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| author | Marijn Tamis <[email protected]> | 2018-09-12 14:12:47 +0200 |
|---|---|---|
| committer | Marijn Tamis <[email protected]> | 2018-09-12 14:12:47 +0200 |
| commit | 7bceea80d4a04649f495f4f4331d7e3bdcdb05ca (patch) | |
| tree | 824767cccf7b516a8c0b32467c6314eaa802f58d /NvCloth/docs/documentation/CollisionDetection | |
| parent | Added missing GenerateProjectsIOS.sh file (diff) | |
| download | nvcloth-7bceea80d4a04649f495f4f4331d7e3bdcdb05ca.tar.xz nvcloth-7bceea80d4a04649f495f4f4331d7e3bdcdb05ca.zip | |
1.1.5 Release (24934621)
Diffstat (limited to 'NvCloth/docs/documentation/CollisionDetection')
4 files changed, 1427 insertions, 0 deletions
diff --git a/NvCloth/docs/documentation/CollisionDetection/Index.html b/NvCloth/docs/documentation/CollisionDetection/Index.html new file mode 100644 index 0000000..61ba480 --- /dev/null +++ b/NvCloth/docs/documentation/CollisionDetection/Index.html @@ -0,0 +1,261 @@ +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" + "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> + + +<html xmlns="http://www.w3.org/1999/xhtml"> + <head> + <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> + <meta http-equiv="x-ua-compatible" content="IE=Edge"/> + + <title>Internal collision detection documentation — NvCloth 1.1.3 documentation</title> + + <link rel="stylesheet" href="../_static/default.css" type="text/css" /> + <link rel="stylesheet" href="../_static/pygments.css" type="text/css" /> + <link rel="stylesheet" href="../_static/breathe.css" type="text/css" /> + <link rel="stylesheet" href="../_static/application.css" type="text/css" /> + <link rel="stylesheet" href="../_static/styleguide.css" type="text/css" /> + + <script type="text/javascript"> + var DOCUMENTATION_OPTIONS = { + URL_ROOT: '../', + VERSION: '1.1.3', + COLLAPSE_INDEX: false, + FILE_SUFFIX: '.html', + HAS_SOURCE: true + }; + </script> + <script type="text/javascript" src="../_static/jquery.js"></script> + <script type="text/javascript" src="../_static/underscore.js"></script> + <script type="text/javascript" src="../_static/doctools.js"></script> + <script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> + <script type="text/javascript" src="../_static/bootstrap.js"></script> + <script type="text/javascript" src="../_static/jquery.cookie.js"></script> + <script type="text/javascript" src="../_static/jquery.storageapi.js"></script> + <link rel="top" title="NvCloth 1.1.3 documentation" href="../index.html" /> + </head> + <body> +<nav class="navbar navbar-inverse navbar-default"> + <div class="row"> + <div 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+<li class="toctree-l1"><a class="reference internal" href="../ReleaseNotes/index.html">Release Notes</a><ul> +<li class="toctree-l2"><a class="reference internal" href="../ReleaseNotes/index.html#id1">1.1.5</a></li> +<li class="toctree-l2"><a class="reference internal" href="../ReleaseNotes/index.html#id2">1.1.4</a></li> +<li class="toctree-l2"><a class="reference internal" href="../ReleaseNotes/index.html#id3">1.1.3</a></li> +<li class="toctree-l2"><a class="reference internal" href="../ReleaseNotes/index.html#id4">1.1.2</a></li> +<li class="toctree-l2"><a class="reference internal" href="../ReleaseNotes/index.html#id5">1.1.1</a></li> +<li class="toctree-l2"><a class="reference internal" href="../ReleaseNotes/index.html#id6">1.1.0</a></li> +<li class="toctree-l2"><a class="reference internal" href="../ReleaseNotes/index.html#id7">1.0.0</a></li> +</ul> +</li> +<li class="toctree-l1"><a class="reference internal" href="../Compiling/index.html">Compiling</a><ul> +<li class="toctree-l2"><a class="reference internal" href="../Compiling/index.html#windows">Windows</a></li> +<li class="toctree-l2"><a class="reference internal" href="../Compiling/index.html#linux">Linux</a></li> +<li class="toctree-l2"><a class="reference internal" href="../Compiling/index.html#mac">Mac</a></li> +<li class="toctree-l2"><a class="reference internal" href="../Compiling/index.html#android">Android</a></li> +</ul> +</li> +<li class="toctree-l1"><a class="reference internal" href="../Modules/Index.html">Modules</a><ul> +<li class="toctree-l2"><a class="reference internal" href="../Modules/Index.html#nvcloth">NvCloth</a></li> +<li class="toctree-l2"><a class="reference internal" href="../Modules/Index.html#nvcloth-extensions">NvCloth extensions</a></li> +</ul> +</li> +<li class="toctree-l1"><a class="reference internal" href="../UserGuide/Index.html">User Guide</a><ul> +<li class="toctree-l2"><a class="reference internal" href="../UserGuide/Index.html#setup">Setup</a><ul> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#initializing-the-library">Initializing the Library</a></li> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#factory">Factory</a></li> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#fabric-cloth">Fabric & Cloth</a><ul> +<li class="toctree-l4"><a class="reference internal" href="../UserGuide/Index.html#fabric">Fabric</a></li> +<li class="toctree-l4"><a class="reference internal" href="../UserGuide/Index.html#cloth">Cloth</a></li> +</ul> +</li> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#solver">Solver</a></li> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#retrieving-simulation-data">Retrieving simulation data</a></li> +</ul> +</li> +<li class="toctree-l2"><a class="reference internal" href="../UserGuide/Index.html#usage">Usage</a><ul> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#common-cloth-properties">Common cloth properties</a></li> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#tethers">Tethers</a></li> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#collision-detection">Collision detection</a></li> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#local-space-simulation">Local space simulation</a></li> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#drag-lift-and-wind">Drag lift and wind</a></li> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#distance-motion-constraints">Distance/Motion