BGimeno <bru### [at] gmailcom> wrote:

> > Le_Forgeron <jgr### [at] freefr> wrote:
> >> If I got the problem exposed in "Smoothing bicubic_patchs, A pain":
> >> * you have four non-coplanar points, each pair of the external 4 sided
> >> figure connected by cubic splines (each pair is adjacent on its spline,
> >> no intermediate point on the spline)
> >>
> >>
> >> Challenge: have a nice surface which joins each splines.
> >>
> >>
> >> For start, an illustration of a possible setting.
> >
> > Yes, and now if you iteratively made that smaller, you'd approach the true
> > surface.  I think with smooth_triangles it would look fairly decent.
> >
> > How you subdivide such rectangles is the challenge, because you have to
> > interpolate along the splines.
> >
>
> http://hof.povray.org/btfh02.html
>
> This is an image that someone made with the first version of my macro
> years ago. Unfortunately, real life has kept me away from this project
> and I did not continue with it, I have left the project abandoned for
> quite some time. In that image smooth_triangle is used and the mesh is
> made from cylinders and spheres. I'm going to try to subdivide the mesh
> and see how it looks. I'm rewriting from scratch and there's still a
> long way to go to implement tessellations like that. Patience.
>
> Bruno Gimeno

I think what you are looking for, given a topologically rectangular array of
points, with coordinates that are in some way geometrically rectangular,
cylindrical or toroidal, is to generate a mesh2 of smooth triangles,
interpolating the points of the grid, with a high apparent level of continuity.
I doubt that an intermediate stage involving Bezier quadrilaterals is the way to
go.

I would think about using Hermite cubics to form the gridlines and Coons patches
with Hermite, rather than linear, interpolation for the surface, though for the
cylindrical and toroidal geometry, I think you would need to specify a curvature
at the joins. Most of the big commercial modellers seem to use NURBS for complex
surfaces so I'm not sure how widely such an idea has been implemented. You need
to solve a tridiagonal set of equations to generate the Hermite cubics,  but
that can be done with for loops.

This should produce e.g a generalised approximate torus using as few as 16
co-ordinates, plus another 8 numbers specifying curvature at the joins.

If you think the idea is interesting, I would do some work on it.

JimT

Post a reply to this message