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E-grāmata: Mathematics for Computer Graphics

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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 calculus

This fifth edition contains over 120 worked examples and over 320 colour illustrations, which are central to the authors 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.

Recenzijas

A broad range of mathematical topics is covered in the 18 chapters of this book. I think this is a very useful book and I highly recommend it as a reference for those who are studying or working in computer-graphics-related fields. (Zhaoqiang Lai, Computing Reviews, April, 2018) The books target audience is computer graphics students and professionals in this area. the information is presented in a minimalist fashion and is very easy to digest. Highly recommended for its crispness and the quality of the material. (Naga Narayanaswamy, Computing Reviews, July, 2018)

1 Introduction
1(4)
1.1 Mathematics for Computer Graphics
1(1)
1.2 Understanding Mathematics
1(1)
1.3 What Makes Mathematics Difficult?
2(1)
1.4 Does Mathematics Exist Outside Our Brains?
2(1)
1.5 Symbols and Notation
3(2)
2 Numbers
5(26)
2.1 Introduction
5(1)
2.2 Background
5(1)
2.3 Counting
5(1)
2.4 Sets of Numbers
6(1)
2.5 Zero
7(1)
2.6 Negative Numbers
8(2)
2.6.1 The Arithmetic of Positive and Negative Numbers
9(1)
2.7 Observations and Axioms
10(1)
2.7.1 Commutative Law
10(1)
2.7.2 Associative Law
10(1)
2.7.3 Distributive Law
11(1)
2.8 The Base of a Number System
11(8)
2.8.1 Background
11(1)
2.8.2 Octal Numbers
12(1)
2.8.3 Binary Numbers
13(1)
2.8.4 Hexadecimal Numbers
14(3)
2.8.5 Adding Binary Numbers
17(1)
2.8.6 Subtracting Binary Numbers
18(1)
2.9 Types of Numbers
19(9)
2.9.1 Natural Numbers
19(1)
2.9.2 Integers
19(1)
2.9.3 Rational Numbers
20(1)
2.9.4 Irrational Numbers
20(1)
2.9.5 Real Numbers
20(1)
2.9.6 Algebraic and Transcendental Numbers
20(1)
2.9.7 Imaginary Numbers
21(4)
2.9.8 Complex Numbers
25(2)
2.9.9 Transcendental and Algebraic Numbers
27(1)
2.9.10 Infinity
27(1)
2.10 Summary
28(1)
2.11 Worked Examples
29(2)
2.11.1 Algebraic Expansion
29(1)
2.11.2 Binary Subtraction
29(1)
2.11.3 Complex Numbers
29(1)
2.11.4 Complex Rotation
30(1)
3 Algebra
31(20)
3.1 Introduction
31(1)
3.2 Background
32(5)
3.2.1 Solving the Roots of a Quadratic Equation
33(4)
3.3 Indices
37(1)
3.3.1 Laws of Indices
37(1)
3.4 Logarithms
38(2)
3.5 Further Notation
40(1)
3.6 Functions
40(6)
3.6.1 Explicit and Implicit Equations
40(1)
3.6.2 Function Notation
41(1)
3.6.3 Intervals
42(1)
3.6.4 Function Domains and Ranges
43(1)
3.6.5 Odd and Even Functions
44(1)
3.6.6 Power Functions
45(1)
3.7 Summary
46(1)
3.8 Worked Examples
46(5)
3.8.1 Algebraic Manipulation
46(1)
3.8.2 Solving a Quadratic Equation
47(2)
3.8.3 Factorising
49(2)
4 Trigonometry
51(14)
4.1 Introduction
51(1)
4.2 Background
51(1)