constraints</a></li> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#attaching-cloth-to-animated-characters">Attaching cloth to animated characters</a></li> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#unit-scaling">Unit scaling</a></li> +</ul> +</li> +<li class="toctree-l2"><a class="reference internal" href="../UserGuide/Index.html#troubleshooting">Troubleshooting</a><ul> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#parts-of-cloth-disappearing-for-single-frame">Parts of cloth disappearing (for single frame)</a></li> +</ul> +</li> +</ul> +</li> +<li class="toctree-l1"><a class="reference internal" href="../CopyRight/Index.html">NVIDIA Copyright Notice</a></li> +</ul> + + </div> +<div id="searchbox" style="display: none"> + <h4>Quick search</h4> + <form class="search form-inline" action="../search.html" method="get"> + <div class="form-group"> + <input type="text" name="q" class="form-control" /> + <input type="submit" value="Search" class="btn btn-primary" /> + </div> + <input type="hidden" name="check_keywords" value="yes" /> + <input type="hidden" name="area" 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collision detection</a></li> +<li class="toctree-l1"><a class="reference internal" href="SelfCollision.html">Self Collision</a></li> +<li class="toctree-l1"><a class="reference internal" href="InterCollision.html">Inter Collision</a></li> +</ul> +</div> +</div> +<div class="section" id="todo"> +<h2>Todo<a class="headerlink" href="#todo" title="Permalink to this headline">¶</a></h2> +<ul class="simple"> +<li>Plane/Convex collision detection</li> +<li>Triangle collision detection</li> +<li>Inter collision detection</li> +</ul> +</div> +</div> + + + </div> + <div class="clearer"></div> + </div> + <div class="col-md-1"></div> +</div> +<div class="masthead"> + <div class="row"> + <ul class="breadcrumb"> + <li><a href="../index.html">NvCloth 1.1.3 documentation</a></li> + </ul> + </div> +</div> + +<footer> + <div class="footer-boilerplate"> + <div class="row"> + <div class="boilerplate"> + Copyright © 2014, NVIDIA Corporation | <a href="http://www.nvidia.com/object/about-nvidia.html" 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\ No newline at end of file diff --git a/NvCloth/docs/documentation/CollisionDetection/InterCollision.html b/NvCloth/docs/documentation/CollisionDetection/InterCollision.html new file mode 100644 index 0000000..4db71f8 --- /dev/null +++ b/NvCloth/docs/documentation/CollisionDetection/InterCollision.html @@ -0,0 +1,288 @@ +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" + "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> + + +<html xmlns="http://www.w3.org/1999/xhtml"> + <head> + <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> + <meta http-equiv="x-ua-compatible" content="IE=Edge"/> + + <title>Inter Collision — NvCloth 1.1.3 documentation</title> + + <link rel="stylesheet" href="../_static/default.css" type="text/css" /> + <link rel="stylesheet" href="../_static/pygments.css" type="text/css" /> + <link rel="stylesheet" href="../_static/breathe.css" type="text/css" /> + <link rel="stylesheet" href="../_static/application.css" type="text/css" 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Of Contents</h4> + <ul> +<li class="toctree-l1"><a class="reference internal" href="../ReleaseNotes/index.html">Release Notes</a><ul> +<li class="toctree-l2"><a class="reference internal" href="../ReleaseNotes/index.html#id1">1.1.5</a></li> +<li class="toctree-l2"><a class="reference internal" href="../ReleaseNotes/index.html#id2">1.1.4</a></li> +<li class="toctree-l2"><a class="reference internal" href="../ReleaseNotes/index.html#id3">1.1.3</a></li> +<li class="toctree-l2"><a class="reference internal" href="../ReleaseNotes/index.html#id4">1.1.2</a></li> +<li class="toctree-l2"><a class="reference internal" href="../ReleaseNotes/index.html#id5">1.1.1</a></li> +<li class="toctree-l2"><a class="reference internal" href="../ReleaseNotes/index.html#id6">1.1.0</a></li> +<li class="toctree-l2"><a class="reference internal" href="../ReleaseNotes/index.html#id7">1.0.0</a></li> +</ul> +</li> +<li class="toctree-l1"><a class="reference internal" href="../Compiling/index.html">Compiling</a><ul> +<li class="toctree-l2"><a class="reference internal" href="../Compiling/index.html#windows">Windows</a></li> +<li class="toctree-l2"><a class="reference internal" href="../Compiling/index.html#linux">Linux</a></li> +<li class="toctree-l2"><a class="reference internal" href="../Compiling/index.html#mac">Mac</a></li> +<li class="toctree-l2"><a class="reference internal" href="../Compiling/index.html#android">Android</a></li> +</ul> +</li> +<li class="toctree-l1"><a class="reference internal" href="../Modules/Index.html">Modules</a><ul> +<li class="toctree-l2"><a class="reference internal" href="../Modules/Index.html#nvcloth">NvCloth</a></li> +<li class="toctree-l2"><a class="reference internal" href="../Modules/Index.html#nvcloth-extensions">NvCloth extensions</a></li> +</ul> +</li> +<li class="toctree-l1"><a class="reference internal" href="../UserGuide/Index.html">User Guide</a><ul> +<li class="toctree-l2"><a class="reference internal" href="../UserGuide/Index.html#setup">Setup</a><ul> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#initializing-the-library">Initializing the Library</a></li> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#factory">Factory</a></li> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#fabric-cloth">Fabric & Cloth</a><ul> +<li class="toctree-l4"><a class="reference internal" href="../UserGuide/Index.html#fabric">Fabric</a></li> +<li class="toctree-l4"><a class="reference internal" href="../UserGuide/Index.html#cloth">Cloth</a></li> +</ul> +</li> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#solver">Solver</a></li> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#retrieving-simulation-data">Retrieving simulation data</a></li> +</ul> +</li> +<li class="toctree-l2"><a class="reference