Preface	5
	Contents	9
	1 Linear Algebra Topics	15
	1.1 Introduction	15
	1.1 Introduction	15
	1.2 Lines	16
	1.3 Angles	19
	1.4 Inner Product Spaces: Orthonormal Bases	21
	1.5 Planes	28
	1.6 Orientation	36
	1.7 Convex Sets	44
	1.8 Principal Axes Theorems	51
	1.9 Bilinear and Quadratic Maps	58
	1.10 The Cross Product Reexamined	64
	1.11 The Generalized Inverse Matrix	67
	1.12 EXERCISES	72
	2 Affine Geometry	77
	2.1 Overview	77
	2.2 Motions	78
	2.2.1 Translations	81
	2.2.2 Rotations in the Plane	82
	2.2.3 Re.ections in the Plane	86
	2.2.4 Motions Preserve the Dot Product	90
	2.2.5 Some Existence and Uniqueness Results	93
	2.2.6 Rigid Motions in the Plane	96
	2.2.7 Summary for Motions in the Plane	99
	2.2.8 Frames in the Plane	101
	2.3 Similarities	108
	2.4 Affine Transformations	109
	2.4.1 Parallel Projections	116
	2.5 Beyond the Plane	119
	2.5.1 Motions in 3-Space	126
	2.5.2 Frames Revisited	132
	2.6 EXERCISES	135
	3 Projective Geometry	140
	3.1 Overview	140
	3.2 Central Projections and Perspectivities	141
	3.3 Homogeneous Coordinates	150
	3.4 The Projective Plane	153
	3.4.1 Analytic Properties of the Projective Plane	157
	3.4.2 Two-dimensional Projective Transformations	166
	3.4.3 Planar Maps and Homogeneous Coordinates	168
	3.5 Beyond the Plane	172
	3.5.1 Homogeneous Coordinates and Maps in 3-Space	175
	3.6 Conic Sections	180
	3.6.1 Projective Properties of Conics	194
	3.7 Quadric Surfaces	204
	3.8 Generalized Central Projections	210
	3.9 The Theorems of Pascal and Brianchon	213
	3.10 The Stereographic Projection	215
	3.11 EXERCISES	219
	4 Advanced Calculus Topics	222
	4.1 Introduction	222
	4.2 The Topology of Euclidean Space	222
	4.3 Derivatives	232
	4.4 The Inverse and Implicit Function Theorem	246
	4.5 Critical Points	254
	4.6 Morse Theory	263
	4.7 Zeros of Functions	266
	4.8 Integration	270
	4.9 Differential Forms	278
	4.9.1 Differential Forms and Integration	287
	4.10 EXERCISES	291
	5 Point Set Topology	295
	5.1 Introduction	295
	5.2 Metric Spaces	296
	5.3 Topological Spaces	303
	5.4 Constructing New Topological Spaces	312
	5.5 Compactness	318
	5.6 Connectedness	322
	5.7 Homotopy	323
	5.8 Constructing Continuous Functions	327
	5.9 The Topology of Pn	329
	5.10 EXERCISES	332
	6 Combinatorial Topology	335
	6.1 Introduction	335
	6.2 What Is Topology?	340
	6.3 Simplicial Complexes	342
	6.4 Cutting and Pasting	347
	6.5 The Classification of Surfaces	352
	6.6 Bordered and Noncompact Surfaces	367
	6.7 EXERCISES	369
	7 Algebraic Topology	372
	7.1 Introduction	372
	7.2 Homology Theory	373
	7.2.1 Homology Groups	373
	7.2.2 Induced Maps	389
	7.2.3 Applications of Homology Theory	398
	7.2.4 Cell Complexes	403
	7.2.5 Incidence Matrices	413
	7.2.6 The Mod 2 Homology Groups	419
	7.3 Cohomology Groups	423
	7.4 Homotopy Theory	426
	7.4.1 The Fundamental Group	426
	7.4.2 Covering Spaces	436
	7.4.3 Higher Homotopy Groups	448
	7.5 Pseudomanifolds	452
	7.5.1 The Degree of a Map and Applications	457
	7.5.2 Manifolds and Poincaré Duality	460
	7.6 Where to Next: What We Left Out	463
	7.7 The CW Complex Pn	467
	7.8 EXERCISES	470
	8 Differential Topology	473
	8.1 Introduction	473
	8.2 Parameterizing Spaces	474
	8.3 Manifolds in Rn	479
	8.4 Tangent Vectors and Spaces	488
	8.5 Oriented Manifolds	497
	8.6 Handle Decompositions	503
