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Jeff Erickson Chambers, Erin Wolf 2015-09-25T20:20:35Z 2015-09-25T20:20:35Z 10000-01-01 2008 2008 Finally, we examine a more fundamental homotopy problem in a different setting. A Rips complex is a simplicial complex defined by a set of points from some metric space where every pair of points within distance 1 is connected by an edge, and every (k + 1)-clique in that graph forms a k-simplex. We prove that the projection map which takes each k-simplex in the Rips complex to the convex hull of the original points in the plane induces an isomorphism between the fundamental groups of both spaces. Since the union of these convex hulls is a polygonal region in the plane, possibly with holes, our result implies that the fundamental group of a planar Rips complex is a free group, allowing us to design efficient algorithms to answer homotopy questions in planar Rips complexes. Made available in DSpace on 2015-09-25T20:20:35Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3337719.pdf: 1717372 bytes, checksum: 34b7aaef52a5130ba92461d1fcec1389 (MD5) Previous issue date: 2008 Embargo set by: Seth Robbins for item 83105 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs U of I Only 98 p. Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2008. http://hdl.handle.net/2142/81824 (MiAaPQ)AAI3337719 eng Computer Science Computing Interesting Topological Features text Computer Science Computer Science University of Illinois at Urbana-Champaign Dissertation Ph.D.