TY - GEN
T1 - Tracing compressed curves in triangulated surfaces
AU - Erickson, Jeff
AU - Nayyeri, Amir
PY - 2012
Y1 - 2012
N2 - A simple path or cycle in a triangulated surface is normal if it intersects any triangle in a finite set of arcs, each crossing from one edge of the triangle to another. We describe an algorithm to "trace" a normal curve in O(min{X,n 2 log X}) time, where n is the complexity of the surface triangulation and X is the number of times the curve crosses edges of the triangulation. In particular, our algorithm runs in polynomial time even when the number of crossings is exponential in n. Our tracing algorithm computes a new cellular decomposition of the surface with complexity O(n); the traced curve appears as a simple path or cycle in the 1-skeleton of the new decomposition. We apply our abstract tracing strategy to two different classes of normal curves: abstract curves represented by normal coordinates, which record the number of intersections with each edge of the surface triangulation, and simple geodesics, represented by a starting point and direction in the local coordinate system of some triangle. Our normal-coordinate algorithms are competitive with and conceptually simpler than earlier algorithms by Schaefer, Sedgwick, and Štefankovic [COCOON 2002, CCCG 2008] and by Agol, Hass, and Thurston [Trans. AMS 2005].
AB - A simple path or cycle in a triangulated surface is normal if it intersects any triangle in a finite set of arcs, each crossing from one edge of the triangle to another. We describe an algorithm to "trace" a normal curve in O(min{X,n 2 log X}) time, where n is the complexity of the surface triangulation and X is the number of times the curve crosses edges of the triangulation. In particular, our algorithm runs in polynomial time even when the number of crossings is exponential in n. Our tracing algorithm computes a new cellular decomposition of the surface with complexity O(n); the traced curve appears as a simple path or cycle in the 1-skeleton of the new decomposition. We apply our abstract tracing strategy to two different classes of normal curves: abstract curves represented by normal coordinates, which record the number of intersections with each edge of the surface triangulation, and simple geodesics, represented by a starting point and direction in the local coordinate system of some triangle. Our normal-coordinate algorithms are competitive with and conceptually simpler than earlier algorithms by Schaefer, Sedgwick, and Štefankovic [COCOON 2002, CCCG 2008] and by Agol, Hass, and Thurston [Trans. AMS 2005].
KW - Computational topology
KW - Geodesics
KW - Normal coordinates
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U2 - 10.1145/2261250.2261270
DO - 10.1145/2261250.2261270
M3 - Conference contribution
AN - SCOPUS:84863956926
SN - 9781450312998
T3 - Proceedings of the Annual Symposium on Computational Geometry
SP - 131
EP - 140
BT - Proceedings of the 28th Annual Symposuim on Computational Geometry, SCG 2012
T2 - 28th Annual Symposuim on Computational Geometry, SCG 2012
Y2 - 17 June 2012 through 20 June 2012
ER -