Wednesday, May 20, 2009

skeletons on non-planar input



three hills

The standard straight skeleton algorithm relies on a sweep plane. By adding points in at different heights we should be able to grow nice houses on hills. We should also be able to preserve semantic data from the skeleton. The naive method has some problems:

naive

It's not too clear in the above, but lower points and edges can intersect with higher edges, because of changes to the points after events. There's a couple of ways out of this - we could say "user beware" and run away, or we could automate something. Turns out this isn't so hard -

[edit: turns out this is hard - it relies on intersecting two parallel vertical planes to a single line, gah! but given a topological approach to the skeleton algorithm, it should eventually work]

robust

We project subterranean edges with a vertical gradient until they come to the surface. this guarantees that the current polygon on the sweep plane.

On a different track, there's a lovely idea of using non-planar sweep planes, such as fields, to define some really interesting geometries. As long as the sweep plane never intersects itself at a previous time, the geometry will remain non-intersecting. If we bound the sweep surface (no polygons allowed outside the limits), or define a Poincare disk/Escher -like vanishing horizon on the surface we could get some really interesting fields. These might be suitable for modelling growth in plants (or animals...?). I'd better find that book on differential geometry.

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