Let's compare some topologies of spheres and see how they behave. This code is what I used to generate them
This code was made for me only, to generate and animate different sphere meshes inside Blender 2.90 for visual representation. This project is made to work as a Blender script that is able to create spheres and dynamically change their properties, so in the code there will be additional steps that are not stricly concerned with the mesh generation. This also means the code is poorly documented and absolutely not bug free. While the code works and can be used in Blender, it comes as is, and it's probably more useful as a reference rather than a solid code base.
Here there is a brief description of the different topologies with the relative pseudocode used to generate its vertices. For more detailed code, feel free to look at the python files, but remember what is was written in the previous disclaimer
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Probably the most intuitive one, because it's built using basic trigonometry. Like the Earth's coordinates it is based on parallels and meridians, which means that this topology requires 2 numbers to define its resolution. Depending on the context, it can be very bad:
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The distribution of vertices is not regular (near the pole there are more and closer together)
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Faces have different shape and size
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Some faces have a different number of vertices from the others (they are quads in the middles, triangles near the poles)
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Having an high number of meridians and low number of parallels (or viceversa) can produce weird results
for p in range(parallels_num): plane_z = (1 - (2 * p) / (parallels_num - 1)) teta = plane_z * pi / 2 for m in range(meridians_num): phi = m / (meridians_num - 1) * 2 * pi addVertex([ radius * cos(phi) * cos(teta), radius * sin(phi) * cos(teta), radius * sin(teta) ])
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Basically take a cube and normalize each vertex (so move it so that its distance from the center is equal to the final radius of the sphere). It's still quite intuitive to picture, not so much to code it. It still has problems:
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Faces have different shape and size
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The distribution of vertices is not regular (near the corners of the originating cube there are more and closer together)
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once you have the cube, it's really easy to calculate
k = resolution + 1 for face in range(6): for j in range(k): for i in range(k): addVertex(getTransformedCoordinates(face, i, j, resolution, radius)) def getTransformedCoordinates(face, i, j, resolution, radius): step = 1 / resolution step3 = np.ones(3) * step j3 = np.ones(3) * j i3 = np.ones(3) * i cube_coords = origin[face] * radius + step3 * (i3 * right[face] + j3 * up[face]) * radius * 2 return normalize(cube_coords, radius)
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Create a base icosahedron, then subdivide each face until you reach the desired resolution. The easiest way to generate the basic mesh is to hard code it (see here), and you get a 12 vertex solid with all identical triangular faces.
Pros:
- All faces are exactly identical
- All vertices are equidistant from their neighbour (constant vertex density)
Cons:
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Each subdivision, from 1 face you create 4 new ones, so the number of faces grows exponentially and you can't easily control it's vertex count. The geometric progression goes like: 20, 80, 320, 1280, 5120, 20480, ...
mesh = generateBaseIcosahedron() for i in range(iterations): for edge in mesh.edges: # get median point position, normalize and create median vertex co = [edge.verts[0].co[i] + edge.verts[1].co[i] for i in range(3)] addVertex(normalize(co, radius))
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It's the popular shape of the football (or soccer ball), generated from an Icosahedron which vertices are cut of and replaced with a pentagonal face. To increase the resolution you can do it in different ways, I chose to truncate a subdivided Icosahedron like shown before, getting this way a Goldberg Polyhedron
It has similar characteristics to the Icosahedron, except that now the faces are not equal to each other (they are pentagons and hexagons), so it's not the best to just approximate a sphere. However, it is esthetically pleasing, and if you want to cover your sphere with hexagons tiles, this is likely the closer you will get
You can also truncate a cube or a tetrahedron, which can increase resolution more granularly, but have more noticeable deformations near the original corners
``` # read data from an Icosahedron to build a separate new mesh for edge in Icosahedron_edges: a = edge.verts[0] b = edge.verts[1] v1 = addVertex((2 * a + b) / 3) v2 = addVertex((a + 2 * b) / 3) ``` -
This is a really simple, yet quite powerful, topology. It's based on the golden angle to place the vertices on a spiral and achieve a very good vertex density The faces can be easily generated with Delauney Triangulation
Pros:
- Very fast to calculate
- Faces have all similar size (but not exactly equal)
- When you animate it it looks absolutely bonkers
- Can be created with any number of vertices (so it's very granular)
Cons:
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Faces still have different shapes
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At the poles the topology loses all regularity
phi = pi * (3 - sqrt(5)) # golden angle (radians) for i in range(res): theta = phi * i z = radius * (- 1 + (i / (res - 1)) * 2) x = radius * cos(theta) * sqrt(1 - z * z) y = radius * sin(theta) * sqrt(1 - z * z) addVertex([x, y, z])
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This is not a good topology: you need many points and the surface has areas of much higher density then others. This could be solved with a stronger pseudo-random generator, but it would still be bad. The faces can be easily generated with Delauney Triangulation
``` for i in range(res): # generate random rotation angle and latitude phi = random() * 2 * pi latitude = radius * 2*random()-1 dist = (radius ** 2 - latitude ** 2) ** 0.5 addVertex([ cos(phi) * dist, sin(phi) * dist, latitude ]) ```
Sadly, I still have some problems in transforming this script in a registerable addon in Blender.
My attemps to do so are in the Addon Branch here on github, if you know what I missed feel free to take a look.
Until then, to use it just place the files in this github in the same folder of your .blend file, in the text windows in Blender load main.py
and then run it. Since it is just a script, if you restart Blender you'll have to rerun the code as well to show again the addon panel/operators.
To create a new sphere, just go in Add > Sphere Topologies and select the topology you want. If you have one of these sphere selected, in the mesh panel you will see a panel called Sphere Topologies, from which you can modify and/or animate some of the settings of the topology.
Remember that if you modify the mesh in edit mode, you shouldn't change those settings anymore
Also, in the current state the animations with keyframes doesn't work6 direi
Very good article on Delauney Triangulation and Voronoi regions here
GitHub post on some basic topologies here