On Circle Intersections by Means of Distance Geometry

  • Bertold BongardtEmail author
Conference paper
Part of the Mechanisms and Machine Science book series (Mechan. Machine Science, volume 73)


The intersection of circles appears as a frequent subproblem in the domains of computational kinematics and geometry. For this reason, the methods for computing its solutions need to be stable and simple. This paper surveys the solution method for circle intersection given by the approach of distance geometry via Cayley–Menger bideterminants. In particular, the equivalence of the squared-quantity method to its linear-quantity counterpart is shown and novel interconnections to related concepts of Non-Euclidean geometry are worked out.


Distance Geometry Cayley–Menger Bideterminants Circle Configurations Squared Distances Computational Kinematics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Bertold Bongardt. “Geometric Characterization of the Workspace of Non-Orthogonal Rotation Axes”. In: Journal of Geometric Mechanics (2014).Google Scholar
  2. [2]
    Bertold Bongardt. “Novel Plüker Operators and a Dual Rodrigues Formula Applied to the IKP of General 3R Chains”. In: Advances in Robot Kinematics. 2018.Google Scholar
  3. [3]
    Paul Bourke. Intersection of two circles. 1997.Google Scholar
  4. [4]
    James W. Cannon, William J. Floyd, Richard Kenyon, and Walter R. Parry. “Hyperbolic Geometry”. In: Flavors of Geometry (1997).Google Scholar
  5. [5]
    Anthony A. Harkin and Joseph B. Harkin. “Geometry of Generalized Complex Numbers”. In: Mathematics Magazine (2004).Google Scholar
  6. [6]
    Timothy F. Havel. “Some examples of the use of distances as coordinates for Euclidean geometry”. In: Journal of Symbolic Computation (1991).Google Scholar
  7. [7]
    Felix Klein. Elementary mathematics from an advanced standpoint: Geometry. Dover, 1939.Google Scholar
  8. [8]
    Jerzy Kocik. A theorem on circle configurations. 2007.Google Scholar
  9. [9]
    Shivesh Kumar, Bertold Bongardt, Marc Simnofske, and Frank Kirchner. “Design and Kinematic Analysis of the Novel Almost Spherical Parallel Mechanism Active Ankle”. In: Journal of Intelligent and Robotic Systems (2018).Google Scholar
  10. [10]
    Richard M. Murray, Zexiang Li, and S. Shankar Sastry. A Mathematical Introduction to Robotic Manipulation. CRC Press, 1994.Google Scholar
  11. [11]
    Josep M. Porta and Federico Thomas. “Yet Another Approach to the Gough-Stewart Platform Forward Kinematics”. In: International Conference on Robotics and Automation. 2018.Google Scholar
  12. [12]
    Nicolás Rojas and Federico Thomas. “Distance-based position analysis of the three seven-link Assur kinematic chains”. In: Mechanism and Machine Theory (2010).Google Scholar
  13. [13]
    Jon M. Selig. Introductory robotics. Prentice Hall, 1992.Google Scholar
  14. [14]
    Jonathan R. Shewchuk. Lecture Notes on Geometric Robustness. Lecture Notes. 2009.Google Scholar
  15. [15]
    Federico Thomas and Lluís Ros. “Revisiting Trilateration for Robot Localization”. In: Transactions on Robotics (2005).Google Scholar
  16. [16]
    Norman J. Wildberger. “A Rational Approach to Trigonometry”. In: Math Horizons (2007).Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Technische Universität Braunschweig, Institut für Robotik und ProzessinformatikBraunschweigGermany

Personalised recommendations