4.3 Units of Angular Measurement
51(1)
4.4 The Trigonometric Ratios
52(3)
4.4.1 Domains and Ranges
55(1)
4.5 Inverse Trigonometric Ratios
55(2)
4.6 Trigonometric Identities
57(1)
4.7 The Sine Rule
58(1)
4.8 The Cosine Rule
58(1)
4.9 Compound-Angle Identities
59(3)
4.9.1 Double-Angle Identities
60(1)
4.9.2 Multiple-Angle Identities
61(1)
4.9.3 Half-Angle Identities
62(1)
4.10 Perimeter Relationships
62(1)
4.11 Summary
63(2)
5 Coordinate Systems
65(12)
5.1 Introduction
65(1)
5.2 Background
65(1)
5.3 The Cartesian Plane
66(1)
5.4 Function Graphs
66(1)
5.5 Shape Representation
67(2)
5.5.1 2D Polygons
67(1)
5.5.2 Area of a Shape
68(1)
5.6 Theorem of Pythagoras in 2D
69(1)
5.7 3D Cartesian Coordinates
69(1)
5.7.1 Theorem of Pythagoras in 3D
70(1)
5.8 Polar Coordinates
70(1)
5.9 Spherical Polar Coordinates
71(1)
5.10 Cylindrical Coordinates
72(1)
5.11 Summary
73(1)
5.12 Worked Examples
73(4)
5.12.1 Area of a Shape
73(1)
5.12.2 Distance Between Two Points
74(1)
5.12.3 Polar Coordinates
74(1)
5.12.4 Spherical Polar Coordinates
75(1)
5.12.5 Cylindrical Coordinates
75(2)
6 Determinants
77(18)
6.1 Introduction
77(1)
6.2 Linear Equations with Two Variables
78(3)
6.3 Linear Equations with Three Variables
81(7)
6.3.1 Sarrus's Rule
87(1)
6.4 Mathematical Notation
88(3)
6.4.1 Matrix
88(1)
6.4.2 Order of a Determinant
89(1)
6.4.3 Value of a Determinant
89(1)
6.4.4 Properties of Determinants
90(1)
6.5 Summary
91(1)
6.6 Worked Examples
91(4)
6.6.1 Determinant Expansion
91(1)
6.6.2 Complex Determinant
92(1)
6.6.3 Simple Expansion
92(1)
6.6.4 Simultaneous Equations
93(2)
7 Vectors
95(24)
7.1 Introduction
95(1)
7.2 Background
95(1)
7.3 2D Vectors
96(3)
7.3.1 Vector Notation
96(1)
7.3.2 Graphical Representation of Vectors
97(1)
7.3.3 Magnitude of a Vector
98(1)
7.4 3D Vectors
99(13)
7.4.1 Vector Manipulation
100(1)
7.4.2 Scaling a Vector
100(1)
7.4.3 Vector Addition and Subtraction
101(1)
7.4.4 Position Vectors
102(1)
7.4.5 Unit Vectors
102(1)
7.4.6 Cartesian Vectors
103(1)
7.4.7 Products
103(1)
7.4.8 Scalar Product
104(1)
7.4.9 The Dot Product in Lighting Calculations
105(1)
7.4.10 The Scalar Product in Back-Face Detection
106(1)
7.4.11 The Vector Product
107(5)
7.4.12 The Right-Hand Rule
112(1)
7.5 Deriving a Unit Normal Vector for a Triangle
112(1)
7.6 Surface Areas
113(2)
7.6.1 Calculating 2D Areas
114(1)
7.7 Summary
115(1)
7.8 Worked Examples
115(4)
7.8.1 Position Vector
115(1)
7.8.2 Unit Vector
115(1)
7.8.3 Vector Magnitude
116(1)
7.8.4 Angle Between Two Vectors
116(1)
7.8.5 Vector Product
116(3)
8 Matrix Algebra
119(34)
8.1 Introduction
119(1)
8.2 Background
119(3)
8.3 Matrix Notation
122(8)
8.3.1 Matrix Dimension or Order
122(1)
8.3.2 Square Matrix
123(1)
8.3.3 Column Vector
123(1)
8.3.4 Row Vector
123(1)
8.3.5 Null Matrix
123(1)
8.3.6 Unit Matrix
124(1)
8.3.7 Trace
124(1)
8.3.8 Determinant of a Matrix
125(1)
8.3.9 Transpose
125(2)
8.3.10 Symmetric Matrix
127(1)
