Question about muller's paper "Position Based Dynamic"

[fishboy82]

Hi:
When I reading the muller's paper "Position Based Dynamic".There is one sentence I don't quite understand, muller said the constraint function C(p1,p2..pn) is independent of rigid body motion when function C is a internal constraint. So the gradient of C is perpendicular to the Rigid mode's translation & rotation. This sentence I can undestand because for any rigid mode's translation & rotation : Deta(C) = gradient(C).Dot(detap1,detap2...detapn). Because Deta(C) is zero ,so the vector (detap1,detap2...detapn) is always perpendicular to gradient(C), but the following sentence confused me for some time : If the correction Deta(p) is chosen to be along gradient(C) both momenta are automatically conserved if all masses are equal. I thought this conclusion for a lot of times and finally I get a naive proof of this:
Because all rigid motion of the N vertices is always perpendicular to the gradient ,let the velocity vector (1,0,0 , 1,0,0, ........1,0,0) is of course a rigid motion vector. So (1,00...1,0,0...1,0,0)Dot(gradient(C)) is Zero. this means all component of gradient(C) corresponding to X component summed to zero, and the same as the y and z component .So if we let all points 's Deta(p) equal to the gradient(C) .the sum of the Deta(p) of the x,y,z component are zero.So this prove the momenta is Zero... I think the proove is naive and I am not very sure whether it is right.So if any body who Have read that paper could give me some opinion in this problem Thanks alot