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Mathematics for Computer Graphics 4th ed. 2014 [Mīkstie vāki]

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  • Formāts: Paperback / softback, 391 pages, height x width: 235x155 mm, weight: 629 g, 14 Tables, black and white; 282 Illustrations, black and white; XVII, 391 p. 282 illus., 1 Paperback / softback
  • Sērija : Undergraduate Topics in Computer Science
  • Izdošanas datums: 20-Dec-2013
  • Izdevniecība: Springer London Ltd
  • ISBN-10: 1447162897
  • ISBN-13: 9781447162896
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Mathematics for Computer Graphics 4th ed. 2014Palielināt
  • Formāts: Paperback / softback, 391 pages, height x width: 235x155 mm, weight: 629 g, 14 Tables, black and white; 282 Illustrations, black and white; XVII, 391 p. 282 illus., 1 Paperback / softback
  • Sērija : Undergraduate Topics in Computer Science
  • Izdošanas datums: 20-Dec-2013
  • Izdevniecība: Springer London Ltd
  • ISBN-10: 1447162897
  • ISBN-13: 9781447162896
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9781447162896.html
Keywords:
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 updated and expanded fourth edition.

The first four chapters revise number sets, algebra, trigonometry and coordinate systems, which are employed in the following chapters on vectors, transforms, interpolation, 3D curves and patches, analytic geometry and barycentric coordinates. Following this, the reader is introduced to the relatively new topic of geometric algebra, and the last two chapters provide an introduction to differential and integral calculus, with an emphasis on geometry.

Mathematics for Computer Graphics covers all of the key areas of the subject, including:





Number sets Algebra Trigonometry Coordinate systems Transforms Quaternions Interpolation Curves and surfaces Analytic geometry Barycentric coordinates Geometric algebra Differential calculus Integral calculus

This fourth edition contains over 120 worked examples and over 270 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

From the book reviews:

This book is undoubtedly a very well written book. The chapters are short, easy to follow and understand. it is a neatly compiled book that many math enthusiasts may want to have on their bookshelves. a really good book both technically as well as pedagogically. (Leslie P. Piegl, zbMATH, Vol. 1295, 2014)

The books 16 chapters cover a broad range of relevant mathematical topics. The logics in the book are presented very clearly, and readers from all levels with an understanding of arithmetic, algebra, trigonometry, and geometry will find this book easy to follow. The worked examples are very helpful in practical projects as well. 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, August, 2014)

