Mathematics for Computer Graphics
John Vince explains a wide range of mathematical techniques and problem-solving strategies associated with computer games, computer animation, virtual reality, CAD and other areas of computer graphics in this completely revised and expanded fifth edition.The first five chapters cover a general intro...
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| Main Author | |
|---|---|
| Format | eBook Book |
| Language | English |
| Published |
London
Springer Nature
2017
Springer Springer London, Limited Springer London |
| Edition | 5 |
| Series | Undergraduate Topics in Computer Science |
| Subjects | |
| Online Access | Get full text |
| ISBN | 9781447173366 1447173368 1447173341 9781447173342 |
| ISSN | 1863-7310 2197-1781 |
| DOI | 10.1007/978-1-4471-7336-6 |
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| Abstract | John Vince explains a wide range of mathematical techniques and problem-solving strategies associated with computer games, computer animation, virtual reality, CAD and other areas of computer graphics in this completely revised and expanded fifth edition.The first five chapters cover a general introduction, number sets, algebra, trigonometry and coordinate systems, which are employed in the following chapters on vectors, matrix algebra, transforms, interpolation, curves and patches, analytic geometry and barycentric coordinates. Following this, the reader is introduced to the relatively new topic of geometric algebra, followed by two chapters that introduce differential and integral calculus. Finally, there is a chapter on worked examples.Mathematics for Computer Graphics covers all of the key areas of the subject, including:· Number sets· Algebra· Trigonometry· Coordinate systems· Determinants· Vectors· Quaternions· Matrix algebra· Geometric transforms· Interpolation· Curves and surfaces· Analytic geometry· Barycentric coordinates· Geometric algebra· Differential calculus· Integral calculusThis fifth edition contains over 120 worked examples and over 320 colour illustrations, which are central to the author's descriptive writing style. Mathematics for Computer Graphics provides a sound understanding of the mathematics required for computer graphics, giving a fascinating insight into the design of computer graphics software and setting the scene for further reading of more advanced books and technical research papers. |
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| AbstractList | John Vince explains a wide range of mathematical techniques and problem-solving strategies associated with computer games, computer animation, virtual reality, CAD and other areas of computer graphics in this completely revised and expanded fifth edition.The first five chapters cover a general introduction, number sets, algebra, trigonometry and coordinate systems, which are employed in the following chapters on vectors, matrix algebra, transforms, interpolation, curves and patches, analytic geometry and barycentric coordinates. Following this, the reader is introduced to the relatively new topic of geometric algebra, followed by two chapters that introduce differential and integral calculus. Finally, there is a chapter on worked examples.Mathematics for Computer Graphics covers all of the key areas of the subject, including:· Number sets· Algebra· Trigonometry· Coordinate systems· Determinants· Vectors· Quaternions· Matrix algebra· Geometric transforms· Interpolation· Curves and surfaces· Analytic geometry· Barycentric coordinates· Geometric algebra· Differential calculus· Integral calculusThis fifth edition contains over 120 worked examples and over 320 colour illustrations, which are central to the author's descriptive writing style. Mathematics for Computer Graphics provides a sound understanding of the mathematics required for computer graphics, giving a fascinating insight into the design of computer graphics software and setting the scene for further reading of more advanced books and technical research papers. |