	8.7 Spherical Modi.cations	511
	8.8 Abstract Manifolds	514
	8.9 Vector Bundles	523
	8.10 The Tangent and Normal Bundles	533
	8.11 Transversality	542
	8.12 Differential Forms and Integration	549
	8.13 The Manifold Pn	562
	8.14 The Grassmann Manifolds	564
	8.15 EXERCISES	566
	9 Differential Geometry	571
	9.1 Introduction	571
	9.2 Curve Length	572
	9.3 The Geometry of Plane Curves	577
	9.4 The Geometry of Space Curves	587
	9.5 Envelopes of Curves	593
	9.6 Involutes and Evolutes of Curves	597
	9.7 Parallel Curves	600
	9.8 Metric Properties of Surfaces	603
	9.9 The Geometry of Surfaces	612
	9.10 Geodesics	634
	9.11 Envelopes of Surfaces	652
	9.12 Canal Surfaces	652
	9.13 Involutes and Evolutes of Surfaces	654
	9.14 Parallel Surfaces	657
	9.15 Ruled Surfaces	659
	9.16 The Cartan Approach: Moving Frames	663
	9.17 Where to Next?	673
	9.18 Summary of Curve Formulas	679
	9.19 Summary of Surface Formulas	681
	9.20 EXERCISES	683
	10 Algebraic Geometry	688
	10.1 Introduction	688
	10.2 Plane Curves: There Is More than Meets the Eye	691
	10.3 More on Projective Space	698
	10.4 Resultants	704
	10.5 More Polynomial Preliminaries	709
	10.6 Singularities and Tangents of Plane Curves	716
	10.7 Intersections of Plane Curves	724
	10.8 Some Commutative Algebra	729
	10.9 Defining Parameterized Curves Implicitly	738
	10.10 Gröbner Bases	742
	10.11 Elimination Theory	759
	10.12 Places of a Curve	761
	10.13 Rational and Birational Maps	778
	10.14 Space Curves	796
	10.15 Parameterizing Implicit Curves	800
	10.16 The Dimension of a Variety	804
	10.17 The Grassmann Varieties	810
	10.18 N-Dimensional Varieties	811
	10.19 EXERCISES	819
	Appendix A Notation	827
	Appendix B Basic Algebra	831
	B.1 Number Theoretic Basics	831
	B.2 Set Theoretic Basics	832
	B.3 Permutations	835
	B.4 Groups	837
	B.5 Abelian Groups	845
	B.6 Rings	849
	B.7 Polynomial Rings	854
	B.8 Fields	861
	B.9 The Complex Numbers	864
	B.10 Vector Spaces	865
	B.11 Extension Fields	869
	B.12 Algebras	873
	Appendix C Basic Linear Algebra	874
	C.1 More on Linear Independence	874
	C.2 Inner Products	876
	C.3 Matrices of Linear Transformations	879
	C.4 Eigenvalues and Eigenvectors	884
	C.5 The Dual Space	887
	C.6 The Tensor and Exterior Algebra	889
	Appendix D Basic Calculus and Analysis	903
	D.1 Miscellaneous Facts	903
	D.2 Series	906
	D.3 Differential Equations	908
	D.4 The Lebesgue Integral	910
	Appendix E Basic Complex Analysis	912
	E.1 Basic Facts	912
	E.2 Analytic Functions	913
	E.3 Complex Integration	916
	E.4 More on Complex Series	917
	E.5 Miscellaneous Facts	919
	Appendix F A Bit of Numerical Analysis	921
	F.1 The Condition Number of a Matrix	921
	F.2 Approximation and Numerical Integration	922
	Bibliography	929
	Abbreviations	929
	Abstract Algebra	929
	Advanced Calculus	929
	Algebraic Curves and Surfaces	929
	Algebraic Geometry	930
	Algebraic Topology	930
	Analytic Geometry	931
	Complex Analysis	931
	Conics	931
	Cyclides	931
	Differential Geometry	932
	Differential Topology	932
	Geodesics	933
	Geometric Modeling	933
	Linear Algebra	933
	Miscellaneous	933
	Numerical Methods	933
	Offset Curves and Surfaces	934
	Projective Geometry and Transformations	934
	Quadrics	934
	Real Analysis	934
	Topology	934
	Index	935
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