Curves, Trees and Surfaces: Applications to Mappings and Meshes,
I will show that any closed curve in plane is encoded by its medial axis, a tree consisting of the centers of all disks enclosed by the curve whose boundary hits the curve in at least two points. For a polygon this is a finite tree. The medial axis gives rise to the iota map from the curve to the unit circles, and this map extends to an 8-QC map of the interiors, independent of the curve. This is really a theorem about surfaces in hyperbolic 3-space: the iota map is extends to a conformal map of the dome of the curve to the unit disk, a fact due to Thurston, Sullivan, Epstein and Marden. We then discuss various applications, including fast numerical conformal mapping and optimal quad-meshing and triangulation of polygonal domains.
Zoom recording: https://www.math.stonybrook.edu/~bishop/lectures/GMT20250530-133351_Recording_3840x2280.mp4
Slides: https://www.math.stonybrook.edu/~bishop/lectures/Berlin.pdf
Event Timeslots (1)
Friday
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