Curves and Surfaces in Geometric Modeling

1 Introduction1.1 Geometric Methods in Engineering1.2 Examples of Problems Using Geometric Modeling2 Basics of Affine Geometry 2.1 Affine Spaces2.2 Examples of Affine Spaces2.3 Chasles' Identity2.4 Affine Combinations, Barycenters2.5 Affine Subspaces2.6 Affine Independence and Affine Frames2.7 Affine Maps2.8 Affine Groups2.9 Affine Hyperplanes3 Introduction to the Algorithmic Geometry of Polynomial Curves3.1 Why Parameterized Polynomial Curves?3.2 Polynomial Curves of degree 1 and 23.3 First Encounter with Polar Forms (Blossoming)3.4 First Encounter with the de Casteljau Algorithm3.5 Polynomial Curves of Degree 33.6 Classification of the Polynomial Cubics3.7 Second Encounter with Polar Forms (Blossoming)3.8 Second Encounter with the de Casteljau Algorith3.9 Examples of Cubics Defined by Control Points4 Multiaffine Maps and Polar Forms4.1 Multiaffine Maps4.2 Affine Polynomials and Polar Forms4.3 Polynomial Curves and Control Points4.4 Uniqueness of the Polar Form of an Affine Polynomial Map4.5 Polarizing Polynomials in One or Several Variables5 Polynomial Curves as Be'zier Curves5.1 The de Casteljau Algorithm5.2 Subdivision Algorithms for Polynomial Curves5.3 The Progressive Version of the de Casteljau Algorithm (the de Boor Algorithm)5.4 Derivatives of Polynomial Curves5.5 Joining Affine Polynomial Functions6 B-Spline Curves6.1 Introduction: Knot Sequences, de Boor Control Points6.2 Infinite Knot Sequences, Open B-Spline Curves6.3 Finite Knot Sequences, Finite B-Spline Curves6.4 Cyclic Knot Sequences, Closed (Cyclic) B-Spline Curves6.5 The de Boor Algorithm6.6 The de Boor Algorithm and Knot Insertion6.7 Polar forms of B-Splines6.8 Cubic Spline Interpolation7 Polynomial Surfaces7.1 Polarizing Polynomial Surfaces7.2 Bipolynomial Surfaces in Polar Form7.3 The de Casteljau Algorithm for Rectangular Surface Patches7.4 Total Degree Surfaces in Polar Form7.5 The de Casteljau Algorithm for Triangular Surface Patches7.6 Directional Derivatives of Polynomial Surfaces8 Subdivision Algorithms for Polynomial Surfaces 2948.1 Subdivision Algorithms for Triangular Patches8.2 Subdivision Algorithms for Rectangular Patches9. Polynomial Spline Surfaces and Subdivision Surfaces9.1 Joining Polynomial Surfaces9.2 Spline Surfaces with Triangular Patches9.3 Spline Surfaces with Rectangular Patches9.4 Subdivision Surfaces10 Embedding an Affine Space in a Vector Space 36910.1 The "Hat Construction", or Homogenizing10.2 Affine Frames of E and Bases of E10.3 Extending Affine Maps to Linear Maps10.4 From Multiaffine Maps to Multilinear Maps10.5 Differentiating Affine Polynomial Functions Using Their Homogenized Polar Forms, Osculating Flats11 Tensor Products and Symmetric Tensor Products 39511.1 Tensor Products11.2 Symmetric Tensor Products11.3 Affine Symmetric Tensor Products11.4 Properties of Symmetric Tensor Products11.5 Polar Forms Revisited12 Appendix 1: Linear Algebra12.1 Vector Spaces12.2 Linear Maps12.3 Quotient Spaces12.4 Direct Sums12.5 Hyperplanes and Linear Forms13 Appendix 2: Complements of Affine Geometry 44213.1 Affine and Multiaffine Maps13.2 Homogenizing Multiaffine Maps13.3 Intersection and Direct Sums of Affine Spaces13.4 Osculating Flats Revisited14 Appendix 3: Topology14.1 Metric Spaces and Normed Vector Spaces14.2 Continuous Functions, Limits14.3 Normed Affine Spaces 46515 Appendix 4: Differential Calculus15.1 Directional Derivatives, Total Derivatives15.2 Jacobian Matrices