| Author | Vince, John |
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| Edition | 5 5th ed. 2017 5th ed. 2017. |
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| ISSN | 1863-7310 |
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| Notes | Includes index |
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| PublicationSeriesTitle | Undergraduate Topics in Computer Science |
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| PublicationYear | 2017 |
| Publisher | Springer Nature Springer Springer London, Limited Springer London |
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| RelatedPersons | Pitts, Andrew Kozen, Dexter C. Breitman, Karin Skiena, Steven S Abramsky, Samson Stewart, Iain Hankin, Chris Mackie, Ian Riis Nielson, Hanne |
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| Snippet | John Vince explains a wide range of mathematical techniques and problem-solving strategies associated with computer games, computer animation, virtual reality,... |
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| SubjectTerms | Computer Graphics Computer graphics -- Mathematics Computer programming, programs, data Computer Science Mathematical Applications in Computer Science |
| TableOfContents | 7.4.12 The Right-Hand Rule -- 7.5 Deriving a Unit Normal Vector for a Triangle -- 7.6 Surface Areas -- 7.6.1 Calculating 2D Areas -- 7.7 Summary -- 7.8 Worked Examples -- 7.8.1 Position Vector -- 7.8.2 Unit Vector -- 7.8.3 Vector Magnitude -- 7.8.4 Angle Between Two Vectors -- 7.8.5 Vector Product -- 8 Matrix Algebra -- 8.1 Introduction -- 8.2 Background -- 8.3 Matrix Notation -- 8.3.1 Matrix Dimension or Order -- 8.3.2 Square Matrix -- 8.3.3 Column Vector -- 8.3.4 Row Vector -- 8.3.5 Null Matrix -- 8.3.6 Unit Matrix -- 8.3.7 Trace -- 8.3.8 Determinant of a Matrix -- 8.3.9 Transpose -- 8.3.10 Symmetric Matrix -- 8.3.11 Antisymmetric Matrix -- 8.4 Matrix Addition and Subtraction -- 8.4.1 Scalar Multiplication -- 8.5 Matrix Products -- 8.5.1 Row and Column Vectors -- 8.5.2 Row Vector and a Matrix -- 8.5.3 Matrix and a Column Vector -- 8.5.4 Square Matrices -- 8.5.5 Rectangular Matrices -- 8.6 Inverse Matrix -- 8.6.1 Inverting a Pair of Matrices -- 8.7 Orthogonal Matrix -- 8.8 Diagonal Matrix -- 8.9 Summary -- 8.10 Worked Examples -- 8.10.1 Matrix Inversion -- 8.10.2 Identity Matrix -- 8.10.3 Solving Two Equations Using Matrices -- 8.10.4 Solving Three Equations Using Matrices -- 8.10.5 Solving Two Complex Equations -- 8.10.6 Solving Three Complex Equations -- 8.10.7 Solving Two Complex Equations -- 8.10.8 Solving Three Complex Equations -- 9 Geometric Transforms -- 9.1 Introduction -- 9.2 Background -- 9.3 2D Transforms -- 9.3.1 Translation -- 9.3.2 Scaling -- 9.3.3 Reflection -- 9.4 Transforms as Matrices -- 9.4.1 Systems of Notation -- 9.5 Homogeneous Coordinates -- 9.5.1 2D Translation -- 9.5.2 2D Scaling -- 9.5.3 2D Reflections -- 9.5.4 2D Shearing -- 9.5.5 2D Rotation -- 9.5.6 2D Scaling -- 9.5.7 2D Reflection -- 9.5.8 2D Rotation About an Arbitrary Point -- 9.6 3D Transforms -- 9.6.1 3D Translation -- 9.6.2 3D Scaling -- 9.6.3 3D Rotation 9.6.4 Gimbal Lock -- 9.6.5 Rotating About an Axis -- 9.6.6 3D Reflections -- 9.7 Change of Axes -- 9.7.1 2D Change of Axes -- 9.7.2 Direction Cosines -- 9.7.3 3D Change of Axes -- 9.8 Positioning the Virtual