Ellipsoid ↩

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created: 2022-04-19 13:09:18
modified: 2022-04-21 15:36:48

An ellipsoid is a special case of the superquadric shape when the blockiness values are both

. This way, the defining implicit equation becomes:

Where , and are the half lengths along the , and axes.

Parametrization

The parametrization is very similar to the way superquadrics are parameterized, we just set . It is just a scaled sphere:

Where and . Warning: even though the parameters are denoted with symbols used for angles, these are not angles, just surface parameters.

Volume

Since the ellipsoid is just a linearly scaled sphere in three dimensions, along the three principal axes, the volume will be the same, but multiplied with the linear scaling factors (, and ):

(See Volume of an n-dimensional sphere.)

Inverse parametrization

Also very similar to how we do it for the sphere (See Spherical coordinate system):