A curve in 4-space


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, special methods must be used.

There is a PDF summary of my current work on space curves.

I first 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.

Then I 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 then be established.

The last part will cover, as examples, various situations I have recently presented in papers and talks.

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.