In the triangle mesh are u_i and u_j the positions of vertex i and vertex j?

Edit: Nvm, I think u is just any scalar function that we're trying to solve the Laplace equation for. It could be displacement / height, vertex positions or something else I guess

dvernet

That's right. In the context of option C of assignment 5 (the water bunny), $u_{i}$ and $u_{j}$ represent the displacement along the normal for vertices $i$ and $j$.

kmcrane

@kapalani: Correct, $u$ denotes whatever type of function you're solving for. For instance, in the demo I showed in class, it was the normal displacement of each vertex (not the vertex position itself). If you instead let (u=(x,y,z)), i.e., if you solve for the vertex coordinate function, then you will modify the geometry itself, rather than some function "sitting on" the surface. The latter leads toward things like mean curvature flow, which we give some links to in the assignment writeup.

PandaX

laplacian = (4a - (b + c + d + e))/h^2 ? Shouldn't it be ((b + c + d + e) - 4a)/h^2 ? reference

kmcrane

@PandaX: Everyone has a different sign convention for Laplace. For every reference you find where it's positive, you can find another where it's negative. Physicists, geometers, engineers, mathematicians... nobody can agree. The most important thing is just that you pick a consistent sign convention and stick with it!

In the triangle mesh are u_i and u_j the positions of vertex i and vertex j?

Edit: Nvm, I think u is just any scalar function that we're trying to solve the Laplace equation for. It could be displacement / height, vertex positions or something else I guess

That's right. In the context of option C of assignment 5 (the water bunny), $u_{i}$ and $u_{j}$ represent the displacement along the normal for vertices $i$ and $j$.

@kapalani: Correct, $u$ denotes whatever type of function you're solving for. For instance, in the demo I showed in class, it was the normal displacement of each vertex (not the vertex position itself). If you instead let (u=(x,y,z)), i.e., if you solve for the vertex coordinate function, then you will modify the geometry itself, rather than some function "sitting on" the surface. The latter leads toward things like mean curvature flow, which we give some links to in the assignment writeup.

laplacian = (4a - (b + c + d + e))/h^2 ? Shouldn't it be ((b + c + d + e) - 4a)/h^2 ? reference

@PandaX:

Everyonehas a different sign convention for Laplace. For every reference you find where it's positive, you can find another where it's negative. Physicists, geometers, engineers, mathematicians... nobody can agree. The most important thing is just that you pick a consistent sign convention and stick with it!