Geodesic icosahedron

Creating geodesic icosahedron given subdivision frequency.

Installation

Install the module using pip install icosphere or clone the repository.

Use

from icosphere import icosphere
nu = 5  # or any other integer
vertices, faces = icosphere(nu)

Examples

Check the examples in icosphere github, python script uses matplotlib for visualization, and one of jupyter notebooks uses plotly. You can also open the notebooks in colab: - with ploty, - with matplotlib.

Why use subdivision frequency?

For a certain subdivision frequency nu, each edge of the icosahedron will be split into nu segments, and each face will be split into nu**2 faces. This is different than a more common approach which recursively applies a subdivision with nu = 2, for example as used in pytorch3d ico_sphere, pymeshlab sphere, trimesh icosphere, and PyMesh generate_icosphere.

The advantage of using the subdivision frequency, compared to the recursive subdivision, is in controlling the mesh resolution. Mesh resolution grows quadratically with subdivision frequencies while it grows exponentially with iterations of the recursive subdivision. To be precise, using the recursive subdivision, the number of vertices and faces in the resulting icosphere grows with iterations iter as nr_vertex = 12 + 10 * (4**iter -1) and nr_face = 10 * 4**iter which gives a sequence of mesh vertices

12, 42, 162, 642, 2562, 10242, 40962, 163842, 655362, 2621442, 10485762...

Notice for example there is no mesh having between 2562 and 10242 vertices. Using subdivision frequency, the number of vertices and faces grows with nu as Notice for example there is no mesh having between 2562 and 10242 vertices. Using subdivision frequency, the number of vertices and faces grows with nu as nr_vertex = 12 + 10 * (nu**2 -1) and nr_face = 20 * nu**2 which gives a sequence of mesh vertices

 12, 42, 92, 162, 252, 362, 492, 642, 812, 1002, 1212, 1442, 1692, 1962,
 2252, 2562, 2892, 3242, 3612, 4002, 4412, 4842, 5292, 5762, 6252, 6762,
 7292, 7842, 8412, 9002, 9612, 10242...

Now there is 15 meshes having between 2562 and 10242 vertices. The advantage is even more pronounced when using higher resolutions.

The code was originally developed for this work.

Reference this work

Dahl, V. A., Dahl, A. B., & Larsen, R. (2014). Surface Detection Using Round Cut. 2014 2nd International Conference on 3D Vision. https://doi.org/10.1109/3dv.2014.60

@inproceedings{Dahl_2014,
	doi = {10.1109/3dv.2014.60},
	url = {https://doi.org/10.1109%2F3dv.2014.60},
	year = 2014,
	month = {dec},
	publisher = {{IEEE}},
	author = {Vedrana A. Dahl and Anders B. Dahl and Rasmus Larsen},
	title = {Surface Detection Using Round Cut},
	booktitle = {2014 2nd International Conference on 3D Vision}
}