internal" href="../UserGuide/Index.html#usage">Usage</a><ul> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#common-cloth-properties">Common cloth properties</a></li> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#tethers">Tethers</a></li> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#collision-detection">Collision detection</a></li> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#local-space-simulation">Local space simulation</a></li> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#drag-lift-and-wind">Drag lift and wind</a></li> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#distance-motion-constraints">Distance/Motion constraints</a></li> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#attaching-cloth-to-animated-characters">Attaching cloth to animated characters</a></li> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#unit-scaling">Unit scaling</a></li> +</ul> +</li> +<li class="toctree-l2"><a class="reference internal" href="../UserGuide/Index.html#troubleshooting">Troubleshooting</a><ul> +<li class="toctree-l3"><a class="reference internal" href="../UserGuide/Index.html#parts-of-cloth-disappearing-for-single-frame">Parts of cloth disappearing (for single frame)</a></li> +</ul> +</li> +</ul> +</li> +<li class="toctree-l1"><a class="reference internal" href="../CopyRight/Index.html">NVIDIA Copyright Notice</a></li> +</ul> + + </div> +<div id="searchbox" style="display: none"> + <h4>Quick search</h4> + <form class="search form-inline" action="../search.html" method="get"> + <div class="form-group"> + <input type="text" name="q" class="form-control" /> + <input type="submit" value="Search" class="btn btn-primary" /> + </div> + <input type="hidden" name="check_keywords" value="yes" /> + <input type="hidden" name="area" value="default" /> + </form> + <p class="searchtip" style="font-size: 90%"> + Enter search terms or a module, class or function name. + </p> +</div> +<script type="text/javascript">$('#searchbox').show(0);</script> +</div> + </div> + <div class="document col-md-8"> + <div class="body"> + + <div class="section" id="inter-collision"> +<h1>Inter Collision<a class="headerlink" href="#inter-collision" title="Permalink to this headline">¶</a></h1> +<div class="section" id="overview"> +<h2>Overview<a class="headerlink" href="#overview" title="Permalink to this headline">¶</a></h2> +<p>Inter collision tries to keep multiple cloth objects from intersecting, but does not keep any cloth from self intersecting. +It does this by keeping the particles from every cloth a minimum distance away from the particles of every other cloth. +This distance can be set using Cloth::setInterCollisionDistance().</p> +</div> +<div class="section" id="broad-phase-collision-detection"> +<h2>Broad phase collision detection<a class="headerlink" href="#broad-phase-collision-detection" title="Permalink to this headline">¶</a></h2> +</div> +<div class="section" id="acceleration-structure"> +<h2>Acceleration structure<a class="headerlink" href="#acceleration-structure" title="Permalink to this headline">¶</a></h2> +<p>The acceleration structures used for Inter Collision are very similar to those used in Self Collision. +The only addition is the broad phase which is used to cull cloth pairs based their bounding boxes, and cull particles from the narrow phase based on the overlapping volumes.</p> +<div class="section" id="id1"> +<h3>Broad phase collision detection<a class="headerlink" href="#id1" title="Permalink to this headline">¶</a></h3> +<p><tt class="code docutils literal"><span class="pre">calculatePotentialColliders()</span></tt> takes care of the broad phase. +It first calculates the individual bounding boxes of each cloth in world space. +The sweep axis is determined using the bounding box enclosing all cloths. +The worldspace bounding boxes are sorted by the lower boundary on the sweep axis. +Sorting this way gives the following properties:</p> +<div class="highlight-python"><pre>true = clothBounds[sortedIndices[i]].mLower <= clothBounds[sortedIndices[i+1]].mLower + +if(clothBounds[sortedIndices[a]].mUpper < clothBounds[sortedIndices[b]].mLower) +{ + true = clothBounds[sortedIndices[a]].mUpper < clothBounds[sortedIndices[b+1]].mLower + // b and all sorted cloths after b do not intersect a +}</pre> +</div> +<p>Now the clothes are tested against the bounds of every other cloth.</p> +<p>For cloth A, all overlapping bounds of the other cloths are stored in temporary memory (in <tt class="code docutils literal"><span class="pre">overlapBounds</span></tt>) in the local space of cloth A. +All the particles of cloth A that overlap with any of those bounds are marked for narrow phase collision (and transformed to world space). +Marking a particle for narrow phase collision is done by storing it in the <tt class="code docutils literal"><span class="pre">mClothIndices</span></tt> and <tt class="code docutils literal"><span class="pre">mParticleIndices</span></tt> arrays. +Now the narrow phase only has to deal with the particles in those lists.</p> +</div> +<div class="section" id="differences-with-self-collision"> +<h3>Differences with self collision<a class="headerlink" href="#differences-with-self-collision" title="Permalink to this headline">¶</a></h3> +<p>Some of the obvious differences:</p> +<blockquote> +<div><ul class="simple"> +<li>Instead of iterating over all particles only particles from the <tt class="code docutils literal"><span class="pre">mClothIndices</span></tt> and <tt class="code docutils literal"><span class="pre">mParticleIndices</span></tt> arrays are used.</li> +<li>The particle cannot be directly accessed, the <tt class="code docutils literal"><span class="pre">getParticle()</span></tt> function is used instead.</li> +<li>Intercollision has no equivalent to self collision indices.</li> +<li>The particles need to be transformed back into local space after collision response.