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<div class="section" id="internal-solver-function-algorithm-documentation">
<h1>Internal solver function/algorithm documentation<a class="headerlink" href="#internal-solver-function-algorithm-documentation" title="Permalink to this headline">¶</a></h1>
<p>This document describes the internal workings of the solver.
We describe how the different parts work independent from platform specific features. Although we might occasionally comment on particular implementation details.</p>
<p>Todo: give this solver a name, so that we can differentiate when a new/different solver is added.</p>
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<div class="section" id="overview">
<h2>Overview<a class="headerlink" href="#overview" title="Permalink to this headline">¶</a></h2>
<p>Overview of the general algorithm goes here.</p>
<ul class="simple">
<li>Integrate particles (integrateParticles())</li>
<li>Simulate wind/drag/lift (applyWind())</li>
<li>Distance constraints (constrainMotion())</li>
<li>Tether constraints (constrainTether())</li>
<li>Solve edge constraints (solveFabric())</li>
<li>Solve separation constraints (constrainSeparation())</li>
<li>Collision (collideParticles())</li>
<li>Self collision (selfCollideParticles())</li>
<li>Update sleep state (updateSleepState())</li>
</ul>
</div>
<div class="section" id="particle-invmass-w-component">
<h2>Particle invMass w component<a class="headerlink" href="#particle-invmass-w-component" title="Permalink to this headline">¶</a></h2>
<p>Todo: check and rewrite:</p>
<div class="highlight-c++"><div class="highlight"><pre><span class="c1">// note on invMass (stored in current/previous positions.w):</span>
<span class="c1">// integrateParticles()</span>
<span class="c1">// - if(current.w == 0) current.w = previous.w</span>
<span class="c1">// constraintMotion()</span>
<span class="c1">// - if(constraint.radius <= 0) current.w = 0</span>
<span class="c1">// computeBounds()</span>
<span class="c1">// - if(current.w > 0) current.w = previous.w</span>
<span class="c1">// collideParticles()</span>
<span class="c1">// - if(collides) current.w *= 1/massScale</span>
<span class="c1">// after simulate()</span>
<span class="c1">// - previous.w: original invMass as set by user</span>
<span class="c1">// - current.w: zeroed by motion constraints and mass-scaled by collision</span>
</pre></div>
</div>
</div>
<div class="section" id="slack">
<h2>Slack<a class="headerlink" href="#slack" title="Permalink to this headline">¶</a></h2>
<p>The position constraint for keeping distance between points a and b is usually written as:</p>
<div class="highlight-c++"><div class="highlight"><pre><span class="n">error</span> <span class="o">=</span> <span class="o">|</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="o">|</span> <span class="o">-</span> <span class="n">restLength</span>
<span class="n">offset</span> <span class="o">=</span> <span class="n">error</span> <span class="o">*</span> <span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="p">)</span><span class="o">/|</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="o">|</span>
<span class="o">=</span> <span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="p">)</span> <span class="o">-</span> <span class="n">restLength</span><span class="o">*</span><span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="p">)</span><span class="o">/|</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="o">|</span>
</pre></div>
</div>
<p>The equation calculating <em>slack</em> pops up often in the solver code.
This does exactly the same as the above:</p>
<div class="highlight-c++"><div class="highlight"><pre><span class="n">slack</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">restLength</span> <span class="o">/</span> <span class="o">|</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="o">|</span>
<span class="n">offset</span> <span class="o">=</span> <span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="p">)</span> <span class="o">*</span> <span class="n">slack</span>
<span class="o">=</span> <span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="p">)</span> <span class="o">-</span> <span class="n">restLength</span><span class="o">*</span><span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="p">)</span><span class="o">/|</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="o">|</span>
</pre></div>
</div>
</div>
<div class="section" id="integration">
<h2>Integration<a class="headerlink" href="#integration" title="Permalink to this headline">¶</a></h2>
<p>The first step of the cloth simulation is the integration.
Explicit Euler integration is used to calculate the new position of the particles.
The velocity of the particles is not stored, instead we use the position from the previous frame to calculate the velocity.
