On Tue, 04 Jan 2000 14:01:29 -0800, Ken wrote:
>Ken <tyl### [at] pacbellnet> wrote:
>
>> If someone knows of some existing code that
>> I could look at or a research paper or two on finding parallels to
>> quadratic curves I would appreciate the help.
>
>I've posted details of my methods for approximating parallel Bezier curves
>in earlier threads this past year.  Summary: since you know the locations of
>the parallel endpoints and midpoint, as well as the slope at start and finish,
>you can compute the locations of the control point(s).  This is a closed form
>solution that works well in "easy cases".

That isn't even close to truly parallel, even for "easy cases."  There are much
better approximations that can be made; I have an article around here somewhere
that talks about it.  Besides, we're not talking about Bezier curves here.

>The general problem is much more difficult:  (1) If the parallel distance
>exceeds the radius of curvature of the parent curve, the parallel curve will
>kink.  This is going to happen *often* in parallels to glyph shapes.
>(2) Parallels to glyphs shapes will often self-cross, elevating the problem
>into constructive planar geometry of curved regions. (3) Etc.  I have
>extended solutions for these cases but they're too involved to describe here.

This, at least, is correct.

>You may want to consider polygon approximations rather than the native curves.
>The computation is slower but it is already solved.

This is what I expected to hear anyway.  All is not lost, though - it's not 
that hard to approximate the bevels with a bunch of smooth_triangles.  It's
just a question of whether it's worth it to put something like that in POV
when there are already external programs that do it so well.

-- 
These are my opinions.  I do NOT speak for the POV-Team.
The superpatch: http://www2.fwi.com/~parkerr/superpatch/
My other stuff: http://www2.fwi.com/~parkerr/traces.html

Post a reply to this message