1 Mathematics
1(2)
1.1 Aims and Objectives of This Book
1(1)
1.2 Who Should Read This Book?
1(1)
1.3 Assumptions Made in This Book
1(1)
1.4 How to Use the Book
2(1)
1.5 Is Mathematics Difficult?
2(1)
2 Numbers
3(8)
2.1 Introduction
3(1)
2.2 Background
3(1)
2.3 Set Notation
3(1)
2.4 Positional Number System
4(1)
2.5 Natural Numbers
4(1)
2.6 Prime Numbers
4(1)
2.7 Integer Numbers
5(1)
2.8 Rational Numbers
5(1)
2.9 Irrational Numbers
6(1)
2.10 Real Numbers
6(1)
2.11 The Number Line
6(1)
2.12 Complex Numbers
6(3)
2.13 Summary
9(2)
3 Algebra
11(12)
3.1 Introduction
11(1)
3.2 Background
11(1)
3.3 Algebraic Laws
12(2)
3.3.1 Associative Law
13(1)
3.3.2 Commutative Law
13(1)
3.3.3 Distributive Law
14(1)
3.4 Solving the Roots of a Quadratic Equation
14(1)
3.5 Indices
15(1)
3.5.1 Laws of Indices
16(1)
3.6 Logarithms
16(2)
3.7 Further Notation
18(1)
3.8 Functions
18(3)
3.8.1 Explicit and Implicit Equations
19(1)
3.8.2 Function Notation
19(1)
3.8.3 Intervals
20(1)
3.8.4 Function Domains and Ranges
21(1)
3.9 Summary
21(2)
4 Trigonometry
23(10)
4.1 Introduction
23(1)
4.2 Background
23(1)
4.3 Units of Angular Measurement
23(1)
4.4 The Trigonometric Ratios
24(3)
4.4.1 Domains and Ranges
26(1)
4.5 Inverse Trigonometric Ratios
27(2)
4.6 Trigonometric Identities
29(1)
4.7 The Sine Rule
29(1)
4.8 The Cosine Rule
30(1)
4.9 Compound Angles
30(1)
4.10 Perimeter Relationships
31(1)
4.11 Summary
31(2)
5 Coordinate Systems
33(10)
5.1 Introduction
33(1)
5.2 Background
33(1)
5.3 The Cartesian Plane
34(1)
5.4 Function Graphs
34(1)
5.5 Geometric Shapes
35(2)
5.5.1 Polygonal Shapes
35(1)
5.5.2 Areas of Shapes
36(1)
5.6 Theorem of Pythagoras in 2D
37(1)
5.7 3D Cartesian Coordinates
37(2)
5.7.1 Theorem of Pythagoras in 3D
38(1)
5.7.2 3D Polygons
39(1)
5.7.3 Euler's Rule
39(1)
5.8 Polar Coordinates
39(1)
5.9 Spherical Polar Coordinates
40(1)
5.10 Cylindrical Coordinates
41(1)
5.11 Summary
42(1)
6 Vectors
43(22)
6.1 Introduction
43(1)
6.2 Background
43(1)
6.3 2D Vectors
44(3)
6.3.1 Vector Notation
44(1)
6.3.2 Graphical Representation of Vectors
45(1)
6.3.3 Magnitude of a Vector
46(1)
6.4 3D Vectors
47(14)
6.4.1 Vector Manipulation
48(1)
6.4.2 Scaling a Vector
48(1)
6.4.3 Vector Addition and Subtraction
49(1)
6.4.4 Position Vectors
49(1)
6.4.5 Unit Vectors
50(1)
6.4.6 Cartesian Vectors
51(1)
6.4.7 Vector Products
52(1)
6.4.8 Scalar Product
52(2)
6.4.9 The Dot Product in Lighting Calculations
54(1)
6.4.10 The Scalar Product in Back-Face Detection
55(1)
6.4.11 The Vector Product
56(4)
6.4.12 The Right-Hand Rule
60(1)
6.5 Deriving a Unit Normal Vector for a Triangle
61(1)
6.6 Areas
61(2)
6.6.1 Calculating 2D Areas
62(1)
6.7 Summary
63(2)
7 Transforms
65(56)
7.1 Introduction
65(1)
7.2 Background
65(1)
7.3 2D Transforms
66(1)
7.3.1 Translation
66(1)
7.3.2 Scaling
66(1)
7.3.3 Reflection
66(1)
7.4 Matrices
67(4)
7.4.1 Systems of Notation
70(1)
7.4.2 The Determinant of a Matrix
70(1)
7.5 Homogeneous Coordinates
71(9)
7.5.1 2D Translation
72(1)
7.5.2 2D Scaling
72(1)
7.5.3 2D Reflections
73(2)
7.5.4 2D Shearing
75(1)
7.5.5 2D Rotation
76(2)
7.5.6 2D Scaling
78(1)
7.5.7 2D Reflection
79(1)
7.5.8 2D Rotation About an Arbitrary Point
79(1)
7.6 3D Transforms
80(7)
7.6.1 3D Translation
80(1)
7.6.2 3D Scaling
81(1)
7.6.3 3D Rotation
81(4)
7.6.4 Gimbal Lock
85(1)
7.6.5 Rotating About an Axis
86(1)
7.6.6 3D Reflections
87(1)
7.7 Change of Axes
87(3)
7.7.1 2D Change of Axes
88(1)
7.7.2 Direction Cosines
89(1)
7.7.3 3D Change of Axes
90(1)
7.8 Positioning the Virtual Camera
90(6)
7.8.1 Direction Cosines
91(2)
7.8.2 Euler Angles
93(3)
7.9 Rotating a Point About an Arbitrary Axis
96(17)
7.9.1 Matrices
96(7)
7.9.2 Quaternions
103(1)
7.9.3 Adding and Subtracting Quaternions
104(1)
7.9.4 Multiplying Quaternions
104(1)
7.9.5 Pure Quaternion
105(1)
7.9.6 The Inverse Quaternion
