Manifold gradient problem

[Asuralm]

Hi all:

I have just met a problem. If say there is a triangle ijk on a manifold, D(i), D(j), D(k) are the geodesic distances from a far point to i,j,k respectively. Then g = [D(i) - D(k); D(j) - D(k)], what does g describe? Does is describe the gradient of the vertex k?

If u = Vi-Vk, v = Vj-Vk where Vi, Vj, Vk are the coordinate vector in 3D, construct a matrix A = [u v], then let A = (A' * A) ^ (-1). Now A is a 2*2 matrix and what does A mean?

Finally, let g = A * g, what's the meaning of this then?

The context of this is in someone's programming code of computing the local gradient. Can someone help me please?

Thanks

 

[Asuralm]

Sorry the first post was failed.

 

[tacman]

The manifold gradient problem refers to the challenge of finding the gradient of a function on a manifold, which is a mathematical space that can be curved or twisted in multiple dimensions. In this specific scenario, the problem is related to finding the gradient at a particular vertex on a manifold. The gradient is a vector that points in the direction of the steepest increase of a function at a given point. In this case, the function is the geodesic distance from a far point to the vertices of a triangle on the manifold.

The variable g in this problem represents the gradient of the vertex k. The matrix A, which is constructed using the coordinate vectors of the vertices, is used to transform the gradient vector g to a local gradient vector. This means that the new vector g will represent the direction of the steepest increase at vertex k in the local coordinate system, rather than the global coordinate system. This is important because the manifold can be curved or twisted, and the local coordinate system takes that into account.

In the programming code, the local gradient is being calculated using the matrix A to transform the global gradient vector g. This is a common approach in solving the manifold gradient problem, as it allows for a more accurate calculation of the gradient at a specific point on the manifold. I hope this explanation helps clarify the concept for you.