"Bald Eagle" <cre### [at] netscapenet> wrote:
>...
> Given that a Bezier patch can be interpreted as a Bezier spline in one dimension
> whose control points slide along Bezier splines oriented in the other dimension,
> and that this can be expressed as a parametiric equation, then
>
> It ought to be possible to define a parametric {} object in POV-Ray based on a
> set of control points and the basis functions of the Bezier splines.
>
> Then, given a point <m, n> (or <u, v>) on the surface, one could place things ON
> the surface of the patch.
>
> This should also allow the generation of _any_ spline between the ordinate /
> cardinal splines defined by the corners.  In theory one should be able to
> simulate a patch by juxtaposing a series of splines from one side to the other
> in either dimension.

If you calculate the two partial derivatives (d/dU and d/dV) for each of the
bivariate functions for a Bezier patch you can use these to reorient objects to
align with the U-, V- and normal directions of the surface.

This article contains some useful information about that:

https://www.scratchapixel.com/lessons/advanced-rendering/bezier-curve-rendering-utah-teapot/bezier-patch-normal

The attached image shows cubes that are reoriented to align with the directions
I mentioned above. It also shows a lot of splines that creates a grid on the
surface of the patch.


>...

--
Tor Olav
http://subcube.com

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