Problem 1935. Spherical surface triangulation

Created by Alfonso Nieto-Castanon

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Create a triangulation of a spherical surface with approximately-uniform coverage<a href = "http://www.alfnie.com/software/print_sphere01.jpg">http://www.alfnie.com/software/print_sphere01.jpg</a>Description:A rather crude way to obtain a triangulation of the spherical surface would be:<pre> [x,y,z] = sphere(64);
 fvc = surf2patch(x,y,z,'triangles');</pre>The function sphere creates a uniform grid in spherical coordinates, so the resulting triangulation produces very small triangles/faces at the poles and considerably larger triangles/faces at the equator (see the sphere at the right in the examples above).In general (with a few exceptions) it is not possible to construct a tessellation of the spherical surface made of triangles of exactly the same size. But we can try...Your function should return a tessellation of the unit-sphere surface formed by triangles with side lengths as similar as possible (as close to identical equilateral triangles as possible). An example of a somewhat reasonable solution would be the sphere at the left in the examples above.Details:Given a number N your function should return an approximately uniform triangulation of the spherical surface with at least N vertices.Uniformity is measured as the ratio between the longest and the shortest edge across your entire triangulation (lower numbers are better). This ratio should be lower or equal to 2 in order to pass this problem (in other words, the longest edge should be at most twice the size of the shortest edge). Your solution will receive a score 1000*(ratio-1). A score of 0 would correspond to identical equilateral triangles.The function output should be a valid PATCH format: a structure with fields vertices (with size Nx3; each row of vertices defining the 3d coordinates of each vertex) and faces (with size Mx3; each row of faces defining the three vertices in each face/triangle as indices into the rows of the matrix vertices).

Create a triangulation of a spherical surface with approximately-uniform coverage

http://www.alfnie.com/software/print_sphere01.jpg

*Description:*

A rather crude way to obtain a triangulation of the spherical surface would be:

[x,y,z] = sphere(64);
 fvc = surf2patch(x,y,z,'triangles');

The function _sphere_ creates a uniform grid in spherical coordinates, so the resulting triangulation produces very small triangles/faces at the poles and considerably larger triangles/faces at the equator (see the sphere at the right in the examples above).

In general (with a few exceptions) it is not possible to construct a tessellation of the spherical surface made of triangles of exactly the same size. But we can try...

Your function should return a tessellation of the unit-sphere surface formed by triangles with side lengths as similar as possible (as close to identical equilateral triangles as possible). An example of a somewhat reasonable solution would be the sphere at the left in the examples above.

*Details:*

Given a number *N* your function should return an approximately uniform triangulation of the spherical surface with at least _N_ vertices.

Uniformity is measured as the ratio between the longest and the shortest edge across your entire triangulation (lower numbers are better). This ratio should be lower or equal to 2 in order to pass this problem (in other words, the longest edge should be at most twice the size of the shortest edge). Your solution will receive a score 1000*(ratio-1). A score of 0 would correspond to identical equilateral triangles.

The function output should be a valid PATCH format: a structure with fields *vertices* (with size Nx3; each row of _vertices_ defining the 3d coordinates of each vertex) and *faces* (with size Mx3; each row of _faces_ defining the three vertices in each face/triangle as indices into the rows of the matrix _vertices_).

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Last Solution submitted on May 21, 2023

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Rafael S.T. Vieira on 29 Jun 2020

The tip here is averages. Catmull & Clark used them to smooth polyhedra for instance.

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Problem 1935. Spherical surface triangulation

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