We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with non-intersecting site-circles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperbolic polyhedra with one vertex on and the remaining vertices beyond the infinite boundary of hyperbolic space. Thus the main theorem states necessary and sufficient conditions for the existence and uniqueness of such polyhedra with prescribed combinatorial type and dihedral angles. More generally, we consider weighted Delaunay triangulations in piecewise flat surfaces, allowing cone singularities with prescribed cone angles in the vertices. The material presented here extends work by Rivin on Delaunay triangulations and ideal polyhedra.
For a comprehensive bibliography on circle packings and circle patterns we refer to Stephenson's monograph @cite_17 . Here, we can only attempt to briefly discuss some of the the most important and most closely related results.
@cite_17
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Part I. An Overview of Circle Packing: 1. A circle packing menagerie 2. Circle packings in the wild Part II. Rigidity: Maximal Packings: 3. Preliminaries: topology, combinatorics, and geometry 4. Statement of the fundamental result 5. Bookkeeping and monodromy 6. Proof for combinatorial closed discs 7. Proof for combinatorial spheres 8. Proof for combinatorial open discs 9. Proof for combinatorial surfaces Part III. Flexibility: Analytic Functions: 10. The intuitive landscape 11. Discrete analytic functions 12. Construction tools 13. Discrete analytic functions on the disc 14. Discrete entire functions 15. Discrete rational functions 16. Discrete analytic functions on Riemann surfaces 17. Discrete conformal structure 18. Random walks on circle packings Part IV: 19. Thurston's Conjecture 20. Extending the Rodin Sullivan theorem 21. Approximation of analytic functions 22. Approximation of conformal structures 23. Applications Appendix A. Primer on classical complex analysis Appendix B. The ring lemma Appendix C. Doyle spirals Appendix D. The brooks parameter Appendix E. Schwarz and buckyballs Appendix F. Inversive distance packings Appendix G. Graph embedding Appendix H. Square grid packings Appendix I. Experimenting with circle packings.