8.3.11 Antisymmetric Matrix
128(2)
8.4 Matrix Addition and Subtraction
130(1)
8.4.1 Scalar Multiplication
130(1)
8.5 Matrix Products
131(4)
8.5.1 Row and Column Vectors
131(1)
8.5.2 Row Vector and a Matrix
132(1)
8.5.3 Matrix and a Column Vector
133(1)
8.5.4 Square Matrices
133(1)
8.5.5 Rectangular Matrices
134(1)
8.6 Inverse Matrix
135(7)
8.6.1 Inverting a Pair of Matrices
141(1)
8.7 Orthogonal Matrix
142(1)
8.8 Diagonal Matrix
143(1)
8.9 Summary
143(1)
8.10 Worked Examples
144(9)
8.10.1 Matrix Inversion
144(1)
8.10.2 Identity Matrix
144(1)
8.10.3 Solving Two Equations Using Matrices
145(1)
8.10.4 Solving Three Equations Using Matrices
146(1)
8.10.5 Solving Two Complex Equations
147(1)
8.10.6 Solving Three Complex Equations
147(1)
8.10.7 Solving Two Complex Equations
148(1)
8.10.8 Solving Three Complex Equations
149(4)
9 Geometric Transforms
153(64)
9.1 Introduction
153(1)
9.2 Background
153(1)
9.3 2D Transforms
154(2)
9.3.1 Translation
154(1)
9.3.2 Scaling
154(1)
9.3.3 Reflection
155(1)
9.4 Transforms as Matrices
156(1)
9.4.1 Systems of Notation
156(1)
9.5 Homogeneous Coordinates
156(11)
9.5.1 2D Translation
158(1)
9.5.2 2D Scaling
158(1)
9.5.3 2D Reflections
159(2)
9.5.4 2D Shearing
161(1)
9.5.5 2D Rotation
162(2)
9.5.6 2D Scaling
164(1)
9.5.7 2D Reflection
165(1)
9.5.8 2D Rotation About an Arbitrary Point
166(1)
9.6 3D Transforms
167(7)
9.6.1 3D Translation
167(1)
9.6.2 3D Scaling
167(1)
9.6.3 3D Rotation
168(4)
9.6.4 Gimbal Lock
172(1)
9.6.5 Rotating About an Axis
173(1)
9.6.6 3D Reflections
174(1)
9.7 Change of Axes
174(3)
9.7.1 2D Change of Axes
174(2)
9.7.2 Direction Cosines
176(1)
9.7.3 3D Change of Axes
177(1)
9.8 Positioning the Virtual Camera
177(6)
9.8.1 Direction Cosines
178(3)
9.8.2 Euler Angles
181(2)
9.9 Rotating a Point About an Arbitrary Axis
183(18)
9.9.1 Matrices
183(7)
9.9.2 Quaternions
190(1)
9.9.3 Adding and Subtracting Quaternions
191(1)
9.9.4 Multiplying Quaternions
192(1)
9.9.5 Pure Quaternion
192(1)
9.9.6 The Inverse Quaternion
193(1)
9.9.7 Unit Quaternion
193(1)
9.9.8 Rotating Points About an Axis
193(4)
9.9.9 Roll, Pitch and Yaw Quaternions
197(2)
9.9.10 Quaternions in Matrix Form
199(1)
9.9.11 Frames of Reference
200(1)
9.10 Transforming Vectors
201(1)
9.11 Determinants
202(2)
9.12 Perspective Projection
204(2)
9.13 Summary
206(1)
9.14 Worked Examples
206(11)
9.14.1 2D Scaling Transform
206(1)
9.14.2 2D Scale and Translate
206(1)
9.14.3 3D Scaling Transform
207(1)
9.14.4 2D Rotation
208(1)
9.14.5 2D Rotation About a Point
209(1)
9.14.6 Determinant of the Rotate Transform
209(1)
9.14.7 Determinant of the Shear Transform
209(1)
9.14.8 Yaw, Pitch and Roll Transforms
210(1)
9.14.9 3D Rotation About an Axis
210(1)
9.14.10 3D Rotation Transform Matrix
211(1)
9.14.11 2D Change of Axes
211(1)
9.14.12 3D Change of Axes
212(1)
9.14.13 Rotate a Point About an Axis
213(1)
9.14.14 Perspective Projection
214(3)
10 Interpolation
217(16)
