Geometric Algebra for Computer Graphics

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1 John Vince Geometric Algebra for Computer Graphics 4u Springer

2 Contents Preface vii 1 Introduction Aims and objectives of this book Mathematics for CGI software The book's structure 2 2 Elementary Algebra 5.; 2.1 Introduction Numbers, variables and arithmetic operators Closure Identity element Inverse element The associative law The commutative law The distributive law Summary 9 3 Complex Algebra Introduction Complex numbers 11 xi

3 xii Contents 3.3 Complex arithmetic The complex plane / as a rotor The product of two complex numbers Powers of complex numbers e, i, sin and cos Logarithm of a complex number Summary 22 4 Vector Algebra Introduction Vector quantities and their graphical representation Vector spaces Linear combinations Spanning sets Linear independence and dependence Standard bases Orthogonal bases Dimension Subspaces Scalar product Vector product Summary 37 5 Quaternion Algebra Introduction Adding quaternions The quaternion product The magnitude of a quaternion The unit quaternion The pure quaternion The conjugate of a quaternion 44

4 Contents xiii 5.8 The inverse quaternion Quaternion algebra Rotating vectors using quaternions Summary 48 6 Geometric Conventions Introduction " Clockwise and anticlockwise Left and right-handed axial systems Summary 54 7 Geometric Algebra Introduction Foundations of geometric algebra Introduction to geometric algebra Length, area and volume The outer product Some algebraic properties Visualizing the outer product Orthogonal bases The outer product in action Area of a triangle The sine rule Intersection of two lines Summary 77 8 The Geometric Product Introduction Clifford's definition of the geometric product Orthogonal vectors Linearly dependent vectors Linearly independent vectors The product of identical basis vectors The product of orthogonal basis vectors The imaginary properties of the outer product The unit bivector pseudoscalar The rotational properties of the pseudoscalar Summary of the products 87

5 xiv Contents 8.5 Multivectors in R The relationship between bivectors, complex numbers and vectors Reversion Rotations in R The vector-bivector product in R Volumes and the trivector The unit trivector pseudoscalar The product of the unit basis vectors in R The product of identical basis vectors The product of orthogonal basis vectors The imaginary properties of the unit bivectors The vector-unit bivector product in R The vector-bivector product in R Unit bivector-bivector products in R Unit vector-trivector product in R Unit bivector-trivector product in R Unit trivector-trivector product in R Higher products in R Blades Duality transformation Summary of products in K Multivectors in R 3 Ill 8.24 Relationship between vector algebra and geometric algebra Relationship between the outer product and the cross product Relationship between geometric algebra and quaternions Inverse of a vector The meet operation Summary Reflections and Rotations Introduction Reflections Reflecting vectors 127

6 Contents xv Reflecting bivectors Reflecting trivectors Rotations Rotating by double reflecting Rotors Rotor matrix Building rotors Interpolating rotors Summary Geometric Algebra and Geometry Introduction Point inside a triangle Point inside a 2D triangle Point inside a 3D triangle 158 ' "' 10.3 The relationship between bivectors and direction cosines Lines and planes 166 ' Relative orientation of a point and a line Relative orientation of a point and a plane Shortest distance from a point to a plane A line intersecting a plane Perspective projection Back-face removal Homogeneous coordinates Introduction Representing 2D lines in 3D homogeneous space Intersection oftwo lines in R Representing 3D lines in 4D homogeneous space Representing lines and planes in 4D homogeneous space Summary Conformal Geometry Introduction Spatial dimension Algebraic underpinning Mathematical language and notation Protection Stereographic projection Signatures and null vectors 204

7 xvi Contents 11.4 The basis blades for the conformal model Representing geometric objects Points Point pair Lines Circles Planes Spheres Conformal transformations Translations Rotations Dilations Reflections Intersections Summary Applications of Geometric Algebra Introduction D Linear transformations Scale transform Refraction transform Rigid-body pose control Ray tracing Summary Programming Tools for Geometric Algebra Introduction Programming implications Programming tools Summary Conclusion 243 References 245 Index 249

Geometric Algebra for Computer Graphics

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