We need to compensate for differences in delta time. Large fluctuations can cause problems, so the delta time is damped.</p>
<p>The following pseudo code describes how this is implemented:</p>
<div class="highlight-c++"><div class="highlight"><pre><span class="c1">//calculate damping constants from user setting 'damping'</span>
<span class="n">logDamping</span> <span class="o">=</span> <span class="n">log_2</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">damping</span><span class="p">)</span> <span class="c1">//from ClothImpl<T>::setDamping</span>
<span class="n">dampExponent</span> <span class="o">=</span> <span class="n">stiffnessFrequency</span> <span class="o">*</span> <span class="n">dt1</span> <span class="c1">//from IterationState::create</span>
<span class="c1">//calculate delta time ajustment</span>
<span class="n">scale</span> <span class="o">=</span> <span class="n">e</span><span class="o">^</span><span class="p">{</span><span class="n">logDamping</span> <span class="o">*</span> <span class="n">dampExponent</span><span class="p">}</span> <span class="o">*</span> <span class="p">{</span><span class="n">dt1</span><span class="o">/</span><span class="n">dt0</span><span class="p">}</span> <span class="c1">//from IterationState::create</span>
<span class="c1">//Do the integration</span>
<span class="n">delta</span> <span class="o">=</span> <span class="p">(</span><span class="n">particle_position1</span><span class="o">-</span><span class="n">particle_position0</span><span class="p">)</span> <span class="o">*</span> <span class="n">scale</span> <span class="o">+</span> <span class="n">acceleration</span>
<span class="n">particle_position1</span> <span class="o">=</span> <span class="n">particle_position1</span> <span class="o">+</span> <span class="n">delta</span>
</pre></div>
</div>
<p>The integration code also replaces the current inverse mass with the previous inverse mass if the current is zero.</p>
<p>Todo: rotating reference frame.</p>
<p>Todo: how does damping work?</p>
</div>
<div class="section" id="wind-simulation">
<h2>Wind simulation<a class="headerlink" href="#wind-simulation" title="Permalink to this headline">¶</a></h2>
<p>Drag and lift simulation is done in the applyWind() function.
The following algorithm is used:</p>
<div class="highlight-c++"><pre>for each triangle
\(p_j\) is the previous and \(c_j\) is the current position and \(m^{-1}_j\) is the inverse mass of the \(j\)th particle in of the triangle.
//Calculate current and previous center of the triangle
\(c_t = \frac{1}{3} \cdot \left( c_0 + c_1 + c_2 \right)\)
\(p_t = \frac{1}{3} \cdot \left( p_0 + p_1 + p_2 \right)\)
//Position difference including wind (same as flow speed times dt)
\(d = c_t - p_t + wind\)
if rotating reference frame
\(d = c_t + R\left(d-c_t\right)\) //where \(R\) is the inverse local space rotation for one solver iteration
//Todo check/explain this
//Unnormalized normal of the triangle
\(n = \left(c_2 - c_0\right) \times \left(c_1 - c_0\right) \)
//Double the area of the triangle
\(a = \left|n\right| \)
\(\Lrg{ \cos\left(\theta\right) = \frac{n \cdot d}{\left|n\right| \cdot \left|d \right|} }\)
\(\Lrg{ \sin\left(\theta\right) = \sqrt{\max(0, 1 - \cos\left(\theta\right)^2)}}\)
//Lift direction is orthogonal to delta, in the delta-normal plane. Its length is \(\left|n\right|\cdot\left|d\right|\)
\(\Lrg{ l_{dir} = \frac{\left( d \times n\right) \times d}{\left|d \right|} }\)
//Calculate the lift and drag impulse
//The lift is at its maximum when \(d\) is at an \(45^\circ\) angle to the triangle
//We use \(\sin\left(\theta\right)\cdot\cos\left(\theta\right) = \frac{1}{2}\sin\left(2\cdot \theta\right)\) to calculate this
\(i_{lift} = c_{lift} \cdot \cos\left(\theta\right) \cdot \sin\left(\theta\right) \cdot l_{dir} \)
\(i_{drag} = c_{drag} \cdot \left|\cos\left(\theta\right)\right| \cdot a \cdot d \)
//\(c_{lift} \) and \(c_{drag}\) are the lift and drag coefficients
//\(c_{drag}\) should be set to compensate for the double area in \(a\)
//combined impulse
\(\Lrg{ i =
\begin{cases}
\epsilon < \left|d\right|^2,& 0\\
\text{else},& i_{lift} + i_{drag}
\end{cases}
}\)
//apply impulses to the particle positions
for each particle \(j = \left\{ 0, 1, 2 \right\} \)
\(c_j = c_j - i \cdot m^{-1}_j \)</pre>
</div>
<p>Note that when we combine the impulses we check if \(\left|d\right|\) is too small as this causes instabilities due to float division inaccuracies.
