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 and, as in my two previous books, the exposition is given largely by Mathematica code. There is emphasis on quadric surfaces, cubic surfaces and quartic surfaces related to the torus. The .pdf file is here

In addition to the .pdf text I have several other files. The global functions used are in the file GlobalFunctionsNS.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.

Left out of the current Surface story is material on lines in Projective Surfaces which will be covered in Section 1.10. You can download this now at Chapter 1 Section 10 (PDF). A new edition of the Surface Story is expected soon with this material and a new Chapter 5.

In Chapter 3 there are two appendices which can be used to check the work in 3.4 and 3.6. These are given only in notebook form. Appendix3A.nb and Appendix3B.nb.

**Note from author:** Originally this was to be called my *Surface Book*. However that name is a trademark owned by Microsoft. Worse, any search engine will give hundreds of links to the Microsoft product if you search for *Surface Book*. So this is instead my *Surface Story*. If you want to get back to this page through a search engine I suggest you search for * Barry Dayton Surface Story*.

Chapter 1 Basic Concepts

- 1.1 Naive Surfaces
- 1.2 Rational Parametric Surfaces
- 1.3 Implicit Equations for rational parametric surfaces
- 1.4 The Torus Story
- 1.5 Curves in Surfaces
- 1.6 Rational Points and Rational Surfaces
- 1.7 Trigonometric Parameterizations
- 1.8 Fractional Linear Transformations
- 1.9 Projective Surfaces
- 1.10 Lines in Projective surfaces through a given point (New)

Chapter 2 Quadric Surfaces in Projective Space

- 2.1 Rulings and Classification
- 2.2 Strategy
- 2.3 Degenerate Case
- 2.4 Case of a single Ruling
- 2.5 Case of no ruling
- 2.6 Case of 2 rulings
- 2.7 Rationalality of quadric surfaces
- 2.8 Transitivity of symmetries on non-singular quadric surfaces
- 2.9 Affine and Projective Symmetries of quadric surfaces

Chapter 3 The 27 Lines on a Smooth Cubic Surfaces

- 3.1 A rational cubic surface
- 3.2 Lines on a cubic surface
- 3.3 The Double 6, Theory
- 3.4 Example of construction of a Double 6
- 3.5 The 15 additional lines
- 3.6 The implicit surface containing a given double 6
- 3.7 Finding lines on a smooth cubic surface, Example 1
- 3.8 Fining Lines and Eckard points on the Clebsch Diagonal Surface

Chapter 4 Fourth degree and related surfaces

- 4.1 Geometric Point Groups and applications
- 4.2 More on the Torus
- 4.3 Gluing Surfaces
- 4.4 Breakfast with Barry.

References