105(1)
7.9.7 Unit Quaternion
106(1)
7.9.8 Rotating Points About an Axis
106(3)
7.9.9 Roll, Pitch and Yaw Quaternions
109(2)
7.9.10 Quaternions in Matrix Form
111(1)
7.9.11 Frames of Reference
112(1)
7.10 Transforming Vectors
113(1)
7.11 Determinants
114(4)
7.12 Perspective Projection
118(2)
7.13 Summary
120(1)
8 Interpolation
121(14)
8.1 Introduction
121(1)
8.2 Background
121(1)
8.3 Linear Interpolation
121(3)
8.4 Non-linear Interpolation
124(6)
8.4.1 Trigonometric Interpolation
124(1)
8.4.2 Cubic Interpolation
125(5)
8.5 Interpolating Vectors
130(3)
8.6 Interpolating Quaternions
133(1)
8.7 Summary
134(1)
9 Curves and Patches
135(24)
9.1 Introduction
135(1)
9.2 Background
135(1)
9.3 The Circle
135(1)
9.4 The Ellipse
136(1)
9.5 Bezier Curves
136(7)
9.5.1 Bernstein Polynomials
136(4)
9.5.2 Quadratic Bezier Curves
140(1)
9.5.3 Cubic Bernstein Polynomials
141(2)
9.6 A Recursive Bezier Formula
143(1)
9.7 Bezier Curves Using Matrices
144(3)
9.7.1 Linear Interpolation
145(2)
9.8 B-Splines
147(5)
9.8.1 Uniform B-Splines
148(2)
9.8.2 Continuity
150(1)
9.8.3 Non-uniform B-Splines
151(1)
9.8.4 Non-uniform Rational B-Splines
151(1)
9.9 Surface Patches
152(5)
9.9.1 Planar Surface Patch
152(1)
9.9.2 Quadratic Bezier Surface Patch
153(2)
9.9.3 Cubic Bezier Surface Patch
155(2)
9.10 Summary
157(2)
10 Analytic Geometry
159(44)
10.1 Introduction
159(1)
10.2 Background
159(8)
10.2.1 Angles
159(1)
10.2.2 Intercept Theorems
160(1)
10.2.3 Golden Section
161(1)
10.2.4 Triangles
161(1)
10.2.5 Centre of Gravity of a Triangle
162(1)
10.2.6 Isosceles Triangle
162(1)
10.2.7 Equilateral Triangle
162(1)
10.2.8 Right Triangle
162(1)
10.2.9 Theorem of Thales
163(1)
10.2.10 Theorem of Pythagoras
163(1)
10.2.11 Quadrilateral
164(1)
10.2.12 Trapezoid
164(1)
10.2.13 Parallelogram
164(1)
10.2.14 Rhombus
165(1)
10.2.15 Regular Polygon
165(1)
10.2.16 Circle
165(2)
10.3 2D Analytic Geometry
167(4)
10.3.1 Equation of a Straight Line
167(1)
10.3.2 The Hessian Normal Form
168(1)
10.3.3 Space Partitioning
169(1)
10.3.4 The Hessian Normal Form from Two Points
170(1)
10.4 Intersection Points
171(3)
10.4.1 Intersecting Straight Lines
171(1)
10.4.2 Intersecting Line Segments
172(2)
10.5 Point Inside a Triangle
174(3)
10.5.1 Area of a Triangle
174(2)
10.5.2 Hessian Normal Form
176(1)
10.6 Intersection of a Circle with a Straight Line
177(2)
10.7 3D Geometry
179(4)
10.7.1 Equation of a Straight Line
179(1)
10.7.2 Intersecting Two Straight Lines
180(3)
10.8 Equation of a Plane
183(8)
10.8.1 Cartesian Form of the Plane Equation
183(3)
10.8.2 General Form of the Plane Equation
186(1)
10.8.3 Parametric Form of the Plane Equation
186(1)
10.8.4 Converting from the Parametric to the General Form
187(2)
10.8.5 Plane Equation from Three Points
189(2)
10.9 Intersecting Planes
191(10)
10.9.1 Intersection of Three Planes
194(2)
10.9.2 Angle Between Two Planes
196(2)
10.9.3 Angle Between a Line and a Plane
198(1)
10.9.4 Intersection of a Line with a Plane
199(2)
10.10 Summary
201(2)
11 Barycentric Coordinates
203(28)
11.1 Introduction
203(1)
11.2 Background
203(1)
11.3 Ceva's Theorem
203(2)
11.4 Ratios and Proportion
205(1)
11.5 Mass Points
206(5)
11.6 Linear Interpolation
211(7)
11.7 Convex Hull Property
218(1)
11.8 Areas
218(8)
11.9 Volumes
226(2)
11.10 Bezier Curves and Patches
228(1)
11.11 Summary
229(2)
12 Geometric Algebra
231(34)
12.1 Introduction
231(1)
12.2 Background
231(1)
12.3 Symmetric and Antisymmetric Functions
231(2)
12.4 Trigonometric Foundations
233(1)
12.5 Vectorial Foundations
234(1)
12.6 Inner and Outer Products
235(1)
12.7 The Geometric Product in 2D
236(2)
12.8 The Geometric Product in 3D
238(2)
12.9 The Outer Product of Three 3D Vectors
240(1)
12.10 Axioms
241(1)
12.11 Notation
242(1)
12.12 Grades, Pseudoscalars and Multivectors
243(1)