This can cause different drag and lift behavior depending on the time step size (or solver frequency).
Smaller time steps result in smaller position deltas which reach the \(\epsilon\) threshold sooner.
This results in less drag/lift at small time steps (high solver frequencies) for slow moving particles.</p>
</div>
<div class="section" id="distance-constraints">
<h2>Distance constraints<a class="headerlink" href="#distance-constraints" title="Permalink to this headline">¶</a></h2>
<p>A distance constraint can limit the movement of a particle to the volume of a sphere.
The constraints are specified with an array of 4 dimensional vectors (physx::PxVec4) where the x, y, z elements define the center and w the radius of the sphere.</p>
<p>The constraints are interpolated between the start and target spheres if both are given.</p>
<p>The constrainMotion() function in the solver is responsible for enforcing these constraints.
The following pseudo code describes how this is implemented:</p>
<div class="highlight-c++"><div class="highlight"><pre><span class="n">delta</span> <span class="o">=</span> <span class="n">sphere_center</span> <span class="o">-</span> <span class="n">particle_position</span>
<span class="n">sqrLength</span> <span class="o">=</span> <span class="n">epsilon</span> <span class="o">+</span> <span class="o">|</span><span class="n">delta</span><span class="o">|^</span><span class="mi">2</span>
<span class="n">radius</span> <span class="o">=</span> <span class="n">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">sphere_radius</span> <span class="o">*</span> <span class="n">scale</span> <span class="o">+</span> <span class="n">bias</span><span class="p">)</span>
<span class="n">slack</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">radius</span> <span class="o">/</span> <span class="n">sqrt</span><span class="p">{</span><span class="n">sqrLength</span><span class="p">}</span>
<span class="k">if</span><span class="p">(</span><span class="n">slack</span><span class="o">></span><span class="mi">0</span><span class="p">)</span>
<span class="p">{</span>
<span class="k">if</span><span class="p">(</span><span class="n">radius</span> <span class="o"><=</span> <span class="mi">0</span><span class="p">)</span>
<span class="n">particle_invMass</span><span class="p">.</span><span class="n">w</span> <span class="o">=</span> <span class="mi">0</span> <span class="c1">//Set the inverse mass to 0 if we are constrained to a zero radius sphere</span>
<span class="n">slack</span> <span class="o">=</span> <span class="n">slack</span> <span class="o">*</span> <span class="n">stiffness</span>
<span class="n">particle</span><span class="p">.</span><span class="n">xyz</span> <span class="o">+=</span> <span class="n">delta</span> <span class="o">*</span> <span class="n">slack</span>
<span class="p">}</span>
</pre></div>
</div>
</div>
<div class="section" id="tether-constraints">
<h2>Tether constraints<a class="headerlink" href="#tether-constraints" title="Permalink to this headline">¶</a></h2>
<p>Tether constraints help with reducing the stretchiness of the cloth without increasing the solver iteration count.
This is done by adding additional max distance constraints connecting simulated particles with their nearest fixed particle(s).
The tether constraints are stored as an index & length pair.
The array of constraints has a multiple of particle count elements.
The constraints in the array are stored in order so that the first particle of the constraint is element%numParticles.
