A Generalized Malfatti Problem

Abstract

Malfatti's problem, first published in 1803, is commonly understood to ask fitting three circles into a given triangle such that they are tangent to each other, externally, and such that each circle is tangent to a pair of the triangles sides. There are many solutions based on geometric constructions, as well as generalizations in which the triangle sides are assumed to be circle arcs. A generalization that asks to fit six circles into the triangle, tangent to each other and to the triangle sides, has been considered a good example of a problem that requires sophisticated numerical iteration to solve by computer. We analyze this problem and show how to solve it quickly.

Keywords: Malfatti's problem, circle packing, geometric constraint solving, GPU programming

Downloads

Download the Paper Download the BiBTeX

Citation

Ching-Shoei Chiang, Christoph M. Hoffmann, and Paul Rosen. A generalized malfatti problem. Computational Geometry: Theory and Applications, 45(8):425--435, October 2012.

Bibtex


@article{Chiang.2012.CGTA,
  author = {Ching-Shoei Chiang and Christoph M. Hoffmann and Paul {\bf
	Rosen}},
  title = {A Generalized Malfatti Problem},
  journal = {Computational Geometry: Theory and Applications},
  year = {2012},
  volume = {45},
  pages = {425-435},
  number = {8},
  month = {October},
  abstract = {Malfatti's problem, first published in 1803, is commonly understood
	to ask fitting three circles into a given triangle such that they
	are tangent to each other, externally, and such that each circle
	is tangent to a pair of the triangles sides. There are many solutions
	based on geometric constructions, as well as generalizations in which
	the triangle sides are assumed to be circle arcs. A generalization
	that asks to fit six circles into the triangle, tangent to each other
	and to the triangle sides, has been considered a good example of
	a problem that requires sophisticated numerical iteration to solve
	by computer. We analyze this problem and show how to solve it quickly.},
  keywords = {Malfatti's problem, circle packing, geometric constraint solving,
	GPU programming}
}