Table Of Contents:
Preface xvii

Geometries for Pedestrians 1(22)

Geometries of Points and Lines 1(8)

Projective Planes 2(3)

Affine Planes 5(1)

Benz Planes--the Original Circle Planes 6(2)

Orthogonal Arrays 8(1)

Geometries on Surfaces 9(11)

(Ideal) Flat Linear Spaces 10(3)

Flat Circle Planes 13(2)

A Network of Relationships 15(1)

Interpolation and the Axiom of Joining 16(1)

Convexity 17(1)

Topological Geometries on Surfaces 18(1)

Classification with Respect to the Group Dimension 19(1)

Definitions of Frequently Used Terms 20(3)

Flat Linear Spaces 23(114)

Models of the Classical Flat Projective Plane 24(4)

The Euclidean Plane Plus Its Line at Infinity 25(1)

The Geometry of Great Circles and the Disk Model 25(2)

The Classical Point Mobius Strip Plane 27(1)

Convexity Theory 28(8)

Convex Curves, Arcs, and Ovals 31(3)

Dependence of the Axioms of Joining and Intersection 34(2)

Continuity of Geometric Operations and the Line Space 36(8)

Isomorphisms, Automorphism Groups, and Polarities 44(9)

Topological Planes and Flat Linear Spaces 53(3)

Classification with Respect to the Group Dimension 56(7)

Flat Projective Planes 57(2)

R2-Planes 59(2)

Mobius Strip Planes 61(2)

Constructions 63(40)

Original Moulton Planes 63(2)

Semi-classical Planes and Generalized Moulton Planes 65(1)

Radial Planes and Radial Moulton Planes 66(1)

Shift Planes and Planar Functions 67(3)

Arc Planes 70(6)

Skew Hyperbolic Planes 76(1)

Cartesian Planes 77(1)

Strambach's SL2(R)-Plane 78(1)

Integrated Foliations 79(1)

Different Ways to Cut and Paste 80(8)

Pasted Planes 88(10)

Semioval Planes 98(5)

The Modified Real Dual Cylinder Plane 103(1)

Planes with Special Properties 103(6)

Compact Groups of Automorphisms 103(3)

More Rigid Planes 106(1)

Differentiable Planes 106(1)

Maximal Flat Stable Planes and the First Nonclassical Flat Linear Space 107(2)

Other Invariants and Characterizations 109(7)

The Lenz-Barlotti Types 110(1)

Groups of Projectivities 111(4)

Semigroups of Continuous Lineations 115(1)

Related Geometries 116(18)

Sharply Transitive Sets 116(4)

Quasi-Sharply-2-Transitive Sets and Abstract Ovals 120(5)

Semibiplanes 125(4)

Pseudoline Arrangements, Universal Planes, Spreads 129(5)

Open Problems 134(3)

Spherical Circle Planes 137(75)

Models of the Classical Flat Mobius Plane 137(7)

The Geometry of Plane Sections 137(1)

The Geometry of Euclidean Lines and Circles 138(2)

Pentacyclic Coordinates 140(1)

The Geometry of Chains 141(2)

The Geometry of the Group of Fractional Linear Maps 143(1)

Derived Planes and Topological Properties 144(10)

Derived R2-Planes 145(1)

Affine Parts 146(1)

Continuity of Geometric Operations 146(3)

Topological Mobius Planes 149(2)

Circle Space and Flag Space 151(3)

Constructions 154(15)

Ovoidal Planes 154(3)

Ewald's Planes 157(2)

Semi-classical Flat Mobius Planes 159(4)

Different Ways to Cut and Paste 163(5)

Integrals of R2-Planes 168(1)

Groups of Automorphisms and Groups of Projectivities 169(16)

Automorphisms and Automorphism Groups 169(6)

Compact Groups of Automorphisms 175(3)

Classification with Respect to Group Dimension 178(5)

Von Staudt's Point of View--Groups of Projectivities 183(2)

The Hering Types 185(10)

q-Translations 186(2)

The Classification 188(2)

Examples 190(5)

Characterizations of the Classical Plane 195(5)

The Locally Classical Plane 196(1)

The Miquelian Plane 196(2)

The Symmetric Plane 198(1)

The Plane with Transitive Group 198(1)

The Plane of Hering Type at Least V 199(1)

Summary 200(1)

Planes with Special Properties 200(2)

Rigid Planes 201(1)

Differentiable Planes 201(1)

Subgeometries and Lie Geometries 202(6)

Recycled Flat Projective Planes 202(2)

Double Covers of R2-Planes and Flat Projective Planes 204(2)

3-Ovals 206(1)

Flocks and Resolutions 206(2)

Open Problems 208(4)

Toroidal Circle Planes 212(77)

Models of the Classical Flat Minkowski Plane 213(15)

The Geometry of Plane Sections 213(1)

The Geometry of Euclidean Lines and Hyperbolas 214(3)

The Pseudo-Euclidean Geometry 217(2)

Pentacyclic Coordinates 219(1)

The Geometry of the Group of Fractional Linear Maps 220(2)

The Geometry of Chains 222(4)

The Beck Model 226(2)

Derived Planes and Topological Properties 228(9)

Derived R2-Planes 229(1)

Affine Parts 230(1)

Standard Representation and Sharply 3-Transitive Sets 231(1)

Continuity of the Geometric Operations 232(2)

Topological Minkowski Planes 234(3)

Circle Space and Flag Space 237(1)

