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Ptolemy's equation and kin

Katie Waddle

Abstract

Three-term relations of the form AB+CD=EF arise in multiple mathematical contexts, including the Ptolemy equation for a cyclic quadrilateral, Casey's theorem on bitangents, Penner's relation for lambda lengths, and Plücker's identity for the maximal minors of a 2x4-matrix. In this note, we explain how these different occurrences of the 3-term relation can be directly obtained from each other.

Ptolemy's equation and kin

Abstract

Three-term relations of the form AB+CD=EF arise in multiple mathematical contexts, including the Ptolemy equation for a cyclic quadrilateral, Casey's theorem on bitangents, Penner's relation for lambda lengths, and Plücker's identity for the maximal minors of a 2x4-matrix. In this note, we explain how these different occurrences of the 3-term relation can be directly obtained from each other.

Paper Structure

This paper contains 12 sections, 7 theorems, 45 equations, 7 figures.

Table of Contents

  1. Four incarnations of the 3-term relation
  2. Each equation implies the others
  3. The torus action
  4. The cross-ratio
  5. Models of the hyperbolic plane
  6. Horocycles and lambda lengths
  7. Proof of Proposition prop: scaling
  8. Generalizations and historical notes
  9. Acknowledgements

Key Result

Theorem 1

Let $A_1,A_2,A_3,A_4$ be four points arranged counterclockwise on a unit circle $\mathbf{S}$ on the Euclidean plane. For $1\le i<j\le 4$, let $d_{ij}$ denote the distance from $A_i$ to $A_j$, see Figure fig: ptolemy. Then

Figures (7)

  • Figure 1: Distances between four points on a circle
  • Figure 2: Bitangent distances between four circles tangent to a fifth circle
  • Figure 3: Hyperbolic distances between horocycles
  • Figure 4: The measurements $d_{ij}$, $t_{ij}$, and $\delta_{ij}$
  • Figure 5: The ideal points $W_1'$ and $W_2'$ used to compute the hyperbolic distance between $W_1$ and $W_2$
  • ...and 2 more figures

Theorems & Definitions (19)

  • Theorem 1: ptolemyAlmagestIntroductionMathematics2014
  • Theorem 2: caseyEquationsProperties11866
  • Theorem 3: penner_decorated_1987
  • Theorem 4: plscriptuscriptcker_new_1865
  • Proposition 6
  • Remark 7
  • Proposition 8
  • proof
  • Definition 9
  • Definition 10
  • ...and 9 more