self</span>::identity()
}
}
<span class="doccomment">/// Builds a rotation from a basis assumed to be orthonormal.
///
/// In order to get a valid rotation matrix, the input must be an
/// orthonormal basis, i.e., all vectors are normalized, and the are
/// all orthogonal to each other. These invariants are not checked
/// by this method.
</span><span class="kw">pub fn </span>from_basis_unchecked(basis: <span class="kw-2">&</span>[Vector3<T>; <span class="number">3</span>]) -> <span class="self">Self </span>{
<span class="kw">let </span>mat = Matrix3::from_columns(<span class="kw-2">&</span>basis[..]);
<span class="self">Self</span>::from_matrix_unchecked(mat)
}
<span class="doccomment">/// Builds a rotation matrix by extracting the rotation part of the given transformation `m`.
///
/// This is an iterative method. See `.from_matrix_eps` to provide mover
/// convergence parameters and starting solution.
/// This implements "A Robust Method to Extract the Rotational Part of Deformations" by Müller et al.
</span><span class="kw">pub fn </span>from_matrix(m: <span class="kw-2">&</span>Matrix3<T>) -> <span class="self">Self
</span><span class="kw">where
</span>T: RealField,
{
<span class="self">Self</span>::from_matrix_eps(m, T::default_epsilon(), <span class="number">0</span>, <span class="self">Self</span>::identity())
}
<span class="doccomment">/// Builds a rotation matrix by extracting the rotation part of the given transformation `m`.
///
/// This implements "A Robust Method to Extract the Rotational Part of Deformations" by Müller et al.
///
/// # Parameters
///
/// * `m`: the matrix from which the rotational part is to be extracted.
/// * `eps`: the angular errors tolerated between the current rotation and the optimal one.
/// * `max_iter`: the maximum number of iterations. Loops indefinitely until convergence if set to `0`.
/// * `guess`: a guess of the solution. Convergence will be significantly faster if an initial solution close
/// to the actual solution is provided. Can be set to `Rotation3::identity()` if no other
/// guesses come to mind.
</span><span class="kw">pub fn </span>from_matrix_eps(m: <span class="kw-2">&</span>Matrix3<T>, eps: T, <span class="kw-2">mut </span>max_iter: usize, guess: <span class="self">Self</span>) -> <span class="self">Self
</span><span class="kw">where
</span>T: RealField,
{
<span class="kw">if </span>max_iter == <span class="number">0 </span>{
max_iter = usize::MAX;
}
<span class="comment">// Using sqrt(eps) ensures we perturb with something larger than eps; clamp to eps to handle the case of eps > 1.0
</span><span class="kw">let </span>eps_disturbance = eps.clone().sqrt().max(eps.clone() * eps.clone());
<span class="kw">let </span><span class="kw-2">mut </span>perturbation_axes = Vector3::x_axis();
<span class="kw">let </span><span class="kw-2">mut </span>rot = guess.into_inner();
<span class="kw">for _ in </span><span class="number">0</span>..max_iter {
<span class="kw">let </span>axis = rot.column(<span class="number">0</span>).cross(<span class="kw-2">&</span>m.column(<span class="number">0</span>))
+ rot.column(<span class="number">1</span>).cross(<span class="kw-2">&</span>m.column(<span class="number">1</span>))
+ rot.column(<span class="number">2</span>).cross(<span class="kw-2">&</span>m.column(<span class="number">2</span>));
<span class="kw">let </span>denom = rot.column(<span class="number">0</span>).dot(<span class="kw-2">&</span>m.column(<span class="number">0</span>))
+ rot.column(<span class="number">1</span>).dot(<span class="kw-2">&</span>m.column(<span class="number">1</span>))
+ rot.column(<span class="number">2</span>).dot(<span class="kw-2">&</span>m.column(<span class="number">2</span>));
<span class="kw">let </span>axisangle = axis / (denom.abs() + T::default_epsilon());
<span class="kw">if let </span><span class="prelude-val">Some</span>((axis, angle)) = Unit::try_new_and_get(axisangle, eps.clone()) {
rot = Rotation3::from_axis_angle(<span class="kw-2">&</span>axis, angle) * rot;
} <span class="kw">else </span>{
<span class="comment">// Check if stuck in a maximum w.r.t. the norm (m - rot).norm()
</span><span class="kw">let </span><span class="kw-2">mut </span>perturbed = rot.clone();
<span class="kw">let </span>norm_squared = (m - <span class="kw-2">&</span>rot).norm_squared();
<span class="kw">let </span><span class="kw-2">mut </span>new_norm_squared: T;
<span class="comment">// Perturb until the new norm is significantly different
</span><span class="kw">loop </span>{
perturbed <span class="kw-2">*</span>=
Rotation3::from_axis_angle(<span class="kw-2">&</span>perturbation_axes, eps_disturbance.clone());
new_norm_squared = (m - <span class="kw-2">&</span>perturbed).norm_squared();
<span class="kw">if </span><span class="macro">abs_diff_ne!</span>(
norm_squared,
new_norm_squared,
epsilon = T::default_epsilon()
) {
<span class="kw">break</span>;
}
}
<span class="comment">// If new norm is larger, it's a minimum
</span><span class="kw">if </span>norm_squared < new_norm_squared {
<span class="kw">break</span>;
}
<span class="comment">// If not, continue from perturbed rotation, but use a different axes for the next perturbation
</span>perturbation_axes = UnitVector3::new_unchecked(perturbation_axes.yzx());
rot = perturbed;
}
}
<span class="self">Self</span>::from_matrix_unchecked(rot)
}
<span class="doccomment">/// Ensure this rotation is an orthonormal rotation matrix. This is useful when repeated
/// computations might cause the matrix from progressively not being orthonormal anymore.
