Solving the problem of Apollonius and other related problems
Graphics Gems III
Spatial tessellations: concepts and applications of Voronoi diagrams
Spatial tessellations: concepts and applications of Voronoi diagrams
Swap conditions for dynamic Voronoi diagrams for circles and line segments
Computer Aided Geometric Design
Voronoi diagram of a circle set from Voronoi diagram of a point set: topology
Computer Aided Geometric Design
Voronoi diagram of a circle set from Voronoi diagram of a point set: geometry
Computer Aided Geometric Design
Finding an Euclidean anti-k-centrum location of a set of points
Computers and Operations Research
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Given a set of three circles in a plane, we want to find a circumcircle to these given circles called generators. This problem is well known as Apollonius Tenth Problem and is often encountered in geometric computations for CAD systems. This problem is also a core part of an algorithm to compute the Voronoi diagram of circles. We show that the problem can be reduced to a simple point-location problem among the regions bounded by two lines and two transformed circles. The transformed circles are produced from the generators via linear fractional transformations in a complex space. Then, some of the lines tangent to these transformed circles corresponds to the desired circumcircle to the generators. The presented algorithm is very simple yet fast. In addition, several degenerate cases are all incorporated into one single general framework.