I need to discretize the surface of prolate spheroid given by the equation

$$\frac{x^2}{L^2} + \frac{y^2}{D^2} + \frac{z^2}{D^2} = \frac{1}{4}$$

The surface has to be divided to 500 equal panels to calculate the velocity potential on it. This means I will need to calculate the normal vector at each panel

Can anyone suggest any methods?

  • 1
    $\begingroup$ Why don't you first define some equally spread anchor points on the spheroid and then recursively subdivide them a couple of times to get the desired subdivision? $\endgroup$ – Tolga Birdal Aug 29 '16 at 16:31

A spheroid is really just a sphere that has been squashed in the different coordinate directions. So to get a mesh for a spheroid is the same as getting a mesh for a sphere: start with the latter, then do the obvious and trivial transform.

So how do you get a mesh for a sphere? If you want quadrilaterals, you start with six quads that cover the surface of the sphere and that are arranged like the faces of a cube. Then refine these by recursively replacing each quad by its four children, where you pull the new points at the midpoints of the edges and the face out onto the sphere. This gives you 24 -> 96 -> 384 -> 1536 quadrilaterals that cover the surface of the sphere.

If you want triangles, then you can either start with the four sides of a tetrahedron, or the 8 sides of an octahedron, or the 20 sides of a icosahedron. You would then recursively refine each triangle into its four children in the same way as above, until you are satisfied that you have enough triangles.


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