Draft talk:Cayley-Menger relations

Article summary.

This page is one in a collection of pages on rigidity theory, distance geometry, and configuration spaces of flexible frameworks. These pages were planned and organized by Dr. Meera Sitharam at the University of Florida and her class of 20+ graduate students during the Spring 2020 semester. Below is the list of contributors from this class. Several of these pages were also heavily edited by William Sims for his PhD qualifying exam under Dr. Meera Sitharam. These pages will continue to be revised over the next 6 months by experts, post-docs, and graduate students participating in the Fields Institute's Thematic Program on Geometric Constraint Systems, Framework Rigidity, and Distance Geometry. Below is a diagram of these pages (colored) showing how they connect to each other and to larger areas (gray).

Usernames/names of contributors from Dr. Meera Sitharam's Spring 2020 graduate course by page:

Automated geometry theorem proving - Alexcooper

Cayley configuration space - Ajha, William Sims, Yichi Zhang

Decomposition Recombination Planning - Rahul Pabhu

Distance geometry - Micstein, William Sims

Distance geometry: Cayley Menger Relations - Abhik18

Geiringer-Laman theorem - William Sims, Vriddhipai

Geometric constraint solving - Kyleshihhuanglo, Rahul Prabhu

Geometric constraint system - Cwphang, Kyuseopark, Rahul Prabhu, William Sims

Geometric rigidity - W.garcia, William Sims

Graph flattenability - William Sims

Pebble game - Adityatharad

Quadratic solvability - Yichi Zhang

Sparsity matroid - Banouge, William Sims, Vriddhipai

Structural rigidity - William sims, Tgandi

Tree-decomposable graph - Bhattabhishek, William Sims

William Sims (talk) 23:42, 25 January 2021 (UTC)

Hidden Comments Copied from Draft Page

My personal comments on the page and your help: I have read your feedback and read the notes you attached in the link below in your 2nd review of my page. I have linked this topic to a couple more pages and did the best I could without adding ambiguity to the page by bringing in new topics like Hilda-Laman graphs and such. I believe it is concise and does not deviate from Cayley-Menger at all and I have linked realizing the EDM algorithm to Cayley-Menger. One formula and one matrix below do not render but I linked a latex compiler below here in the raw text in which they appear. Thank you for all your help

Instructor 2nd: The sphere intersection algorithm and theorem are out of place. It is overkill because it works when d is not given. In your problem, d is given. Also, you haven't mentioned how that algorithm has anything to do with Cayley Menger determinants. You need to look at the lecture notes 1-6 of https://www.cise.ufl.edu/~sitharam/COURSES/GC/GCgood/gc.html where a connection is drawn between Cayley menger determinants and a simpler sphere intersection algorithm. It also shows how a much smaller subset of entries of a distance matrix and their Cayley menger determinants being zero and nonnegative automatically determine what the remaining entries will be, by using this algorithm to determine all the point coordinates and then computing the remaining distance entries of the matrix. The full generality of the algorithm you currently have is more appropriate for the Schoenberg section, both for determining if a matrix is a distance matrix and also for completing a partial distance matrix - when the rank or dimension d is not given.

Michael Rothstein Critique 2 This page is very well constructed and organized. From what I can tell it is just about done and most requirements are met. You seem to have collected a good set of references, some of which I used myself. A good job is done explaining every section. The "Cayley Menger Determinient" section is quite long and might be better broken into 2-3 sections. Overall, the technical information seems accurate and knowledgeable and the page looks clean.