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RV_;!uVa%nf>gw;t1psFL5@P@W diff --git a/docs/documentation.tex b/docs/documentation.tex index 42dab16..ad7954e 100644 --- a/docs/documentation.tex +++ b/docs/documentation.tex @@ -365,7 +365,7 @@ and moving outwards from there. For projecting the constraints, we utilize the method proposed by \citet{MullerPBD}. The details of computing the desired delta positions are -explained in the Appendix \ref{appendix:ConstraintProjection} +explained in Appendix \ref{appendix:ConstraintProjection} \nameref{appendix:ConstraintProjection}. An example derivation is given in Section \ref{section:DistanceConstraint} \nameref{section:DistanceConstraint}. @@ -731,7 +731,111 @@ method and properties. \appendix \section{Constraint Projection} \label{appendix:ConstraintProjection} -asd +Projecting a set of points according to a constraint means moving the points +such that they satisfy the constraint. As we move the points in the +simulation loop directly, the most important issue is the conservation of linear and +angular momentum. Let $\Delta \b{p_i}$ be the displacement of particle $i$ by the +projection. Linear momentum is conserved if + +\begin{equation} + \label{eq:linear_momentum} + \sum\limits_{i} m_i \Delta \b{p_i} = \b{0}, +\end{equation} + +which amounts to conserving the center of mass. Angular momentum is conserved if + +\begin{equation} + \label{eq:angular_momentum} + \sum\limits_{i} \b{r_i} \times m_i \Delta \b{p_i} = 0, +\end{equation} + +where the $\b{r_i}$ are the distances of the $\b{p_i}$ to an arbitrary common +rotation center. If a projection violates either the preservation of +linear or angular momentum, \emph{ghost +forces} get introduced into the system, which act like external forces dragging +and rotating the object. However, only internal constraints need to conserve the +momenta. As we reduce the problem of collision handling to simply moving the +vertex to a valid point, it is somewhat trivial that for example collision +constraints are allowed to have global effects on the object. + +The method that \citet{MullerPBD} propose conserves both momenta for internal +constraints. Let us look at a constraint $C$ with cardinality $n$ on the points +$\b{p_1}, \ldots, \b{p_n}$ with stiffness $k$. We let $\b{p}$ be the +concatenation $[\b{p_1}^T, \ldots, \b{p_n}^T]^T$. For internal constraints, $C$ is +independent of translation and rotation, which means that rotating or +translating the points does not change the value of the constraint function. +Therefore, the gradient $\nabla_{\b{p}} C$ is the direction of maximal change. If the +correction $\Delta \b{p}$ is chosen to be along $\nabla C_{\b{p}}$ both momenta +are automatically conserved if all masses are equal (we handle different masses +later). Given $\b{p}$ we want to find a correction $\Delta \b{p}$ such that +$C(\b{p}+\Delta\b{p})=0$. This equation can be approximated by + +\begin{equation} + C(\b{p}+\Delta\b{p}) \approx C(\b{p}) + \nabla_{\b{p}} C(p)\cdot\Delta + p = 0. + \label{eq:approximation} +\end{equation} + +Restricting $\Delta \b{p}$ to be in the direction of $\nabla_{\b{p}}C$ means +choosing a scalar $\lambda$ such that + +\begin{equation} + \Delta{\b{p}} = \lambda \nabla_{\b{p}}C(\b{p}). + \label{eq:restrictingLambda} +\end{equation} + +Substituing Equation \ref{eq:restrictingLambda} into Equation +\ref{eq:approximation}, solving for $\lambda$ and subsituting back into Equation +\ref{eq:restrictingLambda} gives us the final formula for $\Delta\b{p}$: + +\begin{equation} + \Delta \b{p} = - \frac{C(\b{p})}{\lvert \nabla_{\b{p}} C(\b{p})\rvert^2} + \nabla_{\b{p}}C(\b{p}) +\end{equation} + +which is a regular Newton-Raphson step for the iterative solution of the +non-linear equation given by a single constraint. For the correction of an +individual point $\b{p_i}$ we have + +\begin{equation} + \Delta \b{p_i} = -s\nabla_{\b{p_i}}C(\b{p_1},\ldots,\b{p_n}), +\end{equation} + +where the scaling factor + +\begin{equation} + s = \frac{C(\b{p_1},\ldots,\b{p_n})}{ + \sum\limits_{j} \lvert\nabla_{\b{p_j}}C(\b{p_1},\ldots,\b{p_n})\rvert^2 + } +\end{equation} + +is the same for all points. If the points have individual masses, we weight the +corrections $\Delta \b{p_i}$ by the inverse masses $w_i = 1/m_i$. In this case +a point with infinite mass, i.e. $w_i = 0$ for example does not move, just as +expected. Now Equation \ref{eq:restrictingLambda} is replaced by +$$ +\Delta \b{p_i} = \lambda w_i \nabla_{\b{p_i}}C(\b{p}) +$$ +yielding + +\begin{equation} + s = \frac{ + C(\b{p_1}, \ldots, \b{p_n}) + }{ + \sum\limits_{j}w_j\lvert\nabla_{\b{p_j}}C(\b{p_1},\ldots,\b{p_n})\rvert^2 + } +\end{equation} + +for the scaling factor and for the final correction + +\begin{equation} + \Delta \b{p_i} = -s w_i \nabla_{\b{p_i}}C(\b{p}). +\end{equation} + + +Section \ref{section:DistanceConstraint} gives an example by considering the +distance constraint function $C(\b{p_1},\b{p_2}) = \lvert\b{p_1}-\b{p_2}\rvert +- d$. \section{Supplementary development} -- GitLab