</span><span class="attr">#[inline]
</span><span class="kw">pub fn </span>renormalize(<span class="kw-2">&mut </span><span class="self">self</span>)
<span class="kw">where
</span>T: RealField,
{
<span class="kw">let </span><span class="kw-2">mut </span>c = UnitQuaternion::from(<span class="self">self</span>.clone());
<span class="kw">let _ </span>= c.renormalize();
<span class="kw-2">*</span><span class="self">self </span>= <span class="self">Self</span>::from_matrix_eps(<span class="self">self</span>.matrix(), T::default_epsilon(), <span class="number">0</span>, c.into())
}
}
<span class="doccomment">/// # 3D axis and angle extraction
</span><span class="kw">impl</span><T: SimdRealField> Rotation3<T> {
<span class="doccomment">/// The rotation angle in [0; pi].
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::{Unit, Rotation3, Vector3};
/// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
/// let rot = Rotation3::from_axis_angle(&axis, 1.78);
/// assert_relative_eq!(rot.angle(), 1.78);
/// ```
</span><span class="attr">#[inline]
#[must_use]
</span><span class="kw">pub fn </span>angle(<span class="kw-2">&</span><span class="self">self</span>) -> T {
((<span class="self">self</span>.matrix()[(<span class="number">0</span>, <span class="number">0</span>)].clone()
+ <span class="self">self</span>.matrix()[(<span class="number">1</span>, <span class="number">1</span>)].clone()
+ <span class="self">self</span>.matrix()[(<span class="number">2</span>, <span class="number">2</span>)].clone()
- T::one())
/ <span class="kw">crate</span>::convert(<span class="number">2.0</span>))
.simd_acos()
}
<span class="doccomment">/// The rotation axis. Returns `None` if the rotation angle is zero or PI.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::{Rotation3, Vector3, Unit};
/// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
/// let angle = 1.2;
/// let rot = Rotation3::from_axis_angle(&axis, angle);
/// assert_relative_eq!(rot.axis().unwrap(), axis);
///
/// // Case with a zero angle.
/// let rot = Rotation3::from_axis_angle(&axis, 0.0);
/// assert!(rot.axis().is_none());
/// ```
</span><span class="attr">#[inline]
#[must_use]
</span><span class="kw">pub fn </span>axis(<span class="kw-2">&</span><span class="self">self</span>) -> <span class="prelude-ty">Option</span><Unit<Vector3<T>>>
<span class="kw">where
</span>T: RealField,
{
<span class="kw">let </span>rotmat = <span class="self">self</span>.matrix();
<span class="kw">let </span>axis = SVector::<T, <span class="number">3</span>>::new(
rotmat[(<span class="number">2</span>, <span class="number">1</span>)].clone() - rotmat[(<span class="number">1</span>, <span class="number">2</span>)].clone(),
rotmat[(<span class="number">0</span>, <span class="number">2</span>)].clone() - rotmat[(<span class="number">2</span>, <span class="number">0</span>)].clone(),
rotmat[(<span class="number">1</span>, <span class="number">0</span>)].clone() - rotmat[(<span class="number">0</span>, <span class="number">1</span>)].clone(),
);
Unit::try_new(axis, T::default_epsilon())
}
<span class="doccomment">/// The rotation axis multiplied by the rotation angle.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::{Rotation3, Vector3, Unit};
/// let axisangle = Vector3::new(0.1, 0.2, 0.3);
/// let rot = Rotation3::new(axisangle);
/// assert_relative_eq!(rot.scaled_axis(), axisangle, epsilon = 1.0e-6);
/// ```
</span><span class="attr">#[inline]
#[must_use]
</span><span class="kw">pub fn </span>scaled_axis(<span class="kw-2">&</span><span class="self">self</span>) -> Vector3<T>
<span class="kw">where
</span>T: RealField,
{
<span class="kw">if let </span><span class="prelude-val">Some</span>(axis) = <span class="self">self</span>.axis() {
axis.into_inner() * <span class="self">self</span>.angle()
} <span class="kw">else </span>{
Vector::zero()
}
}
<span class="doccomment">/// The rotation axis and angle in (0, pi] of this rotation matrix.
