Thanks for the references; I'll check them out.
Re conformal maps: continued fractions/linear fractional transformations of the complex plane do some of what might be needed.
The GIS (Geographical Information Systems) and AI/Robotics/Image understanding people do worry about planar region representations. The GIS folks have to worry about non-simple regions with holes and boundaries of arbitrary complexity.
There are some folks looking into "topological logical theories" (TopoLogicLogic = TopoLogic^2 ?!?) that attempt to develop representations which abstract the metric part away from the connectedness/inclusion/topology part. I don't think that they have been terribly successful to date; I think most of their theories have too much computational complexity to be practical. I suppose that knot theory is a non-planar version of some of the same ideas.
My way-too-shallow Google search seems to indicate that good "interval-like" representations for the complex plane would be an interesting PhD thesis topic even today.
One would have to look carefully at existing representations for numerical integrations & other types of numerical analysis applications.
There are already simplistic representations of root loci in terms of unions of circular discs, strips, rectangular regions, etc.
If someone really could develop representations that could handle some fractal applications, that would be terrific!
At 06:08 PM 2/4/2013, David Stoutemyer wrote:
>On-line documentation of the Interval [...] function is at
>http://reference.wolfram.com/mathematica/ref/Interval.html?q=Interval&lang=en
>
>Conceivably one of the Mathematica conference proceedings at
>http://library.wolfram.com/infocenter/Conferences/
>contains additional information. Some of the articles are notebooks, but a free player is downloadable from
>http://www.wolfram.com/products/player/?src=google&136++mathematica+reader&gclid=CJi20JSHnrUCFQhyQgod3CQAGg
>
>Describing 2D regions brings to mind the Swartz-Christoffel transformation, which maps any (closed?) region in the complex plane bounded by straight lines and circular arcs to a half plane.
>
>Perhaps starting with the topmost rightmost point, one could describe such a closed region canonically as a sequence of unions and intersections of half-planes, interiors of circles, discs, and exteriors of discs. Traversing the given boundary in counterclockwise order, use a union for every right turn and an intersection for every left turn. Perhaps also multiply-connected regions could be handled by traversing slits into around then back out of holes. Disjoint regions could be handled by a final union in a lexically determined order. Perhaps an attached Boolean value for each segment could indicate whether the segment is included with the area to its left. Segments where this information is piecewise could be split into smaller segments where it is uniform.
>
>Does Cylindric Algebraic Decomposition have any contribution for a canonical description?
>
>Does Computation Geometry address this issue?
>
>But yes, the first priority is to implement your initial proposal.
>
>-- best regards, david stoutemyer