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I am attempting to compute geodesic distances on manifolds where structural data have been sparsely sampled.

First, off I am not well versed in the mathematics of differential geometry but I do have some knowledge (from an introductory differential geometry class in my undergrad). So forgive me if this question may seem simple.

The structural data that I have includes point locations of the manifold/surface in R^3 along with normals which describe the orientation of the manifold/surface.

In all examples in my DF class notes the type of manifold/surface was always known prior ( e.g. sphere, torus) to solving the geodesic equation and computing the geodesic distance. For these cases, it was easy to compute the metric since these surfaces are easy to parameterize. Now that I am trying to apply what I learned from the class (which was 10 years ago) in a real world scenario I am having a lot of trouble trying to determine how I solve it.

Surely there is a way to compute geodesic distances in this real world scenario. Can someone please help me with this problem?

Any help is greatly appreciated

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# Computing geodesic distances from structural data

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