///
/// Returns `None` if the angle is zero.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::{Rotation3, Vector3, Unit};
/// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
/// let angle = 1.2;
/// let rot = Rotation3::from_axis_angle(&axis, angle);
/// let axis_angle = rot.axis_angle().unwrap();
/// assert_relative_eq!(axis_angle.0, axis);
/// assert_relative_eq!(axis_angle.1, angle);
///
/// // Case with a zero angle.
/// let rot = Rotation3::from_axis_angle(&axis, 0.0);
/// assert!(rot.axis_angle().is_none());
/// ```
</span><span class="attr">#[inline]
#[must_use]
</span><span class="kw">pub fn </span>axis_angle(<span class="kw-2">&</span><span class="self">self</span>) -> <span class="prelude-ty">Option</span><(Unit<Vector3<T>>, T)>
<span class="kw">where
</span>T: RealField,
{
<span class="self">self</span>.axis().map(|axis| (axis, <span class="self">self</span>.angle()))
}
<span class="doccomment">/// The rotation angle needed to make `self` and `other` coincide.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::{Rotation3, Vector3};
/// let rot1 = Rotation3::from_axis_angle(&Vector3::y_axis(), 1.0);
/// let rot2 = Rotation3::from_axis_angle(&Vector3::x_axis(), 0.1);
/// assert_relative_eq!(rot1.angle_to(&rot2), 1.0045657, epsilon = 1.0e-6);
/// ```
</span><span class="attr">#[inline]
#[must_use]
</span><span class="kw">pub fn </span>angle_to(<span class="kw-2">&</span><span class="self">self</span>, other: <span class="kw-2">&</span><span class="self">Self</span>) -> T
<span class="kw">where
</span>T::Element: SimdRealField,
{
<span class="self">self</span>.rotation_to(other).angle()
}
<span class="doccomment">/// Creates Euler angles from a rotation.
///
/// The angles are produced in the form (roll, pitch, yaw).
</span><span class="attr">#[deprecated(note = <span class="string">"This is renamed to use `.euler_angles()`."</span>)]
</span><span class="kw">pub fn </span>to_euler_angles(<span class="self">self</span>) -> (T, T, T)
<span class="kw">where
</span>T: RealField,
{
<span class="self">self</span>.euler_angles()
}
<span class="doccomment">/// Euler angles corresponding to this rotation from a rotation.
///
/// The angles are produced in the form (roll, pitch, yaw).
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::Rotation3;
/// let rot = Rotation3::from_euler_angles(0.1, 0.2, 0.3);
/// let euler = rot.euler_angles();
/// assert_relative_eq!(euler.0, 0.1, epsilon = 1.0e-6);
/// assert_relative_eq!(euler.1, 0.2, epsilon = 1.0e-6);
/// assert_relative_eq!(euler.2, 0.3, epsilon = 1.0e-6);
/// ```
</span><span class="attr">#[must_use]
</span><span class="kw">pub fn </span>euler_angles(<span class="kw-2">&</span><span class="self">self</span>) -> (T, T, T)
<span class="kw">where
</span>T: RealField,
{
<span class="comment">// Implementation informed by "Computing Euler angles from a rotation matrix", by Gregory G. Slabaugh
// https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.371.6578
// where roll, pitch, yaw angles are referred to as ψ, θ, ϕ,
</span><span class="kw">if </span><span class="self">self</span>[(<span class="number">2</span>, <span class="number">0</span>)].clone().abs() < T::one() {
<span class="kw">let </span>pitch = -<span class="self">self</span>[(<span class="number">2</span>, <span class="number">0</span>)].clone().asin();
<span class="kw">let </span>theta_cos = pitch.clone().cos();
<span class="kw">let </span>roll = (<span class="self">self</span>[(<span class="number">2</span>, <span class="number">1</span>)].clone() / theta_cos.clone())
.atan2(<span class="self">self</span>[(<span class="number">2</span>, <span class="number">2</span>)].clone() / theta_cos.clone());
<span class="kw">let </span>yaw =
