How to build a sphere – its more difficult than you think!

Geodesic sphere from wikipedia

Gilgamesh

In my new geometry library Gilgamesh I have started with some real world examples from structural biology namely building solvent excluded and ball-and-stick representations of molecules. For the ball-and-stick models we need to have good low-polygon spheres as big complexes may have hundreds of thousands of atoms.

The sphere class allows you to add transformed and coloured spheres to a mesh. For examples, see basic_mesh and molecules. You can also supply your own lambda function to generate distorted versions of the sphere, for example for building models of moons.

https://github.com/andy-thomason/gilgamesh/blob/master/include/gilgamesh/shapes/sphere.hpp

Spheres have always been and interesting challenge to get right in a geometry library with common problems including uneven sized triangles and erratic mapping.

In the Octet OpenGL framework that I developed for my games students I had several geometric primitives such as spheres and cones. I wanted to present this more formally with a well balanced set of primitives represented by their own classes so that geometric operations such as ray tracing and Constructive Solid Geometry could be done without reference to the mesh class.

Many 3D editing packages offer a choice of Mercator-style spheres and Geospheres.

Mercator spheres divide the sphere into lines of lattitude and longitude using the sin and cosine of these two angles like a Mercator projection on a globe. The problem with these is that they use more triangles towards the poles and have problems if we want to map textures to the sphere.

Geospheres generally look nicer with smaller numbers of triangles. They use triangles that are approximately equilateral in a Buckminster Fuller style.

To construct a geosphere it is possible to inflate a near spherical primitive such as a dodecahedron and add extra triangles to increase curvature. This is what I do in the Octet library. Every triangle in the primitive is split into four or more new triangles with each new vertex extended to sit on the sphere. The problem with this is that the triangles become uneven and we need the coordinates of the dodecahedron to start with. There is an excellent dodecahedron wikipedia article that can help with this, however.

After toying with several methods of generating equally spaced triangles on a sphere I looked at the simple method of adapting the Mercator sphere to have roughly equilateral triangles.

First we have to choose the number of longitude subdivisions and divide Pi by this to get our triangle length. Now for each ring as we go down from the north pole, we generate n vertices where n is the number of lengths in that ring ie floor(2*Pi*r/length).

This gives us our vertices, now how to generate triangles? Each strip has a triangles at the top and b triangles at the bottom. These numbers are mostly different. We walk along the ring adding a triangle to whichever edge has the lowest angle value. This algorithm selects roughly equilateral triangles at every step and fulfils our requirements. One side effect of this is that we always get six triangles at the top and bottom of the sphere forming a hexagon.

The next task is how to UV map the sphere. We need to be careful in three places, at the poles and at the meridian or zero angle joinng the poles. At the poles, choosing one UV coordinate, say (0, 0) will result in distortion of the map, so we may choose to duplicate the vertices at the pole seven times and assign coordinates (0, 0), (1/6, 0) .. (1,0). Why seven and not six? This is related to the meridian mapping problem. We need the final triangle to go from ((n-1)/n, v) to (1, v) so that the texture wraps correctly at the seam. So we must duplicate one vertex at the meridian also. If you don’t understand this, try it yourself!

On unmapped spheres we will want to choose to not duplicate these vertices as they add an overhead. I intend to add this feature later.

Getting started with Gilgamesh.

Start by cloning Gilgamesh and its dependencies (glm and minizip).

git clone https://github.com/andy-thomason/gilgamesh
git submodule init
git submodule update

Make a build directory and use cmake to construct a project (makefiles or visual studio)

mkdir build
cd build
cmake -DCMAKE_BUILD_TYPE=Release ..

Now you can build the examples by running Visual studio or make.

Happy meshing!

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