Apollonius

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Apollonius Tenth Problemas a Point Location
Problem

• The 2001 International Conference
• on Computational Science
• San Francisco, CA, USA
• May 28, 2001
• D.-S. Kim, D. Kim, K. Sugihara and J. Ryu
• Hanyang University, Seoul, Korea
• University of Tokyo, Tokyo, Japan

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Problem Definition
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Previous Researches

• Apollonius (B.C. 262-190)
• Viete, Newton, Euler, Gauss,
• Rokne (1991)
• Technique using Möbius trans. introduced
• Gavrilova and Rokne (1999)
• Möbius trans. used for Dynamic Voronoi Diagram
• Gavrilova and Rokne (1999)
• Analytic and Geometric solutions provided

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Basic Idea

• Shrink given three circles.
• Transform shrunken circles to W-plane via Möbius
  transformation.
• Find tangent lines in W-plane.
• Parse cases.
• Inversely transform the tangent line(s) to
  Z-plane.

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Möbius Trans. (Linear Fractional Trans.)
Z-plane
W-plane

• (a, b, c, d complex or real)
• (ad bc ? 0.)
• Maps disks conformally onto half-planes or disks,
  and vice versa.

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Möbius Trans.

• Transforms
• lines/circles through z3 in Z to lines in W
• circles/lines not through z3 in Z to circles in W
  
• infinity in Z to the origin in W

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Shrunken Circles Möbius Trans.

• Z-plane

W-plane
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W-plane
Z-plane
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Inverse Mapping
W-plane

• Z-plane

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Case Parsing via Point Location Problem
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Cases of Circumcircles
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Degeneracy I
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Cases for Degen I.
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Degeneracy II
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Cases for Degen II.
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Degeneracy III
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Applied to Voronoi Diagram
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Conclusions

• New algorithm for Apollonius 10th Problem
• All cases are explained in a single framework
• Easy to code
• Fast