Essential Mathematics for Computer Graphics fast

Front Cover
Springer Science & Business Media, 2001 - Computers - 229 pages
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Baffled by maths? Then don't give up hope.John Vince will show you how to understand many of the mathematical ideas used in computer animation, virtual reality, CAD, and other areas of computer graphics.In ten chapters you will rediscover - and hopefully discover for the first time a new way of understanding - the mathematical techniques required to solve problems and design computer programs for computer graphic applications. Each chapter explores a specific mathematical topic and takes you forward into more advanced areas until you are able to understand 3D curves and surface patches, and solve problems using vectors.After reading the book, you should be able to refer to more challenging books with confidence and develop a greater insight into the design of computer graphics software.Get to grips with mathematics fast ...- Numbers- Algebra- Trigonometry- Coordinate geometry- Transforms- Vectors- Curves and surfaces- Analytic geometryEssential Mathematics for Computer Graphics fastThe book you will read once, and refer to over and over again!
 

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Contents

Mathematics
1
Is mathematics difficult?
3
Who should read this book?
4
Assumptions made in this book
5
Numbers
7
Natural numbers
8
Integers
9
Irrational numbers
10
Rotating about an axis
96
3D reflections
98
2D change of axes
99
Direction cosines
100
Positioning the virtual camera
102
Direction cosines
103
Euler angles
106
Quaternions
110

Summary
13
Algebra
15
Notation
16
Algebraic laws
17
Associative law
18
Distributive law
19
Indices
20
Logarithms
21
Further notation
23
Trigonometry
25
The trigonometric ratios
26
Example
28
Trigonometric relationships
29
The cosine rule
30
Perimeter relationships
31
Summary
32
Cartesian Coordinates
33
The Cartesian xyplane
34
Function graphs
36
Geometric shapes
37
Areas of shapes
38
Theorem of Pythagoras in 2D
39
3D coordinates
40
3D polygons
41
Summary
42
Vectors
43
2D vectors
45
Magnitude of a vector
47
3D vectors
49
Multiplying a vector by a scalar
50
Position vectors
51
Unit vectors
52
Cartesian vectors
53
Vector multiplication
54
Scalar product
55
Example of the dot product
57
The dot product in lighting calculations
58
The dot product in backface detection
59
The vector product
60
The righthand rule
64
Deriving a unit normal vector for a triangle
65
Areas
66
Calculating 2D areas
67
Summary
68
Transformations
69
2D transformations
70
Reflection
71
Matrices
72
Systems of notation
75
The determinant of a matrix
76
Homogeneous coordinates
77
2D translation
79
2D reflections
80
2D shearing
82
2D rotation
83
2D scaling
86
2D reflections
87
2D rotation about an arbitrary point
88
3D transformations
89
3D scaling
90
Gimbal lock
95
Multiplying quaternions
111
Rotating points about an axis
112
Roll pitch and yaw quaternions
116
Quaternions in matrix form
118
Frames of reference
119
Transforming vectors
120
Determinants
122
Perspective projection
126
Summary
128
Interpolation
129
Linear interpolant
130
Nonlinear interpolation
133
Cubic interpolation
135
Interpolating vectors
141
Interpolating quaternions
145
Summary
147
Curves and Patches
149
The circle
150
The ellipse
151
Bezier curves
152
Quadratic Bezier curves
157
Cubic Bernstein polynomials
158
A recursive Bezier formula
162
Linear interpolation
164
Bsplines
167
Uniform Bsplines
168
Continuity
171
Nonuniform Bsplines
172
Surface patches
173
Quadratic Bezier surface patch
174
Cubic Bezier surface patch
177
Summary
180
Analytic Geometry
181
Review of geometry
182
Intercept theorems
183
Golden section
184
Triangles
185
Isosceles triangle
186
Equilateral triangle
187
Theorem of Pythagoras
188
Trapezoid
189
Rhombus
190
Circle
192
2D analytical geometry
193
The Hessian normal form
194
Space partitioning
197
The Hessian normal form from two points
198
Intersection points
199
Point inside a triangle
202
Hessian normal form
205
Intersection of a circle with a straight line
207
3D geometry
209
Point of intersection of two straight lines
211
Equation of a plane
214
Space partitioning
216
Point of intersection of a line segment and a plane
218
Summary
219
Conclusion
221
References
223
Index
225
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