</li> +</ul> +</div></blockquote> +</div> +</div> +</div> + + + </div> + <div class="clearer"></div> + </div> + <div class="col-md-1"></div> +</div> +<div class="masthead"> + <div class="row"> + <ul class="breadcrumb"> + <li><a href="../index.html">NvCloth 1.1.3 documentation</a></li> + </ul> + </div> +</div> + +<footer> + <div class="footer-boilerplate"> + <div class="row"> + <div class="boilerplate"> + Copyright © 2014, NVIDIA Corporation | <a href="http://www.nvidia.com/object/about-nvidia.html" onclick="s_objectID="http://www.nvidia.com/object/about-nvidia.html_1";return this.s_oc?this.s_oc(e):true">About NVIDIA </a> | <a href="http://www.nvidia.com/object/legal_info.html" onclick="s_objectID="http://www.nvidia.com/object/legal_info.html_1";return 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<input type="hidden" name="area" value="default" /> + </form> + <p class="searchtip" style="font-size: 90%"> + Enter search terms or a module, class or function name. + </p> +</div> +<script type="text/javascript">$('#searchbox').show(0);</script> +</div> + </div> + <div class="document col-md-8"> + <div class="body"> + + <div class="section" id="self-collision"> +<h1>Self Collision<a class="headerlink" href="#self-collision" title="Permalink to this headline">¶</a></h1> +<div class="section" id="overview"> +<h2>Overview<a class="headerlink" href="#overview" title="Permalink to this headline">¶</a></h2> +<p>Self collision tries to keep the cloth from self intersecting. +It does this by keeping the particles a minimum distance away from each other. +This distance can be set using Cloth::setSelfCollisionDistance().</p> +</div> +<div class="section" id="acceleration-structure"> +<h2>Acceleration structure<a class="headerlink" href="#acceleration-structure" title="Permalink to this headline">¶</a></h2> +<p>Testing every particle against all other particle is too slow. +Instead an acceleration structure is used to only test nearby particles.</p> +<p>The acceleration structure is a combination of a regular grid and a sweep. +The longest axes of the AABB around all particles is used for the sweep. +The other two axes are divided in a regular grid of 253x253. +For each particle we test against the current and neighboring cells of the grid in which we sweep along the remaining axis. +The whole process will be described in more detail below.</p> +<div class="section" id="grid-setup"> +<h3>Grid setup<a class="headerlink" href="#grid-setup" title="Permalink to this headline">¶</a></h3> +<p>The AABB is obtained from ClothData (calculated by SwCollision<T4f>::computeBounds()). +The edge lengths are calculated by subtracting the lower bounds from the upper bounds. +The sweep axis index (0 to 2 for x to z so we can use the array operators) is picked by the largest edge length. +The two hash axes are the remaining axes in arbitrary order.</p> +<img src="../_images/SelfCollisionGrid.svg" /><p>The cell size for the hash axes is picked so that it is not smaller than the collision distance set by the user, and large enough that 253 cells cover the whole edge length. +The sweep axis is divided in 65533 cells. +The first and last cells of each axis are reserved as a sentinel. +The amount of sweep cells that need to be checked to cover the collision distance is given by:</p> +<div class="highlight-python"><div class="highlight"><pre><span class="n">numSweepCells</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">gridScale</span> <span class="o">*</span> <span class="n">collisionDistance</span> +</pre></div> +</div> +<p>We calculate the grid scale and grid bias to quickly convert particle coordinates to cell coordinates using \(coordinate \cdot gridScale + gridBias\). +The gridScale is \((cellCount + 1)^{-1}\) and the gridBias is \(-lowerBound \cdot gridScale + 1 \), where \(lowerBound\) is taken from the AABB.</p> +</div> +<div class="section" id="particle-sorting"> +<h3>Particle sorting<a class="headerlink" href="#particle-sorting" title="Permalink to this headline">¶</a></h3> +<p>Particles are sorted based on a key which is constructed from the cell coordinates. +The 16 least significant bits are used to store the sweep axis cell coordinate. +The two other bytes are used to store the hash axis cell coordinates (this is the reason for the specific cell count per axis in the grid).</p> +<p>A radix sort with a radix (bin) size of 256 is used, requiring 4 bins to sort the 32-bit keys.</p> +</div> +<div class="section" id="key-range-sweep"> +<h3>Key range sweep<a class="headerlink" href="#key-range-sweep" title="Permalink to this headline">¶</a></h3> +<p>Individual particles can be processed now that the acceleration structure is setup. +Only 5 cells are processed for each particle, as we don’t want to detect duplicate pairs.</p> +<img src="../_images/SelfCollisionHashAxisKernel.svg" /><p>Particles can be quickly identified in the acceleration structure by looping though the sorted keys. +Cells cannot be directly accessed as they are variable in size (depending on how many particles a cell contains) and stored in a continuous buffer.</p> +<p>The range of keys that need to be tested against the current particles can be calculated using key offsets. +Key offsets are relative key coordinates for the cells marked in yellow in the above diagram.</p> +<p>The first and last keys in the starting cell is calculated using:</p> +<div class="highlight-python"><div class="highlight"><pre><span class="n">firstKey</span> <span class="o">=</span> <span class="n">currentParticleKey</span> <span class="o">-</span> <span class="nb">min</span><span class="p">(</span><span class="n">numSweepCells</span><span class="p">,</span> <span class="n">currentParticleKey</span> <span class="o">&</span> <span class="n">bucketMask</span><span class="p">)</span> +<span class="n">lastKey</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">currentParticleKey</span> <span class="o">+</span> <span class="n">numSweepCells</span><span class="p">,</span> <span class="n">currentParticleKey</span> <span class="o">|</span> <span class="n">bucketMask</span><span class="p">)</span> +</pre></div> +</div> +<p>The key offsets can be added to these values to obtain the range for the other cells. +Note that numSweepCells is not subtracted for firstKey to ensure no duplicate collisions are detected. +Note that bit operations can be used as the different axes are contained in their own bytes within the key.