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2 years ago in Computational Geometry , Numerical Analysis By Itjarz

How can one determine which sub-simplices a point belongs to in a simplex triangulation?

I'm working with simplicial complexes and fine triangulations for numerical analysis. Given a point in the parent simplex, I need a robust algorithm to determine its containing sub-simplex without exhaustive search. This is critical for interpolation and finite element methods, and I'm weighing approaches like barycentric coordinate checks.

 

MathematicalCriteria Triangulation PointLocation

All Answers (2 Answers In All)

By Kumar Answered 1 year ago

For implementation, I consistently recommend using normalized barycentric coordinates relative to the original simplex's vertices. Compute the coordinates of your point; they will be non-negative and sum to 1. The key is that the triangulation pattern (like Kuhn's) corresponds to a unique ordering or permutation of these coordinates at the grid scale 1/q. By sorting or comparing these coordinate values, you can directly map to the specific sub-simplex index without geometric searches. It’s computationally elegant and robust.

By Kushi Gupta Answered 8 months ago

For a simplex with vertices v(1), v(2), …, v(n+1) and a grid of size 1/q, triangulation divides the simplex into sub-simplices. A point belongs to a sub-simplex if it lies within its vertices’ convex combination. This can be checked using barycentric coordinates or geometric inclusion tests.

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