10.1 Introduction
217(1)
10.2 Background
217(1)
10.3 Linear Interpolation
218(2)
10.4 Non-linear Interpolation
220(7)
10.4.1 Trigonometric Interpolation
220(1)
10.4.2 Cubic Interpolation
221(6)
10.5 Interpolating Vectors
227(3)
10.6 Interpolating Quaternions
230(2)
10.7 Summary
232(1)
11 Curves and Patches
233(26)
11.1 Introduction
233(1)
11.2 Background
233(1)
11.3 The Circle
234(1)
11.4 The Ellipse
234(1)
11.5 Bezier Curves
235(7)
11.5.1 Bernstein Polynomials
235(3)
11.5.2 Quadratic Bezier Curves
238(1)
11.5.3 Cubic Bernstein Polynomials
239(3)
11.6 A Recursive Bezier Formula
242(1)
11.7 Bezier Curves Using Matrices
243(4)
11.7.1 Linear Interpolation
244(3)
11.8 B-Splines
247(4)
11.8.1 Uniform B-Splines
247(3)
11.8.2 Continuity
250(1)
11.8.3 Non-uniform B-Splines
251(1)
11.8.4 Non-uniform Rational B-Splines
251(1)
11.9 Surface Patches
251(6)
11.9.1 Planar Surface Patch
251(2)
11.9.2 Quadratic Bezier Surface Patch
253(2)
11.9.3 Cubic Bezier Surface Patch
255(2)
11.10 Summary
257(2)
12 Analytic Geometry
259(48)
12.1 Introduction
259(1)
12.2 Background
259(10)
12.2.1 Angles
260(1)
12.2.2 Intercept Theorems
260(1)
12.2.3 Golden Section
261(1)
12.2.4 Triangles
261(1)
12.2.5 Centre of Gravity of a Triangle
262(1)
12.2.6 Isosceles Triangle
262(1)
12.2.7 Equilateral Triangle
263(1)
12.2.8 Right Triangle
263(1)
12.2.9 Theorem of Thales
263(1)
12.2.10 Theorem of Pythagoras
264(1)
12.2.11 Quadrilateral
265(1)
12.2.12 Trapezoid
265(1)
12.2.13 Parallelogram
265(1)
12.2.14 Rhombus
266(1)
12.2.15 Regular Polygon
266(1)
12.2.16 Circle
267(2)
12.3 2D Analytic Geometry
269(4)
12.3.1 Equation of a Straight Line
269(1)
12.3.2 The Hessian Normal Form
270(2)
12.3.3 Space Partitioning
272(1)
12.3.4 The Hessian Normal Form from Two Points
272(1)
12.4 Intersection Points
273(3)
12.4.1 Intersecting Straight Lines
273(1)
12.4.2 Intersecting Line Segments
274(2)
12.5 Point Inside a Triangle
276(4)
12.5.1 Area of a Triangle
276(2)
12.5.2 Hessian Normal Form
278(2)
12.6 Intersection of a Circle with a Straight Line
280(2)
12.7 3D Geometry
282(4)
12.7.1 Equation of a Straight Line
282(1)
12.7.2 Intersecting Two Straight Lines
283(3)
12.8 Equation of a Plane
286(9)
12.8.1 Cartesian Form of the Plane Equation
286(3)
12.8.2 General Form of the Plane Equation
289(1)
12.8.3 Parametric Form of the Plane Equation
289(2)
12.8.4 Converting from the Parametric to the General Form
291(2)
12.8.5 Plane Equation from Three Points
293(2)
12.9 Intersecting Planes
295(11)
12.9.1 Intersection of Three Planes
298(3)
12.9.2 Angle Between Two Planes
301(1)
12.9.3 Angle Between a Line and a Plane
302(2)
12.9.4 Intersection of a Line with a Plane
304(2)
12.10 Summary
306(1)
13 Barycentric Coordinates
307(30)
13.1 Introduction
307(1)
13.2 Background
307(1)
13.3 Ceva's Theorem
308(1)
13.4 Ratios and Proportion
309(1)
13.5 Mass Points
310(6)
13.6 Linear Interpolation
316(7)
13.7 Convex Hull Property
323(1)
