Computational complexity of combinatorial surfaces
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Constructive Whitney-Graustein theorem: or how to untangle closed planar curves
SIAM Journal on Computing
Determining contractibility of curves
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
Shape Matching: Similarity Measures and Algorithms
SMI '01 Proceedings of the International Conference on Shape Modeling & Applications
Comparison of Distance Measures for Planar Curves
Algorithmica
Area-preserving approximations of polygonal paths
Journal of Discrete Algorithms
Journal of Mathematical Imaging and Vision
Fréchet distance for curves, revisited
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
Exact algorithms for partial curve matching via the Fréchet distance
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
The Computational Geometry of Comparing Shapes
Efficient Algorithms
Homotopic Fréchet distance between curves or, walking your dog in the woods in polynomial time
Computational Geometry: Theory and Applications
SIAM Journal on Discrete Mathematics
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
The frechet distance revisited and extended
Proceedings of the twenty-seventh annual symposium on Computational geometry
Jaywalking your dog: computing the Fréchet distance with shortcuts
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
How to walk your dog in the mountains with no magic leash
Proceedings of the twenty-eighth annual symposium on Computational geometry
Measuring similarity between curves on 2-manifolds via homotopy area
Proceedings of the twenty-ninth annual symposium on Computational geometry
Measuring similarity between curves on 2-manifolds via homotopy area
Proceedings of the twenty-ninth annual symposium on Computational geometry
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Measuring the similarity of curves is a fundamental problem arising in many application fields. There has been considerable interest in several such measures, both in Euclidean space and in more general setting such as curves on Riemannian surfaces or curves in the plane minus a set of obstacles. However, so far, efficiently computable similarity measures for curves on general surfaces remain elusive. This paper aims at developing a natural curve similarity measure that can be easily extended and computed for curves on general orientable 2-manifolds. Specifically, we measure similarity between homotopic curves based on how hard it is to deform one curve into the other one continuously, and define this "hardness" as the minimum possible surface area swept by a homotopy between the curves. We consider cases where curves are embedded in the plane or on a triangulated orientable surface with genus $g$, and we present efficient algorithms (which are either quadratic or near linear time, depending on the setting) for both cases.