Computing geodesic distances from structural data

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SUMMARY

This discussion focuses on computing geodesic distances on manifolds with sparsely sampled structural data in R^3. The user seeks assistance in applying concepts from differential geometry to real-world scenarios where the type of manifold is not predefined. Key challenges include determining the metric for arbitrary surfaces and effectively utilizing point locations and normals for geodesic calculations. The discussion highlights the need for practical methods to compute geodesic distances without prior knowledge of the manifold type.

PREREQUISITES
  • Basic understanding of differential geometry concepts
  • Familiarity with R^3 coordinate systems
  • Knowledge of geodesic equations and metrics
  • Experience with computational geometry tools
NEXT STEPS
  • Research methods for computing geodesic distances on arbitrary manifolds
  • Explore algorithms for metric estimation from point cloud data
  • Learn about tools like CGAL (Computational Geometry Algorithms Library) for geometric computations
  • Investigate numerical methods for solving geodesic equations in non-standard surfaces
USEFUL FOR

This discussion is beneficial for mathematicians, computational geometers, and researchers working with manifold data, particularly those involved in geodesic calculations and surface modeling.

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Greetings,

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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