13.8 Areas
324(9)
13.9 Volumes
333(2)
13.10 Bezier Curves and Patches
335(1)
13.11 Summary
336(1)
14 Geometric Algebra
337(36)
14.1 Introduction
337(1)
14.2 Background
337(1)
14.3 Symmetric and Antisymmetric Functions
338(1)
14.4 Trigonometric Foundations
339(2)
14.5 Vectorial Foundations
341(1)
14.6 Inner and Outer Products
341(2)
14.7 The Geometric Product in 2D
343(2)
14.8 The Geometric Product in 3D
345(2)
14.9 The Outer Product of Three 3D Vectors
347(1)
14.10 Axioms
348(1)
14.11 Notation
349(1)
14.12 Grades, Pseudoscalars and Multivectors
349(2)
14.13 Redefining the Inner and Outer Products
351(1)
14.14 The Inverse of a Vector
351(2)
14.15 The Imaginary Properties of the Outer Product
353(2)
14.16 Duality
355(1)
14.17 The Relationship Between the Vector Product and the Outer Product
356(1)
14.18 The Relationship Between Quaternions and Bivectors
357(1)
14.19 Reflections and Rotations
358(4)
14.19.1 2D Reflections
358(1)
14.19.2 3D Reflections
359(1)
14.19.3 2D Rotations
360(2)
14.20 Rotors
362(3)
14.21 Worked Examples
365(7)
14.21.1 The Sine Rule
365(1)
14.21.2 The Cosine Rule
366(1)
14.21.3 A Point Perpendicular to a Line
367(2)
14.21.4 Reflecting a Vector About a Vector
369(1)
14.21.5 A Point Above or Below a Plane
370(2)
14.22 Summary
372(1)
15 Calculus: Derivatives
373(58)
15.1 Introduction
373(1)
15.2 Background
373(1)
15.3 Small Numerical Quantities
373(2)
15.4 Equations and Limits
375(9)
15.4.1 Quadratic Function
375(1)
15.4.2 Cubic Equation
376(2)
15.4.3 Functions and Limits
378(2)
15.4.4 Graphical Interpretation of the Derivative
380(1)
15.4.5 Derivatives and Differentials
381(1)
15.4.6 Integration and Antiderivatives
382(2)
15.5 Function Types
384(1)
15.6 Differentiating Groups of Functions
385(12)
15.6.1 Sums of Functions
385(2)
15.6.2 Function of a Function
387(4)
15.6.3 Function Products
391(4)
15.6.4 Function Quotients
395(2)
15.7 Differentiating Implicit Functions
397(3)
15.8 Differentiating Exponential and Logarithmic Functions
400(4)
15.8.1 Exponential Functions
400(3)
15.8.2 Logarithmic Functions
403(1)
15.9 Differentiating Trigonometric Functions
404(6)
15.9.1 Differentiating tan
404(2)
15.9.2 Differentiating esc
406(1)
15.9.3 Differentiating sec
406(1)
15.9.4 Differentiating cot
407(1)
15.9.5 Differentiating arcsin, arccos and arctan
408(1)
15.9.6 Differentiating arccsc, arcsec and arccot
409(1)
15.10 Differentiating Hyperbolic Functions
410(3)
15.10.1 Differentiating sinh, cosh and tanh
412(1)
15.11 Higher Derivatives
413(1)
15.12 Higher Derivatives of a Polynomial
413(3)
15.13 Identifying a Local Maximum or Minimum
416(2)
15.14 Partial Derivatives
418(8)
15.14.1 Visualising Partial Derivatives
422(1)
15.14.2 Mixed Partial Derivatives
423(3)
15.15 Chain Rule
426(2)
15.16 Total Derivative
428(1)
15.17 Summary
429(2)
16 Calculus: Integration
431(40)
16.1 Introduction
431(1)
16.2 Indefinite Integral
431(1)
16.3 Integration Techniques
