Real-Time Physics Simulation Forum

if x1 is the particle position then the change of the position in time is the velocity v1 = dx1/dt

(v1 - v2) is the inner derivative (I hope you call it like this in english).
Take the simple example C( x(t) ) = | x(t) |

Then dC/dt = d/dx * | x(t) | * dx(t) / dt (Chain rule)

So,

dC/dt = x(t) / |x(t)| * v




the dC/dz was a typo and I fixed it. It should read dC/dt of course

Ahh, I understand completely now
seems to be working out right for me now

I just have one more question regarding the paper; In equation (3) they describe how to dampen the forces. Inside the sumnation, they have

dC
--- v
dp

As I understand it, dC/dp is a vector but so is velocity (v).
Are they implying a dot product here (x1*x2 + y1*y2 + z1*z2)?
Or, do they imply a component-wise multiplication? (x1*x2, y1*y2, z1*z2)

dC/dp is the Jacobi matrix and in this particlur case it is the gradient of the constraint function, so you better think of it in this case as a row vector. In our example it is.

J = ( Transpose(n) -Transpose(n) )

So in this case you have a 1 x 6 matrix.

Basically the Jacobi matrix stores all gradient vectors. by definition (there are some reasons for this) the gradients are stored in the rows. So this is always a bit confusing the 1D case where you can either interpret it as a vector (the gradient) or as matrix.

Note that this yields the same results:

a * b = Transpose( a ) * b // where a and b are vectors


Does this help?

Ah I see, I wasnt quite sure what exactly a Jacobi was, but I get it now

I got the damping working correctly now, thanks for the help