Shape deformation in continuous map generalization

Given a collection of regions on a map, we seek a method of continuously altering the regions as the scale is varied. This is formalized and brought to rigor as well-defined problems in homotopic deformation. We ask the regions to preserve topology, area-ratios, and relative position as they change over time. A solution is presented using differential methods and computational geometric techniques. Most notably, an application of this method is used to provide an algorithm to obtain cartograms.

Rachel WardEmail:

---

Jeff Danciger   Jeffrey received his undergraduate degree from the College of Creative Studies at UCSB in mathematics and physics. He is currently working on his Ph.D. in mathematics at Stanford University. His research interests include low dimensional topology and geometric analysis. Satyan L. Devadoss   is an Associate Professor of Mathematics at Williams College. His research interests lie in the interplay between topology and geometry, notably in applications to theoretical physics (moduli spaces and string theory) and computer science (cartography and polytopes). John Mugno   received his undergraduate degree from Williams College and is currently continuing his studies in mathematics at the University of Maryland. His areas of interest include combinatorics and topology. Don Sheehy   received his undergraduate degree in Princeton University and is currently pursuing a PhD in Computer Science at Carnegie Mellon University. His research focuses on computational geometry algorithms for meshing. Rachel Ward   received her undergraduate degree at the University of Texas at Austin. She is now a PhD student at Princeton University in the Program in Applied and Computational Mathematics. Her current work is in the area of compressed sensing, combining tools from computational harmonic analysis, probability, and optimization theory.