Projective Geometry: From Foundations to ApplicationsThis book introduces the basic ideas of mathematical proof to students embarking on university mathematics. The emphasis is on helping the reader to understand and construct proofs and write clear mathematics. The authors achieve this by exploring set theory, combinatorics and number theory, which include many fundamental mathematical ideas. This material illustrates how familiar ideas can be formulated rigorously, provides examples demonstrating a wide range of basic methods of proof, and includes some of the all time great classic proofs. The book presents mathematics as a continually developing subject. Material meeting the needs of readers from a wide range of backgrounds is included. The over 250 problems include questions to interest and challenge the most able student but also plenty of routine exercises to help familiarize the reader with the basic ideas. |
Contents
Synthetic geometry | 1 |
12 The axioms of projective geometry | 5 |
13 Structure of projective geometry | 10 |
14 Quotient geometries | 20 |
15 Finite projective spaces | 23 |
16 Affine geometries | 27 |
17 Diagrams | 32 |
efficient communication | 40 |
42 The index of a quadratic set | 141 |
43 Quadratic sets in spaces of small dimension | 144 |
44 Quadratic sets in finite projective spaces | 147 |
45 Elliptic parabolic and hyperbolic quadratic sets | 150 |
46 The Klein quadratic set | 157 |
47 Quadrics | 161 |
48 Plucker coordinates | 165 |
storage reduction for cryptographic keys | 173 |
Exercises | 43 |
True or false? | 50 |
Project | 51 |
You should know the following notions | 53 |
Analytic geometry | 55 |
22 The theorems of Desargues and Pappus | 59 |
23 Coordinates | 65 |
24 The hyperbolic quadric of PG3 F | 69 |
25 Normal rational curves | 74 |
26 The Moulton plane | 76 |
27 Spatial geometries are Desarguesian | 78 |
a communication problem | 81 |
Exercises | 89 |
True or false? | 93 |
The representation theorems or good descriptions of projective and affine spaces | 95 |
32 The group of translations | 104 |
33 The division ring | 110 |
34 The representation theorems | 116 |
35 The representation theorems for collineations | 118 |
36 Projective collineations | 126 |
Exercises | 133 |
True or false? | 136 |
Quadratic sets | 137 |
Exercises | 175 |
True or false? | 178 |
You should know the following notions | 179 |
Applications of geometry to coding theory | 181 |
52 Linear codes | 185 |
53 Hamming codes | 191 |
54 MDS codes | 196 |
55 ReedMuller codes | 203 |
Exercises | 208 |
True or false? | 211 |
You should know the following notions | 212 |
Applications of geometry in cryptography | 213 |
62 Enciphering | 216 |
63 Authentication | 224 |
64 Secret sharing schemes | 233 |
Exercises | 240 |
Project | 242 |
You should know the following notions | 243 |
| 245 | |
Index of notation | 253 |
| 255 | |
Other editions - View all
Projective Geometry: From Foundations to Applications Albrecht Beutelspacher,Ute Rosenbaum No preview available - 1998 |
Common terms and phrases
2-lines 3-dimensional projective space a₁ affine geometry affine plane affine space algorithm arbitrary point assertion attacker authentication system axiom axis H B₁ bijective called central collineation centre codeword coding theory collinear consider contains cryptography define Definition denote diagram distinct points division ring elements enciphering exactly Figure finite projective space fixed follows g and h g₁ g₂ geometry of rank Ham(r Hamming code Hence homogeneous coordinates hyperbolic quadratic set hyperplane at infinity hyperplane H incident induced integer intersects Lemma Let g line g linear code Moulton plane nodes nondegenerate quadratic set number of points order q plaintext plane of order Plücker coordinates point of H point Q points and lines precisely projective geometry projective plane Proof prove quadric quotient geometry rank 2 geometry Reed-Muller code regulus secret sharing schemes set of points space of dimension span subset t-dimensional subspace theorem of Desargues U₁ vector space