[help-3dldf] Re: Rotations

[Hans Aberg]

At 16:45 +0200 2004/08/17, Laurence Finston wrote:
>> My hunch is that this, finding exact intersections, is not a topic for a
>> graphics program, but a wholly independent set of programs, belonging
>> currently to the area of symbolic math and specialty programs for algebraic
>> geometry.
>
>I'll just have to do the best I can then, because I really want this
>functionality in 3DLDF.  I thought it would be a fairly elementary topic,
>but when I looked into it I discovered that intersection theory is
>anything but trivial.

Right, even for rational curves, algebraic geometers spend all their lives
figuring this out, albeit often in a more general setting. There are some
specialty mailing lists for algebraic geometry, perhaps they have been
mentioned in the newsgroup sci.math.research (the charter of this newsgroup
prohibits queries about software, but admits posting announcements of
software of interest to research mathematicians).

>> I do not see why you get so many points: I just thing of the points
>> defining the curves in say a NURBS model. These should be fairly few. These
>> are ones want to agree. Then, once this geometric object has been modelled,
>> the rendering only takes place within what is possible on the output device.
>
>Imagine one of the fancier Archimedian polyhedra in a space with
>multiple reflective surfaces, where the reflections can also be reflected.
>While smooth curves can possibly be represented by spline curves
>using a relatively small number of points, I will need a point
>for each vertex of a polyhedron, and then a point for each reflection of
>the original point.
>
>Surface hiding can also cause the number of points to
>increase rapidly, because one of the methods I will probably have
>to use requires
>objects to be broken down into smaller sub-objects.

You may want to look up ray tracing algorithms. If one wants to emulate
surfaces of different kinds, one way to do it is to assume that it is
composed of a large number of suitably randomly positioned small mirrors.
The actual ray tracing then starts from the eye of the beholder, rather
than from the light source, in order to cut down on computational time.

  Hans Aberg