Parametrization of distorted and dented ellipsoids

Question

My program uses a lot of ellipsoid shaped polygons in 2 and 3 D. So far, I create them by the simple, well-known parametrization about two angles.

Now I'd like to have my ellipses a bit dented and distorted. Currently, I just need some sort of non-perfect ellipsoids, later I'd like to tune it to realisitic production defects. The ellipsoids should roughly look like they were drawn/modeled by a not-so-talented child or like they were made of thin aluminium and someone dented them with its fingers. So, the size of dents is roughly in the order of 1/10 to 1/5 of the ellipsoid size.

Are there any parametrizations to get distorted and dented ellipsoid? Or does this problem have a name to find literature about it?

Answer 1

You can distort the elliposid uniformly by applying a linear map ($\mathbf{F}$, a 3x3 matrix in 3D) such that $$ \mathbf{x}_{\text{distorted}} = \mathbf{F} \cdot \mathbf{x} $$ If the map depends on position ($\mathbf{x}$) you can get more complex deformations. You could search for such maps in the literature but it will probably be easier to find what you want by trial and error.

For more physically based, but computationally tractable approaches, see the paper "Morphology and interaction between lipid domains" by Ursell et al., PNAS, 2009 Aug 11; 106(32): 13301–13306.