platonic_sphere

版本 1.3 (130.6 KB) 作者: David Monteverde
Builds class-1 geodesic polyhedra (based on Platonic solids).
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更新时间 2024/11/29

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The main purpose of this function is to generate points on the surface of a unit sphere that have an optimized uniformity of distribution. A secondary purpose is to facilitate exploration of geodesic polyhedra.
The functional form is:
[polyhedron,properties] = platonic_sphere(symmetry,freq,style)
where inputs are:
symmetry: 'icosahedral', 'octahedral' or 'tetrahedral'
freq: subdivision frequency (natural number)
style: subdivision style ('planar' or 'spherical
The most uniform spherical distribution is produced using icosahedral symmetry. If no input parameters are given, default is icosahedral symmetry with a subdivision frequency of 3 (each face divided as a tetractys) and a planar subdivision style.
Example:
[polyhedron,properties] = platonic_sphere('icosahedral',3)
where output is:
- polyhedron: triangulation object
- properties: structure with fields describing properties of the polyhedron
practicalities:
- visualize: trisurf(polyhedron); axis equal
- make an stl file: stlwrite(polyhedron,'my_psphere')
- make a movie: psphere_movie('octahedral',25)
AUTHOR: David Monteverde
RELEASED: 2024-11-10
UPDATED: 2024-11-29
REV: 1.3
NOTES
1) Surface density of the "platonic sphere" (number of vertices of the subdivided polyhedron) increases quadratically with subdivision frequency:
nvp = (nfb/2)*freq^2 + (neb-3*nfb/2)*freq +( nvb-neb+nfb)
where,
nvp : number of vertices of the subdivided polyhedron
freq : subdivision frequency
nvb : number of vertices of the basis polyhedron
neb : number of edges of the basis polyhedron
nfb : number of faces of the basis polyhedron (platonic solid)
Parenthetically, the above expression is derived from:
nvp = nvb + neb*(freq-1) + nfb*Trinum(freq-2)
where Trinum(n) is the triangular number of n.
For instance, the surface densities for a platonic sphere with icosahedral basis are as follows:
freq: 1 2 3 4 5 6 7 8 9 10 ...
vertices: 12 42 92 162 252 362 492 642 812 1002 ...
2) Additional useful information:
https://en.wikipedia.org/wiki/Geodesic_polyhedron
https://en.wikipedia.org/wiki/List_of_geodesic_polyhedra_and_Goldberg_polyhedra
Pugh, Antony. Polyhedra: a visual approach. University of California Press, 1976

引用格式

David Monteverde (2025). platonic_sphere (https://ww2.mathworks.cn/matlabcentral/fileexchange/175453-platonic_sphere), MATLAB Central File Exchange. 检索时间: .

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版本 已发布 发行说明
1.3

streamlined functions, and implemented spherical style subdivision
1.2

implemented tetrahedral symmetry, added movie creation function
1.1.2

updated description
1.1.1

modified description
1.1

reverted to planar subdivision
1.0.1

added image to description page
1.0.0