On the complexity of some basic problems in computational convexity: I.: containment problems
Discrete Mathematics - Special issue: trends in discrete mathematics
On the area of intersection between two closed 2-D objects
Information Sciences—Informatics and Computer Science: An International Journal
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Handbook of discrete and computational geometry
Handbook of discrete and computational geometry
Minkowski decomposition of convex polygons into their symmetric and asymmetric parts
Pattern Recognition Letters
Computing the Maximum Overlap of Two Convex Polygons Under Translations
ISAAC '96 Proceedings of the 7th International Symposium on Algorithms and Computation
Inscribing an axially symmetric polygon and other approximation algorithms for planar convex sets
Computational Geometry: Theory and Applications
Simulated annealing: Practice versus theory
Mathematical and Computer Modelling: An International Journal
Geometric optimization and sums of algebraic functions
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Geometric optimization and sums of algebraic functions
ACM Transactions on Algorithms (TALG)
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In this paper we solve the following optimization problem: given a simple polygon P, what is the maximum-area polygon that is axially symmetric and is contained in P? This problem pops up in shape reasoning when planar shapes are approximated by simpler shapes, e.g., symmetric shapes, or when they are decomposed hierarchically into simpler shapes. We propose an algorithm for solving the problem, analyze its running time, and describe our implementation of it (for the case of a convex polygon). The algorithm is based on building and investigating a planar map, each cell of which corresponds to a different configuration of the inscribed polygon. We prove that the complexity of the map is O(n^4), where n is the complexity of P. For a convex polygon the complexity is @Q(n^3) in the worst case. A substantial part of the work concentrates on calculation and analysis of arcs of the planar map. Arcs represent topological changes of the structure of the inscribed polygon, and are determined by the geometry of the original polygon. For each face of the map we calculate the area function of the inscribed polygons and look for a global maximum of the compound area function. We achieve this goal by using a numerical method.