See section 3.6 in Part II for information on this graphic.

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.

NEW The second edition is divided into 3 parts for the convenience of readers. The first part covers basic concepts and previews the rest of the book. Most readers will want to read, or at least look at this first. The second part is only for those who want further information on Chapters 2,3 and 4. The third part is Chapter 5 which covers topology of projective hyperboloids with application to the topology of complex projective plane conics and cubics. Part III is independent of Part II, in fact those who are already familiar with the subject of Part III may go directly there.

I do not plan to give Mathematica Notebook versions here. Those who want to try out the code should download my GlobalFunctionsNS.nb Mathematica notebook file dated July 2023 or later. All the global functions you need are here from all parts of this book. This file contains a normal form algorithm for conics and a corrected normal form algorithm for cubics for my Plane Curve Book. In conjuction with this there is an Index to GlobalFunctionsNS (PDF) .

**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
- 1.11 Overview of Book

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.

Chapter 5 Topology and Complex Plane Curves

- 5.1 Chromatic Number
- 5.2 Global Functions for Chapter 5
- 5.3 Topology of Projective Hyperboloids
- 5.4 Normal Form for complex Plane Conics
- 5.5 Normal Form for complex Plane cubics
- 5.6 Topology of Complex Projective solution Space of Plane Conics
- 5.7 Topology of Complex Projective solution Space of Plane cubics

References