Morphing

Thickening

A three-dimensional grid in space comprises the x-, y- and z-coordinates in a parametric equation. Imagine a sphere within this three-dimensional grid. On the sphere itself is another grid, formed by the u- and v-parameters. In order to give a sphere thickness, an additional parameter must be introduced. The w-parameter is like an arrow coming off the surface of a sphere. It relates to w as z relates to x and y (or, to put it a different way, w is like the z-coordinate of u and v).

Because a sphere with thickness has two boundary conditions, an inner and an outer, two different sets of x, y and z must be established. One set describes the inner boundary of the sphere and the second defines its thickness (and therefore implicitly also ...