What Does the Matrix A Represent in Manifold Gradient Calculations?

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SUMMARY

The discussion centers on the interpretation of matrix A in the context of manifold gradient calculations, specifically regarding a triangle formed by points i, j, and k. The matrix A, constructed from vectors u and v derived from the coordinates Vi, Vj, and Vk, represents a transformation related to the geodesic distances D(i), D(j), and D(k). The operation g = A * g indicates a transformation of the gradient vector g, which is essential for understanding local gradient computations in manifold geometry.

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Asuralm
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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
 
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