Constructions 237(18)

The Two Halves of a Toroidal Circle Plane 238(3)

Different Ways to Cut and Paste 241(3)

Integrals of R2-Planes 244(1)

The Generalized Hartmann Planes 244(3)

The Artzy-Groh Planes 247(4)

Modified Classical Planes 251(3)

Proper Toroidal Circle Planes 254(1)

Automorphism Groups and Groups of Projectivities 255(13)

Automorphisms 256(1)

Groups of Automorphisms 257(1)

The Kernels 258(3)

Planes Admitting 3-Dimensional Kernels 261(1)

Classification with Respect to the Group Dimension 262(4)

Von Staudt's Point of View--Groups of Projectivities 266(2)

The Klein-Kroll Types 268(7)

G-Translations 269(2)

q-Translations 271(1)

(p, q)-Homotheties 272(2)

Some Examples 274(1)

Characterizations of the Classical Plane 275(6)

The Locally Classical Plane 275(1)

The Miquelian Plane 275(1)

The Plane with Many Desarguesian Derivations 276(1)

The Plane in Which the Rectangle Configuration Closes 276(1)

The Symmetric Plane 277(2)

The Plane with Flag-Transitive Group 279(1)

The Plane of Klein-Kroll Type at Least V, E, or 21 279(1)

Summary 280(1)

Planes with Special Properties 281(2)

Rigid Planes 281(1)

Differentiable Planes 282(1)

Subgeometries and Lie Geometries 283(4)

Flocks and Resolutions 283(3)

Double Covers of Disk Mobius Strip Planes 286(1)

Open Problems 287(2)

Cylindrical Circle Planes 289(71)

Models of the Classical Flat Laguerre Plane 290(9)

The Geometry of Plane Sections 290(1)

The Geometry of Euclidean Lines and Parabolas 290(2)

The Geometry of Trigonometric Polynomials 292(1)

The Geometry of Oriented Lines and Circles 292(3)

Pentacyclic Coordinates 295(1)

The Geometry of Chains 296(3)

Derived Planes and Topological Properties 299(5)

Derived R2-Planes 299(1)

Affine Parts 300(1)

Continuity of the Geometric Operations 301(1)

Topological Laguerre Planes 302(1)

Circle Space and Flag Space 303(1)

Constructions 304(25)

Ovoidal Planes 304(2)

Semi-classical Flat Laguerre Planes 306(6)

Different Ways to Cut and Paste 312(4)

Integrals of Flat Linear Spaces 316(8)

Planes of Generalized Shear Type 324(1)

Planes of Translation Type 325(1)

The Artzy-Groh Planes 326(2)

Planes of Shift Type 328(1)

Automorphism Groups and Groups of Projectivities 329(13)

Automorphisms 329(3)

The Kernel 332(1)

Planes Admitting an at Least 3-Dimensional Kernel 333(3)

Classification with Respect to the Group Dimension 336(5)

Von Staudt's Point of View--Groups of Projectivities 341(1)

The Kleinewillinghofer Types 342(9)

C-Homologies 343(1)

Laguerre Translations 344(3)

(p, q)-Homotheties 347(2)

Some Examples 349(2)

Characterizations of the Classical Plane 351(1)

Planes with Special Properties 352(1)

Rigid Planes 352(1)

Differentiable Planes 352(1)

Subgeometries and Lie Geometries 353(5)

Recycled Flat Projective Planes 353(2)

Double Covers of R2-Planes and Flat Projective Planes 355(1)

Flocks and Resolutions 356(2)

Open Problems 358(2)

Generalized Quadrangles 360(35)

The Classical Antiregular 3-Dimensional Quadrangle 361(3)

Basic Properties 364(1)

From Circle Planes to Generalized Quadrangles and Back 365(4)

Flat Laguerre Planes 365(2)

Flat Mobius Planes 367(2)

Flat Minkowski Planes 369(2)

Sisters of Circle Planes 371(4)

Sisters of Flat Laguerre Planes 372(1)

Sisters of Flat Mobius Planes 373(1)

Sisters of (Halves of) Flat Minkowski Planes 374(1)

Flat Biaffine Planes and Flat Homology Semibiplanes 375(12)

Flat Biaffine Planes in Flat Laguerre Planes 377(3)

Flat Biaffine Planes in Flat Mobius Planes 380(1)

Flat Homology Semibiplanes in Flat Laguerre Planes 381(2)

Flat Homology Semibiplanes in Flat Mobius Planes 383(2)

Split Semibiplanes 385(2)

Different Ways to Cut and Paste 387(1)

The Apollonius Problem 388(7)

Tubular Circle Planes 395(34)

Unisolvent Sets of Functions 396(10)

Fibrated Circle Planes 397(1)

Models of the Classical Tubular Circle Planes 398(3)

Basic Properties 401(5)

Nested (Ph)unisolvent Sets and Their Circle Planes 406(14)

Unrestricted (Ph)unisolvent Sets 406(2)

Integrating Unisolvent Sets 408(1)

Nested Tubular Circle Planes 408(8)

Automorphisms of Nested Tubular Circle Planes 416(4)

Convexity and Cut-and-Paste Constructions 420(7)

Convexity 421(1)

Different Ways to Cut and Paste 422(5)

Open Problems 427(2)
Appendix 1 Tools and Techniques from Topology and Analysis 429(15)
Appendix 2 Lie Transformation Groups 444(14)
Bibliography 458(25)
Index 483