Re: 3DLDF

[Laurence Finston]

On Sat, 14 Aug 2004, Hans Aberg wrote:

>
> And b-splines are (as opposed to splines or Bexier curves)? And what
> non-rational and uniform mean here?
>

"B" stands for "beta", but I don't know what "beta" refers to.
B-splines are a bit more complex than
Bezier curves, which are also spline curves.  Beyond this, I
don't know the answers off the top of my head.  I believe
b-splines use "weights" in addition to control points, and
possibly an additional property.  I
believe non-rational and uniform refer to characteristics of
the blending functions used to generate the curves.  If
memory serves, there are two different but equivalent ways
of generating spline curves, one of which uses blending
functions.  Sorry, I haven't been working on this lately, so
it's not fresh in my mind.

Good sources of information on this topic are Huw Jones,
_Computer Graphics through Key Mathematics_ and
David Salomon, _Computer Graphics and Geometric Modeling_.
There's also a book called _The NURBS Book_ by Les Piegl and
Wayne Tiller.

NURBS are standard in 3D modelling because of their property
of projective affinity.  In 2D graphics, Bezier curves are
used frequently, but usually not higher order ones, because
acceptable results can be achieved by smoothly blending multiple
Bezier curves of lower order.  Jones goes into this.
I don't remember what order is generally used, 3 or 4 maybe?
Metafont uses Bezier curves with 4 control points, but I
don't know what the relationship is between order and the
number of control points.

>
> It sounds as you are making a program for archaic computers :-): Most
> computers, PC and up, have a FPU which computes with 64-bit IEEE floats
> (typically a "double" in C on a 32-bit computer), and memory is not anymore
> an issue in normal use. And one is now shifting to genuinely 64-bit CPU's
> on the PC level (as the Mac G5).
>

I'd like to implement rendering within 3DLDF, someday
even real-time rendering, so I do have to consider run-time
and memory constraints.  I'd also like it to be runnable on
less powerful platforms.  I believe this is in the spirit of
the GNU Coding Standards.

>
> The quaternions were created in order to describe 3D-rotations, but the
> underlying math will be the same as if you work with matrices. So I am not
> sure you want to get out there.

I thought it might at least help me to spread the error more
evenly.  I'd really appreciate it if you (or anyone else)
could suggest something that might work better.

> And quaternions do not work in dimensions >
> 3. (There is an analogue in 7 dimensions; its 8-dimensional Cayley algebra
> is not associative. But there is no analogue in other, higher dimensions.)
>

This isn't that important at the moment.

Laurence