Camera -- 9.8.1 Direction Cosines -- 9.8.2 Euler Angles -- 9.9 Rotating a Point About an Arbitrary Axis -- 9.9.1 Matrices -- 9.9.2 Quaternions -- 9.9.3 Adding and Subtracting Quaternions -- 9.9.4 Multiplying Quaternions -- 9.9.5 Pure Quaternion -- 9.9.6 The Inverse Quaternion -- 9.9.7 Unit Quaternion -- 9.9.8 Rotating Points About an Axis -- 9.9.9 Roll, Pitch and Yaw Quaternions -- 9.9.10 Quaternions in Matrix Form -- 9.9.11 Frames of Reference -- 9.10 Transforming Vectors -- 9.11 Determinants -- 9.12 Perspective Projection -- 9.13 Summary -- 9.14 Worked Examples -- 9.14.1 2D Scaling Transform -- 9.14.2 2D Scale and Translate -- 9.14.3 3D Scaling Transform -- 9.14.4 2D Rotation -- 9.14.5 2D Rotation About a Point -- 9.14.6 Determinant of the Rotate Transform -- 9.14.7 Determinant of the Shear Transform -- 9.14.8 Yaw, Pitch and Roll Transforms -- 9.14.9 3D Rotation About an Axis -- 9.14.10 3D Rotation Transform Matrix -- 9.14.11 2D Change of Axes -- 9.14.12 3D Change of Axes -- 9.14.13 Rotate a Point About an Axis -- 9.14.14 Perspective Projection -- 10 Interpolation -- 10.1 Introduction -- 10.2 Background -- 10.3 Linear Interpolation -- 10.4 Non-linear Interpolation -- 10.4.1 Trigonometric Interpolation -- 10.4.2 Cubic Interpolation -- 10.5 Interpolating Vectors -- 10.6 Interpolating Quaternions -- 10.7 Summary -- 11 Curves and Patches -- 11.1 Introduction -- 11.2 Background -- 11.3 The Circle -- 11.4 The Ellipse -- 11.5 Bézier Curves -- 11.5.1 Bernstein Polynomials -- 11.5.2 Quadratic Bézier Curves -- 11.5.3 Cubic Bernstein Polynomials -- 11.6 A Recursive Bézier Formula -- 11.7 Bézier Curves Using Matrices -- 11.7.1 Linear Interpolation 13.4 Ratios and Proportion Intro -- Preface -- Contents -- 1 Introduction -- 1.1 Mathematics for Computer Graphics -- 1.2 Understanding Mathematics -- 1.3 What Makes Mathematics Difficult? -- 1.4 Does Mathematics Exist Outside Our Brains? -- 1.5 Symbols and Notation -- 2 Numbers -- 2.1 Introduction -- 2.2 Background -- 2.3 Counting -- 2.4 Sets of Numbers -- 2.5 Zero -- 2.6 Negative Numbers -- 2.6.1 The Arithmetic of Positive and Negative Numbers -- 2.7 Observations and Axioms -- 2.7.1 Commutative Law -- 2.7.2 Associative Law -- 2.7.3 Distributive Law -- 2.8 The Base of a Number System -- 2.8.1 Background -- 2.8.2 Octal Numbers -- 2.8.3 Binary Numbers -- 2.8.4 Hexadecimal Numbers -- 2.8.5 Adding Binary Numbers -- 2.8.6 Subtracting Binary Numbers -- 2.9 Types of Numbers -- 2.9.1 Natural Numbers -- 2.9.2 Integers -- 2.9.3 Rational Numbers -- 2.9.4 Irrational Numbers -- 2.9.5 Real Numbers -- 2.9.6 Algebraic and Transcendental Numbers -- 2.9.7 Imaginary Numbers -- 2.9.8 Complex Numbers -- 2.9.9 Transcendental and Algebraic Numbers -- 2.9.10 Infinity -- 2.10 Summary -- 2.11 Worked Examples -- 2.11.1 Algebraic Expansion -- 2.11.2 Binary Subtraction -- 2.11.3 Complex Numbers -- 2.11.4 Complex Rotation -- 3 Algebra -- 3.1 Introduction -- 3.2 Background -- 3.2.1 Solving the Roots of a Quadratic Equation -- 3.3 Indices -- 3.3.1 Laws of Indices -- 3.4 Logarithms -- 3.5 Further Notation -- 3.6 Functions -- 3.6.1 Explicit and Implicit Equations -- 3.6.2 Function Notation -- 3.6.3 Intervals -- 3.6.4 Function Domains and Ranges -- 3.6.5 Odd and Even Functions -- 3.6.6 Power Functions -- 3.7 Summary -- 3.8 Worked Examples -- 3.8.1 Algebraic Manipulation -- 3.8.2 Solving a Quadratic Equation -- 3.8.3 Factorising -- 4 Trigonometry -- 4.1 Introduction -- 4.2 Background -- 4.3 Units of Angular Measurement -- 4.4 The Trigonometric Ratios -- 4.4.1 Domains and Ranges 4.5 Inverse Trigonometric Ratios -- 4.6 Trigonometric Identities -- 4.7 The Sine Rule -- 4.8 The Cosine Rule -- 4.9 Compound-Angle Identities -- 4.9.1 Double-Angle Identities -- 4.9.2 Multiple-Angle Identities -- 4.9.3 Half-Angle Identities -- 4.10 Perimeter Relationships -- 4.11 Summary -- 5 Coordinate Systems -- 5.1 Introduction -- 5.2 Background -- 5.3 The Cartesian Plane -- 5.4 Function Graphs -- 5.5 Shape Representation -- 5.5.1 2D Polygons -- 5.5.2 Area of a Shape -- 5.6 Theorem of Pythagoras in 2D -- 5.7 3D Cartesian Coordinates -- 5.7.1 Theorem of Pythagoras in 3D -- 5.8 Polar Coordinates -- 5.9 Spherical Polar Coordinates -- 5.10 Cylindrical Coordinates -- 5.11 Summary -- 5.12 Worked Examples -- 5.12.1 Area of a Shape -- 5.12.2 Distance Between Two Points -- 5.12.3 Polar Coordinates -- 5.12.4 Spherical Polar Coordinates -- 5.12.5 Cylindrical Coordinates -- 6 Determinants -- 6.1 Introduction -- 6.2 Linear Equations with Two Variables -- 6.3 Linear Equations with Three Variables -- 6.3.1 Sarrus's Rule -- 6.4 Mathematical Notation -- 6.4.1 Matrix -- 6.4.2 Order of a Determinant -- 6.4.3 Value of a Determinant -- 6.4.4 Properties of Determinants -- 6.5 Summary -- 6.6 Worked Examples -- 6.6.1 Determinant Expansion -- 6.6.2 Complex Determinant -- 6.6.3 Simple Expansion -- 6.6.4 Simultaneous Equations -- 7 Vectors -- 7.1 Introduction -- 7.2 Background -- 7.3 2D Vectors -- 7.3.1 Vector Notation -- 7.3.2 Graphical Representation of Vectors -- 7.3.3 Magnitude of a Vector -- 7.4 3D Vectors -- 7.4.1 Vector Manipulation -- 7.4.2 Scaling a Vector -- 7.4.3 Vector Addition and Subtraction -- 7.4.4 Position Vectors -- 7.4.5 Unit Vectors -- 7.4.6 Cartesian Vectors -- 7.4.7 Products -- 7.4.8 Scalar Product -- 7.4.9 The Dot Product in Lighting Calculations -- 7.4.10 The Scalar Product in Back-Face Detection -- 7.4.11 The Vector Product 11.8 B-Splines -- 11.8.1 Uniform B-Splines -- 11.8.2 Continuity -- 11.8.3 Non-uniform B-Splines -- 11.8.4 Non-uniform Rational B-Splines -- 11.9 Surface Patches -- 11.9.1 Planar Surface Patch -- 11.9.2 Quadratic Bézier Surface Patch -- 11.9.3 Cubic Bézier Surface Patch -- 11.10 Summary -- 12 Analytic Geometry -- 12.1 Introduction -- 12.2 Background -- 12.2.1 Angles -- 12.2.2 Intercept Theorems -- 12.2.3 Golden Section -- 12.2.4 Triangles -- 12.2.5 Centre of Gravity of a Triangle -- 12.2.6 Isosceles Triangle -- 12.2.7 Equilateral Triangle -- 12.2.8 Right Triangle -- 12.2.9 Theorem of Thales -- 12.2.10 Theorem of Pythagoras -- 12.2.11 Quadrilateral -- 12.2.12 Trapezoid -- 12.2.13 Parallelogram -- 12.2.14 Rhombus -- 12.2.15 Regular Polygon -- 12.2.16 Circle -- 12.3 2D Analytic Geometry -- 12.3.1 Equation of a Straight Line -- 12.3.2 The Hessian Normal Form -- 12.3.3 Space Partitioning -- 12.3.4 The Hessian Normal Form from Two Points -- 12.4 Intersection Points -- 12.4.1 Intersecting Straight Lines -- 12.4.2 Intersecting Line Segments -- 12.5 Point Inside a Triangle -- 12.5.1 Area of a Triangle -- 12.5.2 Hessian Normal Form -- 12.6 Intersection of a Circle with a Straight Line -- 12.7 3D Geometry -- 12.7.1 Equation of a Straight Line -- 12.7.2 Intersecting Two Straight Lines -- 12.8 Equation of a Plane -- 12.8.1 Cartesian Form of the Plane Equation -- 12.8.2 General Form of the Plane Equation -- 12.8.3 Parametric Form of the Plane Equation -- 12.8.4 Converting from the Parametric to the General Form -- 12.8.5 Plane Equation from Three Points -- 12.9 Intersecting Planes -- 12.9.1 Intersection of Three Planes -- 12.9.2 Angle Between Two Planes -- 12.9.3 Angle Between a Line and a Plane -- 12.9.4 Intersection of a Line with a Plane -- 12.10 Summary -- 13 Barycentric Coordinates -- 13.1 Introduction -- 13.2 Background -- 13.3 Ceva's Theorem |
| Title | Mathematics for Computer Graphics |
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