Re: [PyX-devel] Re: GlyphInterchangeFormat

[Michael S. <m-s...@us...>]

Hi,

As far as I know this is what people use:

1. For the interpolation problem: Curves passing through given points:

   (A) smooth curves built by piecewise polynoms (parameterized by the
       spacial coordinates). This requires to solve quite large systems
       of linear equations -- not a problem in principle, but the
       outcome is often poor because the polynoms are too restrictive.
> IIRC Michael Schindler played with some smooth curve thru some given
> points before already. I'm not sure whether any code is laying around
> somewhere, which could be usefull in that respect. But in principle
> you're right: it might be added on top of the existing PyX.

       for the case of equal spacing this problem needs no numeric
       solver, and I am planning to implement this as a
       graph.style.curve

   (B) parametric curves. In principle one gets non-linear equations
       for the parameter values  -- a really hard problem.
       This requires a lot of information on the specific geometric
       case.

       Especially, using boundary constraints like a certain tangent
       of a curve or so is not straight-forward formulated in terms of
       a general parameterization.
       This is maybe what Magnus Lie Hetland asks for.
> > The curve construction reminds me of a feature (or set of features)
> > from MetaPost that I believe is not available in PyX, but which can be
> > quite nice... Basically, it's the spline stuff that tries to fit a
> > curve to the points you give (without you having to give the Bezier
> > contour points explicitly). Also, you can give such (very useful)
> > constraints as the direction of the curve at a given point, and the
> > spline will adapt to that. I assume this could quite easily be built
> > on *top* of PyX -- I just thought it might be nice to have it as an
> > integrated part of the path mechanism? (I don't know the details of
> > the splines used by MP/MF, but they look fine, IMO.)
       I do not know how metafont is doing that, but I think MF knows
       some generic cases and uses a kind of "optimal"
       parameterization -- depending on the problem it has to solve.
       I do not even think that it is really an interpolating program
       -- rather a mixture of interpolating and optimisation...

       Another example is finding the outline of brushes or enlarging
       arbitrarily shaped boxes.
> > And another thing... I have no idea how it's done in MP (in MF I can
> > sort of understand it) but MP allows you to use custom-made paths as
> > brushes in other parhts. Very nice for calligraphic-like looks, for
> > example. (I guess one heavy-handed way of doing it would be simply to
> > replicate the brush all along the path, with the appropriate spacing
> > calcilated somehow, perhaps even dynamically along the path.)


2. Optimization problem: Approximating curves

   (C) Mainly used are extended bezier curves that have a
       well-specified starting and ending point and try to pass
       through several other points on their way. To me this never
       looked very useful ...

Maybe there is the "quadrature of the circle": The bezier curves
depend linearly on their endpoints. For them one can build up a linear
system of equations -- again with some (linear) constraints ...
This would merge the advantages of the (ordinary) splines and the
parametric curves ...
Does anyone know the name of this method or where to find this?
I would say that this might be the most promising way to build a
general algorithm in PyX for these non-standard path operations.

Michael.

-- 
"A mathematician is a device for turning coffee into theorems"
  Paul Erdös.