</p> +<p>Collision detection/response can begin once the first/last keys are known.</p> +<p>Note that the first/last keys from the previous processed particle are reused for optimizations that can be done because the keys are sorted.</p> +</div> +<div class="section" id="collision-detection-and-response"> +<h3>Collision detection and response<a class="headerlink" href="#collision-detection-and-response" title="Permalink to this headline">¶</a></h3> +<p>Near phase collision detection can start once two possibly colliding keys are identified. +The pointers to the keys are converted to the particles and the rest particles.</p> +<p>A basic distance check between the particles and the user specified collision distance is done, culling non-colliding particles.</p> +<p>When available the distance is also checked against the rest distance to cull particles that are supposed to be close in the mesh. +This makes it possible to set the collision distance larger than the shortest constraint in the fabric without providing manual collision indices.</p> +<p>Next a position delta is calculated if the particle collision is not culled:</p> +<div class="highlight-python"><pre>diff = p1-p0 +ratio = collisionDistance / |diff| +scale = stiffness / (epsilon + w0 + w1) +delta = scale * (diff - diff * ratio +p0 += delta * w0 +p1 -= delta * w1</pre> +</div> +<p>TODO: format nicer</p> +</div> +</div> +</div> + + + </div> + <div class="clearer"></div> + </div> + <div class="col-md-1"></div> +</div> +<div class="masthead"> + <div class="row"> + <ul class="breadcrumb"> + <li><a href="../index.html">NvCloth 1.1.3 documentation</a></li> + </ul> + </div> +</div> + +<footer> + <div class="footer-boilerplate"> + <div class="row"> + <div class="boilerplate"> + Copyright © 2014, NVIDIA Corporation | <a href="http://www.nvidia.com/object/about-nvidia.html" onclick="s_objectID="http://www.nvidia.com/object/about-nvidia.html_1";return this.s_oc?this.s_oc(e):true">About NVIDIA </a> | <a href="http://www.nvidia.com/object/legal_info.html" onclick="s_objectID="http://www.nvidia.com/object/legal_info.html_1";return this.s_oc?this.s_oc(e):true">Legal Information </a> | <a href="http://www.nvidia.com/object/privacy_policy.html" 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<input type="hidden" name="area" value="default" /> + </form> + <p class="searchtip" style="font-size: 90%"> + Enter search terms or a module, class or function name. + </p> +</div> +<script type="text/javascript">$('#searchbox').show(0);</script> +</div> + </div> + <div class="document col-md-8"> + <div class="body"> + + <div class="section" id="sphere-capsule-collision-detection"> +<h1>Sphere Capsule collision detection<a class="headerlink" href="#sphere-capsule-collision-detection" title="Permalink to this headline">¶</a></h1> +<p>Sphere and capsule collision is divided in 3 steps:</p> +<blockquote> +<div><ul class="simple"> +<li>Sphere/capsule generation</li> +<li>Acceleration structure generation</li> +<li>Collision detection</li> +</ul> +</div></blockquote> +<div class="section" id="sphere-capsule-generation"> +<h2>Sphere/ Capsule generation<a class="headerlink" href="#sphere-capsule-generation" title="Permalink to this headline">¶</a></h2> +<p>The sphere data set by the user doesn’t need to be converted before it can be used for collision detection. +However, the data will be interpolated between frames when continuous collision detection is enabled.</p> +<p>The capsules need more setup work as the representation set by the user is not directly usable for collision detection. +Some values are precomputed (in generateCones()) so they won’t have to be computed unnecessarily in the collision detection routine. +All the computations are explained in the cone collision detection documentation. Refer to the code to see which values are precomputed.</p> +</div> +<div class="section" id="sphere-acceleration-structure"> +<h2>Sphere acceleration structure<a class="headerlink" href="#sphere-acceleration-structure" title="Permalink to this headline">¶</a></h2> +<p>We use a small sweep and prune (SAP) acceleration structure to avoid testing each particle against every sphere. +We begin by calculating the axis aligned bounding volume containing all particles and collision spheres (SwCollision<T4f>::buildAcceleration()). +That space is then divided into an 8x8x8 grid. +The span of each sphere along the 3 primary axes are stored in bitmasks where each bit is for a different sphere. +So we have 8 bitmasks for each axis. +We store the info twice to ensure that particles don’t skip over marked grid cells (when using continuous collision detection). +The first set contains spans that are extended to the maximum, while the spans in the second set are extended to the minimum (see image below). +So if the sphere was contained in cells 2, 3, and 4 along the x axis, the cells 5, 6, and 7 would also be marked for the first set.</p> +<img src="../_images/SphereAcceleration.svg" /><p>Thick bordered bitmask cells indicate that the sphere is inside. +The cells that contain the spheres can be obtained by combining both sets using binary AND. +This is done when continuous collision detection is disabled (SwCollision<T4f>::mergeAcceleration()).</p> +<p>We use both bit mask sets when continuous collision detection is enabled to test if the span from the particle movement overlaps the sphere span stored in the bitmasks. +For each axis, we test the highest of the two position values against the bitmask from the first set and the lowest against the second set. +We don’t have to test the cells in between because the bitmasks are extended (e.g. if the particle moves from cell 5 to cell 1 along the x axis both bitmasks will test positive, even though the particle skipped past the sphere. However, if the particle moved from cell 5 to 6 only one of the cells tests positive, culling the collision).