For a more applied course: What about Jean Gallier: Geometric Methods andApplications.
From the preface:
"Novelties: As far as we know, there is no fully developed modern exposition integrating the basic concepts of affine geometry, projective geometry, Euclidean geometry, Hermitian geometry, basics of Hilbert spaces with a touch of Fourier series, basics of Lie groups and Lie algebras, as well as a presentation of curves and surfaces both from the standard differential point of view and from the algorithmic point of view in terms of control points (in the polynomial and rational case).
From the table of contents:
Preface 1Introduction
1.1 Geometries: Their Origin, Their Uses
1.2 Prerequisites and Notation2Basics of Affine Geometry
2.1 Affine Spaces
2.2 Examples of Affine Spaces
2.3 Chasles's Identity
2.4 Affine Combinations, Barycenter
2.5 Affine Subspaces
2.6 Affine Independence and Affine Frames
2.7 Affine Maps2.8 Affine Groups
2.9 Affine Geometry: A Glimpse
2.10 Affine Hyperplanes
2.11 Intersection of Affine Spaces
2.12 Problems
3Properties of Convex Sets: A Glimpse
3.1 Convex Sets3.2 Caratheodory's Theorem
3.3 Radon's and Helly's Theorems Contents
3.4 Problems
4Embedding an Affine Space in a Vector Space
4.1 The "Hat Construction," or Homogenizing
4.2 Affine Frames of E and Bases of Ё
4.3 Another Construction of E
4.4 Extending Affine Maps to Linear Map
4.5 Problems
5 Basics of Projective Geometry
5.1 Why Projective Spaces?
5.2 Projective Spaces
5.3 Projective Subspaces
5.4 Projective Frames
5.5 Projective Maps
5.6 Projective Completion of an Affine Space, AffinePatches
5.7 Making Good Use of Hyperplanes at Infinity
5.8 The Cross-Ratio
5.9 Duality in Projective Geometry
5.10 Cross-Ratios of Hyperplanes
5.11 Complexification of a Real Projective Space
5.12 Similarity Structures on a Projective Space
5.13 Some Applications of Projective Geometry
5.14 Problems
6Basics of Euclidean Geometry
6.1 Inner Products, Euclidean Spaces
6.2 Orthogonality, Duality, Adjoint of a Linear Map
6.3 Linear Isometries (Orthogonal Transformations)
6.4 The Orthogonal Group, Orthogonal Matrices
6.5 Qi?-Decomposition for Invertible Matrices
6.6 Some Applications of Euclidean Geometry
6.7 Problems
7The Cartan-Dieudonne Theorem
7.1 Orthogonal Reflections
7.2 The Cartan-Dieudonne Theorem for Linear Isometries
7.3 (^-Decomposition Using Householder Matrices
7.4 Affine Isometries (Rigid Motions)
7.5 Fixed Points of Affine Maps
7.6 Affine Isometries and Fixed Points
7.7 The Cartan-Dieudonne Theorem for Affine Isometries
7.8 Orientations of a Euclidean Space, Angles
7.9 Volume Forms, Cross Products
7.10 Problems
8The Quaternions and the Spaces S3, SUB), SOC),and RP3
8.1 The Algebra M of Quaternions
8.2 Quaternions and Rotations in SOC)
8.3 Quaternions and Rotations in SOD)
8.4 Applications of Euclidean Geometry to MotionInterpolation
8.5 Problems
9Dirichlet—Voronoi Diagrams and DelaunayTriangulations
9.1 Dirichlet-Voronoi Diagrams
9.2 Simplicial Complexes and Triangulations
9.3 Delaunay Triangulations
9.4 Delaunay Triangulations and Convex Hulls
9.5 Applications of Voronoi Diagrams and DelaunayTriangulations
9.6 Problems10 Basics of Hermitian Geometry
10.1 Sesquilinear and Hermitian Forms, Pre-Hilbert Spacesand Hermitian Spaces
10.2 Orthogonality, Duality, Adjoint of a Linear Map
10.3 Linear Isometries (Also Called UnitaryTransformations)
10.4 The Unitary Group, Unitary Matrices
10.5 Problems11 Spectral Theorems in Euclidean and Hermitian Spaces
11.1 Introduction: What's with Lie Groups and LieAlgebras?
11.2 Normal Linear Maps
11.3 Self-Adjoint, Skew Self-Adjoint, and OrthogonalLinear Maps
11.4 Normal, Symmetric, Skew Symmetric, Orthogonal,Hermitian, Skew Hermitian, and Unitary Matrices ....
11.5 Problems
12 Singular Value Decomposition (SVD) and Polar Form
12.1 Polar Form
12.2 Singular Value Decomposition (SVD)
12.3 Problems
13 Applications of Euclidean Geometry to VariousOptimization Problems
13.1 Applications of the SVD and Qi^-Decomposition toLeast Squares Problems :
13.2 Minimization of Quadratic Functions UsingLagrange Multipliers
13.3 Problems
14 Basics of Classical Lie Groups: The Exponential Map,Lie Groups, and Lie Algebras
14.1 The Exponential Map
14.2 The Lie Groups GL(n,i), SL(n,M), O(n), SO(n),the Lie Algebras gZ(rc, R), sl(n,R), o(n), so(n), and theExponential Map
14.3 Symmetric Matrices, Symmetric Positive DefiniteMatrices, and the Exponential Map
14.4 The Lie Groups GL(n, C), SL(n, C), U(n), SU(n),the Lie Algebras gZ(rc, C), sZ(n,C), u(n), su(n),and the Exponential Map14.5 Hermitian Matrices, Hermitian Positive DefiniteMatrices, and the Exponential Map
14.6 The Lie Group SE(n) and the Lie Algebra se(n)14.7 Finale: Lie Groups and Lie Algebras14.8 Applications of Lie Groups and Lie Algebras
14.9 Problems
15 Basics of the Differential Geometry of Curves
15.1 Introduction: Parametrized Curves15.2 Tangent Lines and Osculating Planes15.3 Arc Length15.4 Curvature and Osculating Circles (Plane Curves) ....15.5 Normal Planes and Curvature CD Curves)15.6 The Frenet Frame CD Curves)15.7 Torsion CD Curves)15.8 The Frenet Equations CD Curves)15.9 Osculating Spheres CD Curves)15.10 The Frenet Frame for nD Curves (n > 4)15.11 Applications15.12 Problems
16 Basics of the Differential Geometry of Surfaces16.1 Introduction16.2 Parametrized Surfaces16.3 The First Fundamental Form (Riemannian Metric) . . .16.4 Normal Curvature and the Second Fundamental Form 16.5 Geodesic Curvature and the Christoffel Symbols16.6 Principal Curvatures, Gaussian Curvature, MeanCurvature16.7 The Gauss Map and Its Derivative dN16.8 The Dupin Indicatrix16.9 The Theorema Egregium of Gauss, the Equationsof Codazzi-Mainardi, and Bonnet's Theorem16.10 Lines of Curvature, Geodesic Torsion, AsymptoticLines16.11 Geodesic Lines, Local Gauss-Bonnet Theorem16.12 Applications16.13 Problems
17 Appendix17.1 Hyperplanes and Linear Forms17.2 Metric Spaces and Normed Vector Spaces
