Computational Geometry for Shape Matching and Shape
Approximation  Literature and Supplementary Material
Sequence of five lectures given by Günter Rote as part of the
2nd Winter School on Computational Geometry,
Amirkabir University of Technology, Tehran,
March 2–6, 2010.
General Literature
H. Alt and L. J. Guibas. Discrete
geometric shapes: Matching, interpolation, and approximation. In
J.R. Sack and J. Urrutia, editors, Handbook of Computational
Geometry, pages 121–153. Elsevier Science Publishers
B.V. NorthHolland, Amsterdam, 2000.
doi:10.1016/B9780444825377/500048
Congruence and similarity of geometric figures (Day 1)
The most basic question in shape matching is to test whether two
objects are equal, possibly after translation, rotation, scaling, or
other geometric transformations. We show how to test whether two point
sets in the plane or in space can be tested for exact congruence.

H. Alt, K. Mehlhorn, H. Wagener and E. Welzl, Congruence, similarity
and symmetries of geometric objects, Discrete Comput. Geom. 3
(1988), pp. 237–256,
doi:10.1007/BF02187910
 M.D. Atkinson, An optimal algorithm for geometrical congruence, J. Algorithms 8 (1987), 159–172.
doi:10.1016/01966774(87)900368

K. Sugihara, An n log n algorithm for
determining the congruity of polyhedra. J. Comput. System Sci. 29 (1984), 36–47.
doi:10.1016/00220000(84)900114
Further Reading

Peter Braß, Christian Knauer: Testing congruence and symmetry
for general 3dimensional objects. Comput. Geom. 27(1): 311
(2004).
doi:10.1016/j.comgeo.2003.07.002
 The fastest known congruencetesting algorithm in higher
dimensions, as of 2009:
Peter Braß, Christian Knauer: Testing the Congruence of dDimensional Point Sets. Int. J. Comput. Geometry Appl. 12(12): 115124 (2002).
doi:10.1142/S0218195902000761
The Hausdorff distance between geometric figures (Day 2)
The Hausdorff distance is the most basic measure for comparing two
point sets. We will discuss how the Hausdorff distance between two
plane geometric objects can be computed efficiently, using the
planesweep technique and Voronoi diagrams of sets of line segments.
 Helmut Alt, Bernd Behrends, Johannes Blömer: Approximate
Matching of Polygonal
Shapes. Ann. Math. Artif. Intell. 13(34): 251265
(1995). doi:10.1007/BF01530830
The Fréchet distance (Day 3)
The Fréchet distance is a natural distance between curves that does
not just regard them as point sets but takes their order into
account. We will see how the Fréchet distance can be computed.
 H. Alt and M. Godau.
Computing the Fréchet distance between two polygonal curves.
Internat. J. Comput. Geom. Appl. 5:75–91, 1995.
doi:10.1142/S0218195995000064
Further References

Helmut Alt, Christian Knauer, and Carola Wenk.
Matching polygonal curves with respect to the
Fréchet distance.
in: Proc. 18th Annual Symp. Theoret. Aspects of Computer Science (STACS 2001), Dresden, Germany, February 15–17, 2001,
A. Ferreira and H. Reichel (eds.), LNCS 2010, pp. 6374,
2001.
doi:10.1007/3540446931

Günter Rote:
Computing the Fréchet distance between piecewise smooth curves.
Computational Geometry, Theory and Applications 37 (2007),
162174. (Special issue for the 20th European Workshop on
Computational Geometry)
doi:10.1016/j.comgeo.2005.01.004

Helmut Alt, Alon Efrat, Günter Rote, and Carola Wenk: Matching planar
maps. Journal of Algorithms 49 (2003), pp. 262283. doi:10.1016/S01966774(03)000853.
See also: Carola Wenk, Helmut Alt, Alon Efrat, Lingeshwaran
Palaniappan, and Günter Rote: Finding a curve in a map (Video). in:
Video and Multimedia Review of Computational Geometry, Proceedings of
the Nineteenth Annual Symposium on Computational Geometry, San Diego,
June 810, 2003. Association for Computing Machinery, 2003,
pp. 384385.
 Kevin Buchin, Maike Buchin, Christian Knauer, Günter Rote and Carola
Wenk.
How difficult
is it to walk the dog?
In: Abstracts of the 23rd European Workshop on Computational Geometry, Graz,
March 2007, pp. 170–173.
Convex Approximation (day 5A)
A basic data processing task is to replace a complicated geometric
object by a simpler approximation. We will discuss and analyze the
Sandwich algorithm for a simultaneous inner and outer approximation of
convex plane shapes by polygons.
Further References
 Survey article: E. M. Bronstein. Approximation of convex sets by polytopes (in
Russian).
Sovremennaya Matematika. Fundamental'nye Napravleniya (Contemporary Mathematics.
Fundamental Directions), Vol. 22, Geometry, 2007.
English translation:
Journal of Mathematical Sciences, Vol. 153, No. 6, 2008,
727–762.
 G. K. Kamenev.
The Initial Convergence Rate of Adaptive Methods
for Polyhedral Approximation of Convex Bodies
(in Russian).
Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 2008, Vol. 48, No. 5, pp. 763–778.
English translation: Computational Mathematics and
Mathematical Physics, 2008, Vol. 48, No. 5, pp. 724–738.
 G. K. Kamenev.
Duality Theory of Optimal Adaptive Methods for Polyhedral
Approximation of Convex Bodies
(in Russian).
Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 2008, Vol. 48, No. 3, pp. 397–417.
English translation: Computational Mathematics and
Mathematical Physics, 2008, Vol. 48, No. 3, pp. 376–394.
Pseudotriangulations (Day 5B)
Pseudotriangulations are a nice geometric structure and also a useful
data structure. We will review the basic definitions and properties of
Pseudotriangulations, and see how they arise in applications such as
ray shooting, kinetic collision detection, lower approximation and
locally convex hulls.

Slides
 Günter Rote, Francisco Santos and Ileana Streinu: Pseudotriangulations
— a survey.
In: Surveys on Discrete and Computational GeometryTwenty
Years Later. Editors: Jacob E. Goodman, János Pach
and Richard Pollack, Contemporary Mathematics, Band 453,
American Mathematical Society, 2008, pp. 343–410. arXiv:math/0612672 [math.CO]
Günter Rote
Last modified: Mon Mar 15 16:27:36 CET 2010