</p> +</div> +<div class="section" id="collideparticles"> +<h2>collideParticles()<a class="headerlink" href="#collideparticles" title="Permalink to this headline">¶</a></h2> +<p>TODO mass scaling +CollideParticles iterates through the cloth particles to handle non-continuous collisions with spheres and capsules. +The capsule cones are handled first so that the corresponding spheres can be ignored, ensuring that no double collisions are registered (in the case where a particle intersects both the cone and sphere of the capsule). +The sphere and cone collision detection is discussed in the sections below. +The results of the collision detection routines are stored in the ImpulseAccumulator which keeps track of the depenetration delta sum, friction velocity sum (multiplied by the frame time) and collision count (to calculate the averages).</p> +<p>The average depenetration delta is directly added to the current particle position.</p> +<p>The friction impulse (calculated by calculateFrictionImpulse()) is directly applied to the previous particle position so that only the velocity is changed.</p> +</div> +<div class="section" id="capsule-collision-detection"> +<h2>Capsule collision detection<a class="headerlink" href="#capsule-collision-detection" title="Permalink to this headline">¶</a></h2> +<div class="section" id="cone-collision-detection"> +<h3>Cone collision detection<a class="headerlink" href="#cone-collision-detection" title="Permalink to this headline">¶</a></h3> +<p>Capsule collision detection is split in two parts: Sphere and cone collision detection. +Here we describe the cone collision detection. Spheres are described in the next subsection.</p> +<p>The cone needs to be constructed in a way that it correctly fills the space between two spheres with centers \(c_i\) and radii \(r_i\), where \(i \in \left\{ 1,2 \right\} \). +The capsule is a continuous surface, so the cone needs to intersect both spheres at a tangent. +The problem can be solved in 2d by projecting it onto a plane that contains both sphere centers. +We use an approach similar as described <a class="reference external" href="http://jwilson.coe.uga.edu/emt669/Student.Folders/Kertscher.Jeff/Essay.3/Tangents.html">here</a>.</p> +<img src="../_images/CapsuleCircleIntersectionDiagram.svg" /><p>Two circles are constructed: circle m (green) with center on midpoint \(m = \frac{1}{2}\left(c_1+c_2\right)\) and radius \(r_m=\frac{1}{2}|c_2-c_1|\) +and circle 3 (black) with center \(c_1\) and radius \(r_3=r_1-r_2\). +The intersection between circle 3 and m gives us the tangent point \(t_1\) (see <a class="reference external" href="http://paulbourke.net/geometry/circlesphere/">here</a> for more details on circle intersections). +The tangent line intersects points \(c_2\) and \(t_1\).</p> +<p>\(t_2\) is obtained by offsetting \(t_1\) by \(\frac{t_1-c_1}{r_3}r_2\), and similar with \(c_2\) to obtain \(t_3\).</p> +<p>The cone length can be derived from the diagram: +$$ +V_l = \left|t_3-t_2\right| = \left|t_1-c_2\right| = \sqrt{\left|c_1-c_2\right|^2-\left(r_1-r_2\right)^2} +$$ +The axis length of the cone is +$$ +A_l = \left|c_2 - c_1\right| +$$ +The axis of the cone is +$$ +A = \frac{\left(c_2 - c_1\right)}{A_l} +$$</p> +<img src="../_images/CapsuleRadiusDiagram.svg" /><p>The radius of the cone above point \(m\) (in blue) is +$$ +V_r = \left(\frac{1}{2}\left(r_2 - r_1\right) + r_1\right) \frac{A_l}{V_l} = \frac{1}{2}\left(r_2 + r_1\right) \frac{A_l}{V_l} +$$ +as the triangles made from the red-blue line pairs are similar. +The slope of the cone (the rate of change of the radius) is +$$ +V_s = \frac{r_2 - r_1}{V_l} +$$</p> +<p>Detecting if point p lies inside the cone is done in two steps: detecting if it lies within the infinite cone and checking if it is contained within the span of \(t_2\) and \(t_3\).</p> +<img src="../_images/CapsulePointCollisionDetection.svg" /><p>Point P is first projected onto the cone axis \(A\) giving the distance (dotted red in diagram) to \(m\) along \(A\): +$$ +d = \left(p-m\right)\cdot A +$$ +The radius below p (thick blue) is +$$ +r_p = d V_s + V_r +$$ +The distance of \(p\) to \(A\) is given by: +$$ +d_{\bot} = \sqrt{\left|p-m\right|^2 - d^2} +$$ +The point is contained within the infinite cone when \(r_p> d_{\bot}\).</p> +<img src="../_images/CapsulePointCollisionDetection2.svg" /><p>The distance \(d_o\) (blue dashed line) between \(p\) and \(m’\) is given by: +$$ +d_o = d + V_s d_{\bot} +$$ +\(d_o\) is less than \(\frac{1}{2}A_l\) if point \(p\) lies inside the finite capped cone: +$$ +\frac{1}{2}A_l < d_o$$</p> +<p>Todo: collision response</p> +<p>The (un-normalized) collision normal \(n\) is given by: +$$ +n = (p-m) - d_o A +$$</p> +<img src="../_images/CapsulePointCollisionResponse.svg" /><p>The collision response scale is calculated using: +$$ +s = \frac{r_p}{d_{\bot}}cos (\alpha)^2 - cos (\alpha)^2 +$$ +where \(cos(\alpha)^2\) is calculated using: +$$ +cos(\alpha)^2 = 1 - \left(\frac{r_2-r_1}{A_l}\right)^2 = 1 - sin(\alpha)^2 = \left(\frac{\left|t_1-c_2\right|}{\left|c_1-c_2\right|}\right)^2 +$$</p> +<p>\(\frac{r_p}{d_{\bot}} \) is 1 when the particle is exactly on the surface of the cone, bigger than 1 when the particle has penetrated the capsule. +\(cos (\alpha)^2\) ???