(<span class="self">self</span>[(<span class="number">1</span>, <span class="number">0</span>)].clone() / theta_cos.clone()).atan2(<span class="self">self</span>[(<span class="number">0</span>, <span class="number">0</span>)].clone() / theta_cos);
(roll, pitch, yaw)
} <span class="kw">else if </span><span class="self">self</span>[(<span class="number">2</span>, <span class="number">0</span>)].clone() <= -T::one() {
(
<span class="self">self</span>[(<span class="number">0</span>, <span class="number">1</span>)].clone().atan2(<span class="self">self</span>[(<span class="number">0</span>, <span class="number">2</span>)].clone()),
T::frac_pi_2(),
T::zero(),
)
} <span class="kw">else </span>{
(
-<span class="self">self</span>[(<span class="number">0</span>, <span class="number">1</span>)].clone().atan2(-<span class="self">self</span>[(<span class="number">0</span>, <span class="number">2</span>)].clone()),
-T::frac_pi_2(),
T::zero(),
)
}
}
}
<span class="attr">#[cfg(feature = <span class="string">"rand-no-std"</span>)]
</span><span class="kw">impl</span><T: SimdRealField> Distribution<Rotation3<T>> <span class="kw">for </span>Standard
<span class="kw">where
</span>T::Element: SimdRealField,
OpenClosed01: Distribution<T>,
T: SampleUniform,
{
<span class="doccomment">/// Generate a uniformly distributed random rotation.
</span><span class="attr">#[inline]
</span><span class="kw">fn </span>sample<<span class="lifetime">'a</span>, R: Rng + <span class="question-mark">?</span>Sized>(<span class="kw-2">&</span><span class="self">self</span>, rng: <span class="kw-2">&mut </span>R) -> Rotation3<T> {
<span class="comment">// James Arvo.
// Fast random rotation matrices.
// In D. Kirk, editor, Graphics Gems III, pages 117-120. Academic, New York, 1992.
// Compute a random rotation around Z
</span><span class="kw">let </span>twopi = Uniform::new(T::zero(), T::simd_two_pi());
<span class="kw">let </span>theta = rng.sample(<span class="kw-2">&</span>twopi);
<span class="kw">let </span>(ts, tc) = theta.simd_sin_cos();
<span class="kw">let </span>a = SMatrix::<T, <span class="number">3</span>, <span class="number">3</span>>::new(
tc.clone(),
ts.clone(),
T::zero(),
-ts,
tc,
T::zero(),
T::zero(),
T::zero(),
T::one(),
);
<span class="comment">// Compute a random rotation *of* Z
</span><span class="kw">let </span>phi = rng.sample(<span class="kw-2">&</span>twopi);
<span class="kw">let </span>z = rng.sample(OpenClosed01);
<span class="kw">let </span>(ps, pc) = phi.simd_sin_cos();
<span class="kw">let </span>sqrt_z = z.clone().simd_sqrt();
<span class="kw">let </span>v = Vector3::new(pc * sqrt_z.clone(), ps * sqrt_z, (T::one() - z).simd_sqrt());
<span class="kw">let </span><span class="kw-2">mut </span>b = v.clone() * v.transpose();
b += b.clone();
b -= SMatrix::<T, <span class="number">3</span>, <span class="number">3</span>>::identity();
Rotation3::from_matrix_unchecked(b * a)
}
}
<span class="attr">#[cfg(feature = <span class="string">"arbitrary"</span>)]
</span><span class="kw">impl</span><T: SimdRealField + Arbitrary> Arbitrary <span class="kw">for </span>Rotation3<T>
<span class="kw">where
</span>T::Element: SimdRealField,
Owned<T, U3, U3>: Send,
Owned<T, U3>: Send,
{
<span class="attr">#[inline]
</span><span class="kw">fn </span>arbitrary(g: <span class="kw-2">&mut </span>Gen) -> <span class="self">Self </span>{
<span class="self">Self</span>::new(SVector::arbitrary(g))
}
}
</code></pre></div>
</section></div></main><div id="rustdoc-vars" data-root-path="../../../" data-static-root-path="../../../static.files/" data-current-crate="nalgebra" data-themes="" data-resource-suffix="" data-rustdoc-version="1.67.1 (d5a82bbd2 2023-02-07)" data-search-js="search-444266647c4dba98.js" data-settings-js="settings-bebeae96e00e4617.js" data-settings-css="settings-af96d9e2fc13e081.css" ></div></body></html>