Re: [PyX-devel] Re: [PyX-user] Some geometry thoughts

[Andre W. <wo...@us...>]

Hi,

On 27.07.04, Magnus Lie Hetland wrote:
> > >     Equal(b, a + Point(10, 0))
> > 
> > I just couldn't resist: A linear equation solver can be easily build
> > like this.
> 
> I know -- I've thought about this myself in the past, and I've even
> found code that does this sort of thing. I don't remember exactly
> where, but it had a variable class and did all the linear equations
> behind the scenes.
> 
> Also, numarray (and Numeric) has support for the equation solving
> part; I guess they could be optional mechanisms for speed, if this is
> indeed the way to go.

BTW: I didn't wrote a linear equation solver, but what I did was
exactly a "variable class" and the "linear equations behind the
scenes", i.e. building up linear terms. To actually solve the linear
equations I used Numeric in my sample code already.

> Certainly. It would also be neat to be able to use parametrized curves
> in the same system, as in MetaPost; but I guess it's not quite that
> critical, as you already have support for finding intersections
> in PyX.

Finding intersections etc. is usually a nonlinear operation. But you
can define points by that ... sure. I guess it will be similar to
MetaPost in that respect (though I'm not quite sure right now).

> > My grasp tells me, that a transformation contains another set of
> > variables (6 for a 2d->2d affine transformation). When multiplying
> > with a (lazy) point (two variables) the system becomes non-linear.
> > I'm kind of confused at the moment ... ;-)
> 
> Hm. No, I don't think so... If you say that one point, transformed by
> an affine transform, equals another point, all you have is a linear
> set of two equations in two variables -- no?

Say p and p' are (2d) points, and t is a transformation (i.e. a 2x2
matrix + a 2d vector). Then you can write

p' = t(p)

Once you say p' and p are lazy vectors (i.e. they contain 2 variables)
as well as t is a lazy transformation (i.e. it contain 6 variables),
than the equation above is non-linear. There are non-linear terms in
matrix times vector when they both contain variables.

So I think what one has to do is to postpone this non-linear equation
until the non-linear terms are solved by other constraints. I guess,
thats what MetaPost does as well ... as soon as p or t gets defined
somewhere else, the equation above becomes linear.

> Anyway, writing a system like this based on linear equations is a bit
> more work than a one-way function-based one. I'll have a look but I
> can't promise anything.

Sure, nobody does. But we might have a starting point now ... and
maybe somebody is really interested in this. Its not top priority for
me, but well, on the other hand, I already like it ... we can get
something really funny quite quickly I guess.


André

-- 
by  _ _      _    Dr. André Wobst
   / \ \    / )   wo...@us..., http://www.wobsta.de/
  / _ \ \/\/ /    PyX - High quality PostScript figures with Python & TeX
 (_/ \_)_/\_/           visit http://pyx.sourceforge.net/