12.13 Redefining the Inner and Outer Products
244(1)
12.14 The Inverse of a Vector
244(2)
12.15 The Imaginary Properties of the Outer Product
246(2)
12.16 Duality
248(1)
12.17 The Relationship Between the Vector Product and the Outer Product
249(1)
12.18 The Relationship between Quaternions and Bivectors
249(1)
12.19 Reflections and Rotations
250(4)
12.19.1 2D Reflections
250(1)
12.19.2 3D Reflections
251(1)
12.19.3 2D Rotations
252(2)
12.20 Rotors
254(3)
12.21 Applied Geometric Algebra
257(6)
12.22 Summary
263(2)
13 Calculus: Derivatives
265(54)
13.1 Introduction
265(1)
13.2 Background
265(1)
13.3 Small Numerical Quantities
265(2)
13.4 Equations and Limits
267(8)
13.4.1 Quadratic Function
267(1)
13.4.2 Cubic Equation
268(2)
13.4.3 Functions and Limits
270(2)
13.4.4 Graphical Interpretation of the Derivative
272(1)
13.4.5 Derivatives and Differentials
273(1)
13.4.6 Integration and Antiderivatives
273(2)
13.5 Function Types
275(1)
13.6 Differentiating Groups of Functions
276(11)
13.6.1 Sums of Functions
276(2)
13.6.2 Function of a Function
278(3)
13.6.3 Function Products
281(5)
13.6.4 Function Quotients
286(1)
13.7 Differentiating Implicit Functions
287(4)
13.8 Differentiating Exponential and Logarithmic Functions
291(4)
13.8.1 Exponential Functions
291(2)
13.8.2 Logarithmic Functions
293(2)
13.9 Differentiating Trigonometric Functions
295(5)
13.9.1 Differentiating tan
295(1)
13.9.2 Differentiating CSC
296(1)
13.9.3 Differentiating sec
296(2)
13.9.4 Differentiating cot
298(1)
13.9.5 Differentiating arcsin, arccos and arctan
298(1)
13.9.6 Differentiating arccsc, arcsec and arccot
299(1)
13.10 Differentiating Hyperbolic Functions
300(2)
13.10.1 Differentiating sinh, cosh and tanh
301(1)
13.11 Higher Derivatives
302(1)
13.12 Higher Derivatives of a Polynomial
303(2)
13.13 Identifying a Local Maximum or Minimum
305(2)
13.14 Partial Derivatives
307(7)
13.14.1 Visualising Partial Derivatives
311(1)
13.14.2 Mixed Partial Derivatives
312(2)
13.15 Chain Rule
314(2)
13.16 Total Derivative
316(1)
13.17 Summary
317(2)
14 Calculus: Integration
319(42)
14.1 Introduction
319(1)
14.2 Indefinite Integral
319(1)
14.3 Integration Techniques
320(24)
14.3.1 Continuous Functions
320(1)
14.3.2 Difficult Functions
320(1)
14.3.3 Trigonometric Identities
321(3)
14.3.4 Exponent Notation
324(1)
14.3.5 Completing the Square
325(1)
14.3.6 The Integrand Contains a Derivative
326(3)
14.3.7 Converting the Integrand into a Series of Fractions
329(1)
14.3.8 Integration by Parts
330(7)
14.3.9 Integration by Substitution
337(4)
14.3.10 Partial Fractions
341(3)
14.4 Area Under a Graph
344(1)
14.5 Calculating Areas
345(7)
14.6 Positive and Negative Areas
352(2)
14.7 Area Between Two Functions
354(1)
14.8 Areas with the y-Axis
355(1)
14.9 Area with Parametric Functions
356(2)
14.10 Bernhard Riemann
358(2)
14.10.1 Domains and Intervals
358(1)
14.10.2 The Riemann Sum
359(1)
14.11 Summary
360(1)
15 Worked Examples
361(18)
15.1 Introduction
361(1)
15.2 Area of Regular Polygon
361(1)
15.3 Area of Any Polygon
362(1)
15.4 Dihedral Angle of a Dodecahedron
363(1)
15.5 Vector Normal to a Triangle
364(1)
15.6 Area of a Triangle Using Vectors
365(1)
15.7 General Form of the Line Equation from Two Points
365(1)
15.8 Angle Between Two Straight Lines
366(1)
15.9 Test if Three Points Lie on a Straight Line
367(1)
15.10 Position and Distance of the Nearest Point on a Line to a Point
367(3)
15.11 Position of a Point Reflected in a Line
370(2)
15.12 Intersection of a Line and a Sphere
372(4)
15.13 Sphere Touching a Plane
376(2)
15.14 Summary
378(1)
16 Conclusion
379(2)
Appendix A Limit of (sinθ)/θ 381(4)
Appendix B Integrating cosnθ 385(2)
Index 387
Prof 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:





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)