# 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

### 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}
}