432(21)
16.3.1 Continuous Functions
432(1)
16.3.2 Difficult Functions
432(2)
16.3.3 Trigonometric Identities
434(2)
16.3.4 Exponent Notation
436(1)
16.3.5 Completing the Square
437(2)
16.3.6 The Integrand Contains a Derivative
439(1)
16.3.7 Converting the Integrand into a Series of Fractions
440(1)
16.3.8 Integration by Parts
441(5)
16.3.9 Integration by Substitution
446(4)
16.3.10 Partial Fractions
450(3)
16.4 Area Under a Graph
453(1)
16.5 Calculating Areas
453(9)
16.6 Positive and Negative Areas
462(1)
16.7 Area Between Two Functions
463(2)
16.8 Areas with the y-Axis
465(1)
16.9 Area with Parametric Functions
466(2)
16.10 The Riemann Sum
468(2)
16.11 Summary
470(1)
17 Worked Examples
471(20)
17.1 Introduction
471(1)
17.2 Area of Regular Polygon
471(1)
17.3 Area of Any Polygon
472(1)
17.4 Dihedral Angle of a Dodecahedron
473(1)
17.5 Vector Normal to a Triangle
474(1)
17.6 Area of a Triangle Using Vectors
475(1)
17.7 General Form of the Line Equation from Two Points
475(1)
17.8 Angle Between Two Straight Lines
476(1)
17.9 Test if Three Points Lie on a Straight Line
477(1)
17.10 Position and Distance of the Nearest Point on a Line to a Point
478(2)
17.11 Position of a Point Reflected in a Line
480(3)
17.12 Intersection of a Line and a Sphere
483(4)
17.13 Sphere Touching a Plane
487(2)
17.14 Summary
489(2)
18 Conclusion
491(2)
Appendix A Limit of (sinθ)/θ 493(4)
Appendix B Integrating cosnθ 497(2)
Index 499
Professor John Vince began working in computer graphics at Middlesex Polytechnic in 1968. His research activities centered on computer animation software and resulted in the PICASO and PRISM animation systems. Whilst at Middlesex, he designed the UKs first MSc course in Computer Graphics and developed a popular program of short courses in computer animation for television designers. In 1986 he joined Rediffusion Simulation as a Research Consultant and worked on the development of real-time computer systems for commercial flight simulators. In 1992 he was appointed Chief Scientist of Thomson Training Simulation Ltd. In 1995 he was appointed Professor of Digital Media at the National Centre for Computer Animation at Bournemouth University and in 1999 he was made Head of Academic Group for Computer Animation. He was awarded a DSc by Brunel University in recognition of his work in computer graphics. He has written and edited over 40 books on computer graphics, computer animation and virtual reality, including the following Springer titles: Mathematics for Computer Graphics (2014)

Calculus for Computer Graphics (2013)

Matrix Transforms for Computer Games and Animation (2012)

Expanding the Frontiers of Visual Analytics and Visualization (2012)

Quaternions for Computer Graphics (2011)

Rotation Transforms for Computer Graphics (2011
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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