"Solid Simulation with Oriented Particle" Problem

[apapaxionga]

When I read this paper "Solid Simulation with Oriented Particle"(http://matthias-mueller-fischer.ch/publ ... ticles.pdf), I have some problems. I hope someone familiar with this paper can give me some help.

1. For equation 6, "We shift the center of mass of each Ai to the global one...", Why they do it like that? If add Ai, will A still be the most opitmal least square match?
2. For equation (7)and (8), " To compute the moment matrix of a particle with orthonormal oritentation matrix R,........", How to calculate R ? Can someone explain it to me?

[Remotion]

Hi,

> 2. For equation (7)and (8), " To compute the moment matrix of a particle with orthonormal oritentation matrix R,........", How to calculate R ? Can someone explain it to me?

This is about Generalized Shape Matching so one get R the same way as in Shape Matching by polar decomposition of Matrix A.

//compute rotation matrix R and symmetrix matrix S so that R*S=A
polar_decomposition(A, R, S); //in A, out R and S

[apapaxionga]

Hi,

> 2. For equation (7)and (8), " To compute the moment matrix of a particle with orthonormal oritentation matrix R,........", How to calculate R ? Can someone explain it to me?

This is about Generalized Shape Matching so one get R the same way as in Shape Matching by polar decomposition of Matrix A.

//compute rotation matrix R and symmetrix matrix S so that R*S=A
polar_decomposition(A, R, S); //in A, out R and S

So in order to get R, A should be calculated firstly. However, In order to get A, Ai should be calculated firstly(equation 7). But Ai contains the matrix R(such as Asphere and Aellipsoid in equation 8 and 9). This is a deadlock loop procedure. How to solve this problem?

[Remotion]

So in order to get R, A should be calculated firstly. However, In order to get A, Ai should be calculated firstly(equation 7). But Ai contains the matrix R(such as Asphere and Aellipsoid in equation 8 and 9). This is a deadlock loop procedure. How to solve this problem?

I think the point is that first R (just below equation 3) is the shape matching R that can be computed using polar decomposition.

The second R that is used in equation (8) and (9) is just simple rotation matrix of the simple particle.
It can be computed form particles Quaternion. (Quaternion is stored and updated for every particle)

Something like this:
polar_decomposition(A, R1, S); //in A, out R and S

Matrix R2 = quaternionToMatrix(q); //get the rotation matrix of the particle