Using Tensor Diagrams to Represent and Solve Geometric Problems

Understanding the relation between algebra and geometry is an important tool in computer graphics as well as many other disciplines. Unfortunately the algebraic equations describing various geometric situations can get incredibly complicated. This complication can be mitigated by using a notational tool callled a Tensor Diagram. The earliest use of this notation can be traced back to Sylvester in about 1900, and it has popped up sporatically in various publications since then. It has not recieved the attention it deserves though, partly due to ambiguities in early usage and partly due to the difficulty of drawing and publishing the diagrams. With the advent of computer aids for drawing and publishing I hope to get this notation more into the mainstream of mathematics, both to simplify and to beautify calculation.

This docment contains the notes handed out at a tutorial given at Siggraph in 2002. (It is a revision of the notes for a simiar tutorial in 2001). Several chapters are reprints of earlier papers and other chapters are rough drafts of new material. The document represents a checkpoint of my understanding of the topic at that time, but is a bit dated given more recent work on the subject. It is, however, the only documentation on some of these results. I am in actively engaged in updating and revising it.

Particular errata:
The chapter on the Theorem of Pappus actually refers to Pascal’s Theorem (I always get those names confused)