</p> +<p>$$ +s = \cos\left(\alpha\right)^2 \left(\frac{r_p}{d_{\bot}}-1\right) +$$</p> +<p>Variables in <tt class="code docutils literal"><span class="pre">generateCones()</span></tt>:</p> +<div class="highlight-python"><pre>center = \(m\) +axis.xyz = \(\frac{1}{2}(c_2-c_1) = \frac{1}{2}A_l A\) +cIt->axis = \(A\) +axis.w = \(-\frac{1}{2}r_3\) +sqrAxisHalfLength = \((\frac{1}{2}A_l)^2\) +sqrConeHalfLength = \((\frac{1}{2}V_l)^2\) +slope = \(\frac{\frac{1}{2}(r_2-r_1)}{\frac{1}{2}V_l} = \frac{(r_2-r_1)}{V_l} = V_s\) +cIt->radius = \((\frac{1}{2}(r_2-r_1)+r_1)\frac{A_l}{V_l} = V_r \)</pre> +</div> +<p>Variables in <tt class="code docutils literal"><span class="pre">cloth::SwCollision<T4f>::collideCones()</span></tt>:</p> +<div class="highlight-python"><div class="highlight"><pre><span class="n">todo</span> +</pre></div> +</div> +</div> +<div class="section" id="sphere-collision-detection"> +<h3>Sphere collision detection<a class="headerlink" href="#sphere-collision-detection" title="Permalink to this headline">¶</a></h3> +<p>We calculate the following values for the regular particle against sphere collision:</p> +<div class="highlight-python"><div class="highlight"><pre><span class="n">delta</span> <span class="o">=</span> <span class="n">particle</span> <span class="o">-</span> <span class="n">sphereCenter</span> +<span class="n">sqDeltaLegnth</span> <span class="o">=</span> <span class="n">epsilon</span> <span class="o">+</span> <span class="n">dot</span><span class="p">(</span><span class="n">delta</span><span class="p">,</span> <span class="n">delta</span><span class="p">)</span> +<span class="n">negativeScale</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">sphereRadius</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">sqDeltaLegnth</span><span class="p">)</span> +</pre></div> +</div> +<p>NegativeScale will be negative when the particle does not collide with the sphere, at which point we can skip the rest of the work. +We apply a depenetration impulse to the particle when it does collide:</p> +<div class="highlight-python"><div class="highlight"><pre><span class="n">particleDelta</span> <span class="o">-=</span> <span class="n">delta</span> <span class="o">*</span> <span class="n">negativeScale</span> +</pre></div> +</div> +</div> +<div class="section" id="sphere-ccd"> +<h3>Sphere CCD<a class="headerlink" href="#sphere-ccd" title="Permalink to this headline">¶</a></h3> +<p>Continuous collision detection between cloth and spheres calculate the time of impact between cloth particles and moving spheres. +Both particles and spheres move linearly during a solver iteration, so this problem is the same as a line sphere intersection (making the particle movement relative to the sphere).</p> +<img src="../_images/SphereCCD.svg" /><p>We start with the following equations: +\begin{align} +x =& p_0 + (p_1-p_0)t\\ +y =& c_0 + (c_1-c_0)t\\ +\left|x-y\right|^2 =& (r_0+(r_1-r_0)t)^2 +\end{align}</p> +<p>where \(p_0\) and \(p_1\) are the beginning and end positions of the particle, \(c_0\) and \(c_1\) are the beginning and end positions of the sphere, and \(r_0\) and \(r_1\) are the beginning and end radii of the sphere.</p> +<p>We simplify the equations by working relative to the sphere: +\begin{align} +q_0 =& p_0 - c_0\\ +q_1 =& p_1 - c_1\\ +x-y =& q_0 + (q_1-q_0)t\\ +\end{align}</p> +<p>Expanding this gives us: +\begin{align} +q_0^2 + 2q_0(q_1-q_0)t + (q_1-q_0)^2t^2 =& r_0^2 + 2r_0(r_1-r_0)t + (r_1-r_0)^2t^2\\ +q_0^2 + (2q_0q_1-2q_0^2)t + (q_1^2-2q_1q_0+q_0^2)t^2 =& r_0^2 + (2r_0r_1-2r_0^2)t + (r_1^2-2r_1r_0+r_0^2)t^2\\ +\end{align} +We rewrite this as a quadratic polynomial: +\begin{align} +q_0^2-r_0^2 + (2q_0q_1-2q_0^2)t-(2r_0r_1-2r_0^2)t + (q_1^2-2q_1q_0+q_0^2)t^2-(r_1^2-2r_1r_0+r_0^2)t^2 =0\\ +q_0^2-r_0^2 + (2q_0q_1-2r_0r_1-2q_0^2+2r_0^2)t + (q_1^2-r_1^2-2q_1q_0+2r_1r_0+q_0^2-r_0^2)t^2 =0\\ +m_{00} + (2m_{01}-2m_{00})t + (m_{11}-2m_{01}+m_{00})t^2 =0\\ +\end{align}</p> +<p>Using the following definitions for readability: +\begin{align} +m_{00} =& q_0^2 - r_0^2\\ +m_{11} =& q_1^2 - r_1^2\\ +m_{01} =& q_0q_1 - r_0r_1\\ +\end{align}</p> +<p>We can now use the quadratic formula to solve for \(t\): +\begin{align} +a =& m_{11}-2m_{01}+m_{00}\\ +b =& 2m_{01}-2m_{00}\\ +c =& m_{00}\\ +d =& b^2-4ac = 4m_{01}^2-4m_{01}m_{00}+4m_{00}^2 -4m_{00}m_{11}+4m_{00}m_{01}-4m_{00}^2\\ +d =& 4m_{01}^2-4m_{00}m_{11}\\ +t =& \frac{-b+\sqrt{d}}{2a} = \frac{-m_{01}+m_{00} + \sqrt{\left(m^2_{01}-m_{00}m_{11}\right)}}{m_{11}-2m_{01}+m_{00}}\\ +\end{align}</p> +<p>The variables used in <tt class="code docutils literal"><span class="pre">SwCollision<T4f>::collideSpheres</span></tt> are very similar:</p> +<div class="highlight-python"><pre>dotPrevPrev = \(m_{00}\) +dotPrevCur = \(m_{01}\) +dotCurCur = \(m_{11}\) //(epsilon is omitted here) +discriminant = \(\frac{1}{4}d\) +sqrtD = \(\frac{1}{2}\sqrt{\left(d\right)}\) +halfB = -\(\frac{1}{2}b\) +minusA = \(-a\)</pre> +</div> +<p>Continuous collision detection can be skipped if one (relative) sphere is contained in the other (relative) sphere, or \(\left|q_0-q_1\right|^2 < (r_0-r_1)^2\). +We can reuse \(a\) for this check. If \(a > 0\) Continuous collision detection is executed (TODO: check if this makes sense. it looks like discrete collision detection never runs when CCD is enabled. Are all cases really handled when a<0?): +\begin{align} +a >& 0\\ +a =& m_{11} - 2m_{01} + m_{00}\\ +=& q_1^2-r_1^2 - 2q_0q_1 +2r_0r_1 + q_0^2-r_0^2\\ +=& q_0^2+q_1^2- 2q_0q_1 -r_1^2 + 2r_0r_1 -r_0^2\\ +=& (q_1-q_0)^2 - (r_1-r_0)^2 < 0\\ +=& (p_1 - c_1 - p_0 + c_0)^2 - (r_1-r_0)^2\\ +=& ((p_1 - p_0) - (c_1 - c_0))^2 - (r_1-r_0)^2 < 0\\ +\end{align} +TODO: check code if we can skip both if sphere 0 is \(\in\) sphere 1 or sphere 1 is \(\in\) sphere 0 as we only do collision detection with sphere 1 in the end.</p> +<p>Collision will happen within the current frame when \( 0 < t < 1\). +The following position correction is applied to stop the particle from penetrating the sphere: +\begin{align} +p’_1 =& c_1 + q_1 + \left(q_0-q_1\right)\left(1 - t\right)\\ +=& p_1 + \left(q_0-q_1\right)\left(1 - t\right) +\end{align} +Note that \(q_1 + \left(q_0-q_1\right)\left(1 - t\right)\) is the toi position relative to the toi sphere center. +This offset is added to the current sphere center \(c_1\), this is correct if the particle sticks to the sphere for the rest of the iteration step duration.</p> +<p>The rest of the steps are the same as the non-continuous collision detection (but using the distance from the toi calculation).