The index stored in the constraint defines the other (anchor) particle of the constraint.</p>
<p>The constraint for one particle is solved as follows:</p>
<div class="highlight-c++"><div class="highlight"><pre><span class="n">offset</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">each</span> <span class="n">tether</span> <span class="n">connecting</span> <span class="n">pb</span>
<span class="c1">//pa is the anchor particle</span>
<span class="n">delta</span> <span class="o">=</span> <span class="n">pa</span> <span class="o">-</span> <span class="n">pb</span>
<span class="n">radius</span> <span class="o">=</span> <span class="n">tetherLength</span> <span class="o">*</span> <span class="n">scale</span>
<span class="n">slack</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">radius</span> <span class="o">/</span> <span class="p">(</span><span class="o">|</span><span class="n">delta</span><span class="o">|</span> <span class="o">+</span> <span class="n">epsilon</span><span class="p">)</span>
<span class="n">offset</span> <span class="o">+=</span> <span class="n">delta</span> <span class="o">*</span> <span class="n">max</span><span class="p">(</span><span class="n">slack</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
<span class="n">pb</span> <span class="o">+=</span> <span class="n">offset</span> <span class="o">*</span> <span class="n">stiffness</span>
</pre></div>
</div>
<p>The stiffness is calculated as follows:</p>
<div class="highlight-c++"><div class="highlight"><pre><span class="n">stiffness</span> <span class="o">=</span> <span class="n">tetherConstraintStiffness</span> <span class="o">*</span> <span class="n">numParticles</span> <span class="o">/</span> <span class="n">numTethers</span>
</pre></div>
</div>
</div>
<div class="section" id="edge-constraints">
<h2>Edge constraints<a class="headerlink" href="#edge-constraints" title="Permalink to this headline">¶</a></h2>
<p>Algorithm:</p>
<div class="highlight-c++"><div class="highlight"><pre><span class="n">r</span> <span class="o">=</span> <span class="n">restlength</span>
<span class="n">pi</span> <span class="o">=</span> <span class="n">particle</span> <span class="n">i</span>
<span class="n">piw</span> <span class="o">=</span> <span class="n">inv</span> <span class="n">weight</span> <span class="n">of</span> <span class="n">particle</span> <span class="n">i</span>
<span class="n">h</span> <span class="o">=</span> <span class="n">pb</span><span class="o">-</span><span class="n">pa</span>
<span class="n">e2</span> <span class="o">=</span> <span class="n">epsilon</span> <span class="o">+</span> <span class="o">|</span><span class="n">h</span><span class="o">|^</span><span class="mi">2</span>
<span class="n">er</span> <span class="o">=</span> <span class="n">r</span><span class="o">></span><span class="mi">0</span> <span class="o">?</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">r</span> <span class="o">/</span> <span class="n">sqrt</span><span class="p">{</span><span class="n">e2</span><span class="p">})</span> <span class="o">:</span> <span class="mi">0</span>
<span class="k">if</span><span class="p">(</span><span class="n">useMultiplier</span><span class="p">)</span>
<span class="p">{</span>
<span class="c1">//from PhaseConfig.h cloth::transform</span>
<span class="n">multiplierC</span> <span class="o">=</span> <span class="n">log2</span><span class="p">(</span><span class="n">stiffnessMultiplier</span><span class="p">)</span>
<span class="n">compressionLimitC</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">compressionLimit</span>
<span class="n">strechLimitC</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">stretchLimit</span>
<span class="n">er</span> <span class="o">-=</span> <span class="n">multiplierC</span> <span class="o">*</span> <span class="n">max</span><span class="p">(</span><span class="n">compressionLimitC</span><span class="p">,</span> <span class="n">min</span><span class="p">(</span><span class="n">er</span><span class="p">,</span> <span class="n">stretchLimitC</span><span class="p">))</span>
<span class="p">}</span>
<span class="n">stiffnessExponent</span> <span class="o">=</span> <span class="n">stiffnessFrequency</span> <span class="o">*</span> <span class="n">dt</span><span class="o">/</span><span class="n">iterations</span>
<span class="n">stiffness</span> <span class="o">=</span> <span class="n">log2</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">stiffness</span><span class="p">)</span> <span class="c1">//check when this happens //from PhaseConfig.h cloth::transform</span>
<span class="n">stiffnessC</span> <span class="o">=</span> <span class="mi">1</span><span class="o">-</span><span class="mi">2</span><span class="o">^</span><span class="p">{</span><span class="n">stiffness</span> <span class="o">*</span> <span class="n">stiffnessExponent</span><span class="p">}</span>
<span class="n">ex</span> <span class="o">=</span> <span class="n">er</span> <span class="o">*</span> <span class="n">stiffnessC</span> <span class="o">/</span> <span class="p">(</span><span class="n">pbw</span><span class="o">+</span><span class="n">paw</span><span class="p">)</span>
<span class="n">pa</span> <span class="o">=</span> <span class="n">pa</span> <span class="o">+</span> <span class="n">h</span><span class="o">*</span><span class="n">ex</span> <span class="o">*</span> <span class="n">paw</span>
<span class="n">pb</span> <span class="o">=</span> <span class="n">pb</span> <span class="o">-</span> <span class="n">h</span><span class="o">*</span><span class="n">ex</span> <span class="o">*</span> <span class="n">pbw</span>
</pre></div>
</div>
</div>
<div class="section" id="separation-constraints">
<h2>Separation constraints<a class="headerlink" href="#separation-constraints" title="Permalink to this headline">¶</a></h2>
<p>Separation constraints do exactly the opposite of distance constraints.
