This is my start of an exposition on surfaces using Mathematica code. Click on the title above for the .pdf 13MB file. This story covers naive implicit surfaces and rational parametric surfaces in real 3 space. By naive implicit surfaces I mean surfaces given by a single polynomial equation in 3 real variables with integer or Mathematica machine number coefficients. Most of the algorithms are numerical.

In addition to the .pdf text I have several other files. The global functions used are in the file GlobalFunctionsS.nb (nb. file) This file contains also most of the functions for Plane curves and Space Curves under suffixes 2D 3D and MD. New files may have suffixes S,NS or RS for general surface functions, naive surface functions or rational surface functions respectively. Here is the index to surface Global Functions (pdf) A warning is that several of the old functions have new code.

In addition for Mathematica users, are my appendices for Section 1.9 Appendix A (.nb file), Appendix B (.nb file) and Appendix C (.nb file). These are made to be executed, not so much to be read, and are given separately.

- 1.1 Naive Surfaces
- 1.1.1 Definition and Examples
- 1.1.2 Regular and Smooth Surfaces

- 1.2 Rational Parametric Surfaces
- 1.3 Implicit Equations for rational parametric surfaces
- 1.3.1 Theoretical Method
- 1.3.2 Direct Method

- 1.4 The Torus Story
- 1.4.1 Preliminaries
- 1.4.2 The Torus

- 1.5 Curves in surfaces
- 1.5.1 Curves in Parametric surfaces
- 1.5.2 Curves in implicit surfaces
- 1.5.3 Parametric curves in implicit surfaces
- 1.5.4 Some new code for old functions
- 1.5.5 Ovals and Pseudo-lines

- 1.6 Rational points and Rational surfaces
- 1.7 Quadric Surfaces
- 1.8 Trigonometric parameterization
- 1.8.1 Quadric Surfaces
- 1.8.2 Other parametric suffaces given by trigonometry
- 1.8.3 The Klein Bottle

- 1.9 Twenty seven Lines on a cubic surface
- 1.9.1 The Double 6 configuration
- 1.9.2 Hilbert's Theory
- 1.9.3 The problems
- 1.9.4 Construction of an explicit double 6
- 1.9.5 15 additional lines
- 1.9.6 Implict equation of constructed cubic surface
- 1.9.7 Finding lines on a smooth implicit cubic
- 1.9.8 Lines and Eckart points on the Clebsch Diagonal Surface

- References

**Preview of revised Section 7 on quadric surfaces, April 2022.**
In a future version of this book Chapter 1 section 7 will be revised and enlarged and moved to Chapter 2 section 3. Here is a preview of this new section (PDF) Some material that will appear in Sections 1 and 2 of Chapter 2 may be missing but probably not critical. A preliminary GlobalFunctionsSC2.nb contains any new functions used as well as all old functions.