Sphere Shape Approximation Using Triangles?

Stas B. · 2009-01-31 23:34:43

Hi there! I was wondering if anyone could help me out with this problem:
What is the optimal method for tessellating a sphere using triangles? That is, if I can construct some X number of triangles in 3D space, how do I construct them in such a way that they'll create a closed shape as closely approximating a sphere as possible?
I can think of a few possible solutions, but they're probably not optimal.
I don't think I really need the most optimal solution, but I'm asking out of curiosity, since I don't really know how to even approach this problem.
Could anyone please help me out?
Thanks in advance.
Stas B.

mathsyperson · 2009-02-01 01:34:38

An icosahedron would do a fairly good job. That's the only one I can come up with without thinking too much.

You could bend the rules by building a 2D plane and then claiming that it's a sphere with infinite radius.

JaneFairfax · 2009-02-01 01:40:08

mathsyperson wrote:

You could bend the rules by building a 2D plane and then claiming that it's a sphere with infinite radius.

integer · 2009-02-01 05:02:22

Either
the Dodecahedron (12 Faces, 20 Vertices , 30 Edges )
or
the Icosahedron (20 Faces , 12 Vertices, 30 Edges )

For the icosahedron this is my method of choice:
As you increase the size of the "sphere" the distance
from the center of the sphere to a vertex will become
larger than the distance from the center of the sphere
to the midpoint of an edge. Obvious. If will remain in
a constant proportion. However, when that distance
exceeds a preset number of pixels, replace each
equilateral triangle with 4 equilateral triangles, and
then reset the vertices to equal distances.

The decision depends on the resolution of the screen.
The calculations are done only when the difference
between the arc of the sphere and the edge (chord)
are noticeable.

mathsyperson · 2009-02-01 06:17:48

Could you elaborate on that method?
The way I'm interpreting it, you wouldn't be able to arrange a set of 4 triangles in any way other than the position the one big triangle was in originally.

MathsIsFun · 2009-02-01 09:04:50

You could pick a point at the center of the large triangle, and then "raise" it to the radius of the sphere. Then turn the big triangle into 3 smaller triangles using that point. There may be some other complication I am not thinking about, though.

Math Is Fun Forum

#1 2009-01-31 23:34:43

Sphere Shape Approximation Using Triangles?

#2 2009-02-01 01:34:38

Re: Sphere Shape Approximation Using Triangles?

#3 2009-02-01 01:40:08

Re: Sphere Shape Approximation Using Triangles?

#4 2009-02-01 05:02:22

Re: Sphere Shape Approximation Using Triangles?

#5 2009-02-01 06:17:48

Re: Sphere Shape Approximation Using Triangles?

#6 2009-02-01 09:04:50

Re: Sphere Shape Approximation Using Triangles?

Board footer