These constraints ensure that the particles remain outside the defined spheres.</p>
<p>The constraints are interpolated between the start and target spheres if both are given.</p>
<p>The constrainSeparation() function in the solver is responsible for enforcing these constraints.
Solving a single constraint is done as follows:</p>
<div class="highlight-c++"><div class="highlight"><pre><span class="c1">//p is the particle position</span>
<span class="c1">//c is the sphere center</span>
<span class="c1">//r is the sphere radius</span>
<span class="n">d</span> <span class="o">=</span> <span class="n">c</span><span class="o">-</span><span class="n">p</span>
<span class="n">l</span> <span class="o">=</span> <span class="o">|</span><span class="n">d</span><span class="o">|</span>
<span class="n">slack</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">r</span><span class="o">/</span><span class="n">l</span>
<span class="k">if</span><span class="p">(</span><span class="n">slack</span> <span class="o"><</span> <span class="mi">0</span><span class="p">)</span>
<span class="n">p</span> <span class="o">+=</span> <span class="n">d</span> <span class="o">*</span> <span class="n">slack</span>
</pre></div>
</div>
</div>
<div class="section" id="fabric-data-structure">
<h2>Fabric data structure<a class="headerlink" href="#fabric-data-structure" title="Permalink to this headline">¶</a></h2>
<p>The fabric data structure contains the constraint information that can be reused by multiple cloth instances.
The constraints are divided in different phases.
Each phase usually contains a different type of constraints such as (horizontal/vertical) stretching, searing, bending constraints.</p>
<p>mPhases contains indices indicating which set from mSets belongs to a phase.</p>
<p>mSets contains a list of start (and end) indices for mRestvalues and mIndices (if multiplied by 2).
The first rest value of set s is mRestvalues[mSets[s]] and the last is mRestvalues[mSets[s+1]-1].</p>
<p>mRestvalues contains the rest lengths / edge lengths of the constraints.</p>
<p>mIndices contains pairs of indices connected by a constraint. (mIndices.size()*2 == mRestvalues.size())</p>
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<h3><a href="../NvCloth.html">Table Of Contents</a></h3>
<ul>
<li><a class="reference internal" href="#">Internal solver function/algorithm documentation</a><ul>
<li><a class="reference internal" href="#overview">Overview</a></li>
<li><a class="reference internal" href="#particle-invmass-w-component">Particle invMass w component</a></li>
<li><a class="reference internal" href="#slack">Slack</a></li>
<li><a class="reference internal" href="#integration">Integration</a></li>
<li><a class="reference internal" href="#wind-simulation">Wind simulation</a></li>
<li><a class="reference internal" href="#distance-constraints">Distance constraints</a></li>
<li><a class="reference internal" href="#tether-constraints">Tether constraints</a></li>
<li><a class="reference internal" href="#edge-constraints">Edge constraints</a></li>
<li><a class="reference internal" href="#separation-constraints">Separation constraints</a></li>
<li><a class="reference internal" href="#fabric-data-structure">Fabric data structure</a></li>
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