</p> +<p>TODO: check the other collisionMask calculations and comments:</p> +<div class="highlight-python"><pre>// skip continuous collision if the (un-clamped) particle +// trajectory only touches the outer skin of the cone. +T4f rMin = prevRadius + halfB * minusA * (curRadius - prevRadius); +collisionMask = collisionMask & (discriminant > minusA * rMin * rMin * sSkeletonWidth);</pre> +</div> +<p>\begin{align} +r_{min} =& r_0 - \frac{1}{2}ba(r_1-r_0)\\ +\frac{1}{4}d >& -a r_{min}^2 s\\ +\frac{1}{4}d >& -a (r_0 - \frac{1}{2}ba(r_1-r_0))^2 s\\ +\frac{1}{4}d >& -asr_0^2 - as\frac{1}{4}b^2a^2(r_1-r_0)^2\\ +\frac{1}{4}d >& -asr_0^2 - as\frac{1}{4}b^2a^2(r_1-r_0)^2\\ +d >& -4asr_0^2 - 4as\frac{1}{4}b^2a^2(r_1-r_0)^2\\ +d >& 4a(-sr_0^2 - s\frac{1}{4}b^2a^2(r_1-r_0)^2)\\ +\frac{d}{4a} >& -sr_0^2 - s\frac{1}{4}b^2a^2(r_1-r_0)^2\\ +\frac{d}{4a} >& -s(r_0^2 + \frac{1}{4}b^2a^2(r_1-r_0)^2)\\ +\end{align}</p> +<p>\begin{align} +y =& at^2 + bt + c\\ +t =& -b/(2a)\\ +y_{min} =& c - \frac{b^2}{4a} = -\frac{d}{4a} = \text{vertex}_y\\ +\end{align}</p> +<p>\begin{align} +b =& 2m_{01}-2m_{00}\\ +b =& 2q_0q_1 - 2r_0r_1 - 2q_0^2 + 2r_0^2\\ +b =& 2q_0q_1 - 2q_0^2 + 2r_0^2 - 2r_0r_1\\ +b =& 2(p_0-c_0)(p_1-c_1) - 2(p_0-c_0)^2 + 2r_0^2 - 2r_0r_1\\ +b =& 2(p_0p_1-p_0c_1-c_0p_1+c_0c_1) - 2(p_0^2+c_0^2 - 2 p_0c_0) + 2r_0^2 - 2r_0r_1\\ +\end{align}</p> +<p>TODO: what is going on:</p> +<div class="highlight-python"><pre>// reduce ccd impulse if (clamped) particle trajectory stays in sphere skin, +// i.e. scale by exp2(-k) or 1/(1+k) with k = (tmin - toi) / (1 - toi) +T4f minusK = sqrtD * recip(minusA * oneMinusToi) & (oneMinusToi > gSimd4fEpsilon); +oneMinusToi = oneMinusToi * recip(gSimd4fOne - minusK);</pre> +</div> +</div> +<div class="section" id="cone-ccd"> +<h3>Cone CCD<a class="headerlink" href="#cone-ccd" title="Permalink to this headline">¶</a></h3> +<p>Cone continuous collision detection works in a similar way as the sphere CCD.</p> +<blockquote> +<div>At roughly double the cost of discrete collision, the linearized trajectory of the particle relative to the capsule is tested for intersection. The 6th order polynomial is approximated by a quadratic equation, assuming the capsule length and slope stay constant.</div></blockquote> +<p>– From <a class="reference external" href="https://wiki.nvidia.com/engwiki/index.php/PhysX/sdk/project/clothing3x/Pipeline">Engwiki, csigg</a></p> +<p>We use the cross product <a class="reference external" href="http://www.qc.edu.hk/math/Advanced%20Level/Point_to_line.htm">method</a> instead of dot products to calculate the point line distance: +$$ +\text{distance}_\bot = \left|(p-c) \times a\right| +$$ +where \(c\) is a point on the line, \(a\) is the unit direction of the line (axis) and \(p\) is the point.</p> +<p>We start with the following equations: +\begin{align} +x =& \left(p_0-c_0\right) \times A_0 + \left(\left(p_1-c_1\right)\times A_1-\left(p_0-c_0\right)\times A_0\right)t\\ +\left|x\right|^2 =& (r_0+(r_1-r_0)t)^2 +\end{align} +where \(p_0\) and \(p_1\) are the beginning and end positions of the particle, +\(c_0\) and \(c_1\) are the beginning and end positions of the cone center, +\(A_0\) and \(A_1\) are the beginning and end cone axes, +and \(r_0\) and \(r_1\) are the beginning and end radii of the cone +below the particle (like \(r_p\) from the discrete cone section above): +\begin{align} +r = ((p-c) \cdot A) V_s + V_r +\end{align} +where \(V_s\) is the cone slope and \(V_r\) is the radius above the cone mid point.</p> +<p>Note that \(x\) is very similar to the \(x-y\) from the sphere ccd section above, only adding the cross products.</p> +<p>We simplify the equations using the following definitions: +\begin{align} +q_0 =& \left(p_0 - c_0\right) \times A_0\\ +q_1 =& \left(p_1 - c_1\right) \times A_1\\ +x =& q_0 + (q_1-q_0)t\\ +\end{align}</p> +<p>We can now solve for \(t\) in the same way as described in the sphere ccd section above. +The checks determining if collision occurred, and the skipping conditions are also the same as with sphere ccd.</p> +<p>So far, all checks were against the infinite cone. Next, we check if the collision falls on the cone section between the spheres. +We interpolate the following values to the time of impact:</p> +<p>\begin{align} +p_{toi} =& p_0-c_0 - (p_0-c_0 - p_1-c_1)t\\ +A_{ltoi} =& \left|A_1A_{l1} - (A_1A_{l1} - A_0A_{l0})(1-t)\right|\\ +A_{toi} =& \left(A_1A_{l1} - (A_1A_{l1} - A_0A_{l0})(1-t)\right)A_{ltoi}^{-1}\\ +\end{align} +Where \(A_{l0}\) is the axis length at the start and \(A_{l1}\) at the end.</p> +<p>We calculate the point line distance, and the distance along the axis: +\begin{align} +d_{\bot} =& \sqrt{p_{toi}^2 - \left(p_{toi} \cdot A_{toi}\right)^2}\\ +d_| = & p_{toi} \cdot A_{toi} +\end{align}</p> +<p>Now we can calculate \(d_o\) again: +$$ +d_o = d_| + V_s d_{\bot} +$$ +which leaves us at with the same conditions used in the non-continuous capsule section.</p> +</div> +</div> +<div class="section" id="calculatefrictionimpulse"> +<h2>calculateFrictionImpulse()<a class="headerlink" href="#calculatefrictionimpulse" title="Permalink to this headline">¶</a></h2> +<p>The <tt class="code docutils literal"><span class="pre">calculateFrictionImpulse</span></tt> function is used to do the friction calculations after the collisions have been calculated. +The friction calculation is simplified by assuming the particle only touches one flat surface which approximates the average collision surface. +The normal of this surface is in the same direction as the collision delta calculated from the collision detection functions (<tt class="code docutils literal"><span class="pre">mDeltaXYZ</span></tt> in the accumulator).</p> +<p>The velocity of the surface is calculated each time a collision is detected. +The sum of these velocities also stored in the accumulator (<tt class="code docutils literal"><span class="pre">mVelXYZ</span></tt>). +The velocity of the friction surface is the average of all those velocities.</p> +<p>The friction impulse will be along the tangential part of the relative velocity between the particle and the surface. +The friction impulse magnitude is proportional to the friction coefficient and the collision delta magnitude. 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