A curve in 4-space

Volume 2: A Wolfram Language approach to Real Algebraic

Space Curves

Space curves in `n`-dimensional affine or projective space present a major challenge that we did not need to deal with for plane curves, instead of a single equation a system of `n`-1 or more equations are needed. This system is far from unique and, in many cases, may be over-determined. Since I allow numerical coefficients and, in general, numerical over-determined systems are to be avoided, specical methods must be used.

The first two Chapters will discuss *naive* space curves in 3-space, that is, curves defined by a system of two equations in 3 variables. Not only is this case familiar to what we learned in multivariable calculus but many methods in the plane curve case such as critical points and path tracing are still available.

The next few chapters develop the methods from numerical linear algebra that will be needed, Macaulay and Sylvester matrices and duality. The basic definitions and techniques for the general case will be established.

The last few chapters will cover, as examples, various situations I have recently presented in papers and talks. These are listed at the end of the left sidebar of this webpage.

For more information, a here is a 30 page PDF summary of this volume. In addition here is a Mathematica 11.2 notebook with code and examples for the global functions used in this book along with an alphabetical list of the fuctions (PDF) with their syntax.