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Contents |
6 |
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Preface |
12 |
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1 Mathematics |
14 |
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1.1 Is Mathematics Difficult? |
15 |
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1.2 Who should Read this Book? |
15 |
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1.3 Aims and Objectives of this Book |
16 |
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1.4 Assumptions Made in this Book |
16 |
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1.5 How to Use the Book |
16 |
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2 Numbers |
18 |
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2.1 Natural Numbers |
18 |
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2.2 Prime Numbers |
19 |
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2.3 Integers |
19 |
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2.4 Rational Numbers |
19 |
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2.5 Irrational Numbers |
19 |
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2.6 Real Numbers |
20 |
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2.7 The Number Line |
20 |
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2.8 Complex Numbers |
20 |
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2.9 Summary |
22 |
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3 Algebra |
23 |
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3.1 Notation |
23 |
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3.2 Algebraic Laws |
24 |
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3.3 Solving the Roots of a Quadratic Equation |
26 |
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3.4 Indices |
27 |
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3.5 Logarithms |
27 |
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3.6 Further Notation |
28 |
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3.7 Summary |
28 |
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4 Trigonometry |
29 |
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4.1 The Trigonometric Ratios |
30 |
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4.2 Example |
30 |
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4.3 Inverse Trigonometric Ratios |
31 |
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4.4 Trigonometric Relationships |
31 |
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4.5 The Sine Rule |
32 |
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4.6 The Cosine Rule |
32 |
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4.7 Compound Angles |
32 |
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4.8 Perimeter Relationships |
33 |
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4.9 Summary |
34 |
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5 Cartesian Coordinates |
35 |
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5.1 The Cartesian xy-plane |
35 |
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5.2 3D Coordinates |
40 |
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5.3 Summary |
41 |
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6 Vectors |
42 |
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6.1 2D Vectors |
43 |
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6.2 3D Vectors |
45 |
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6.3 Deriving a Unit Normal Vector for a Triangle |
58 |
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6.4 Areas |
59 |
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6.5 Summary |
60 |
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7 Transformation |
61 |
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7.1 2D Transformations |
61 |
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7.2 Matrices |
63 |
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7.3 Homogeneous Coordinates |
67 |
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7.4 3D Transformations |
76 |
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7.5 Change of Axes |
83 |
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7.6 Direction Cosines |
85 |
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7.7 Rotating a Point about an Arbitrary Axis |
93 |
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7.8 Transforming Vectors |
108 |
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7.9 Determinants |
109 |
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7.10 Perspective Projection |
113 |
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7.11 Summary |
115 |
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8 Interpolation |
116 |
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8.1 Linear Interpolant |
116 |
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8.2 Non-Linear Interpolation |
119 |
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8.3 Interpolating Vectors |
125 |
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8.4 Interpolating Quaternions |
128 |
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8.5 Summary |
130 |
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9 Curves and Patches |
131 |
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9.1 The Circle |
131 |
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9.2 The Ellipse |
132 |
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9.3 Bézier Curves |
133 |
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9.4 A recursive Bézier Formula |
141 |
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9.5 Bézier Curves Using Matrices |
141 |
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9.6 B-Splines |
145 |
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9.7 Surface Patches |
149 |
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9.8 Summary |
154 |
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10 Analytic Geometry |
155 |
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10.1 Review of Geometry |
155 |
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10.2 2D Analytical Geometry |
164 |
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10.3 Intersection Points |
169 |
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10.4 Point Inside a Triangle |
172 |
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10.5 Intersection of a Circle with a Straight Line |
176 |
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10.6 3D Geometry |
177 |
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10.7 Equation of a Plane |
181 |
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10.8 Intersecting Planes |
189 |
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10.9 Summary |
199 |
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11 Barycentric Coordinates |
200 |
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11.1 Ceva’s Theorem |
200 |
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11.2 Ratios and Proportion |
202 |
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11.3 Mass Points |
203 |
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11.4 Linear Interpolation |
209 |
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11.5 Convex Hull Property |
215 |
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11.6 Areas |
216 |
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11.7 Volumes |
224 |
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11.8 Bézier Curves and Patches |
227 |
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11.9 Summary |
228 |
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12 Worked Examples |
229 |
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12.1 Calculate the Area of a Regular Polygon |
229 |
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12.2 Calculate the Area of any Polygon |
230 |
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12.3 Calculate the Dihedral Angle of a Dodecahedron |
230 |
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12.4 Vector Normal to a Triangle |
232 |
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12.5 Area of a Triangle using Vectors |
233 |
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12.6 General Form of the Line Equation from Two Points |
233 |
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12.7 Calculate the Angle between Two Straight Lines |
234 |
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12.8 Test If Three Points Lie On a Straight Line |
235 |
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12.9 Find the Position and Distance of the Nearest Point on a Line to a Point |
236 |
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12.10 Position of a Point Re.ected in a Line |
238 |
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12.11 Calculate the Intersection of a Line and a Sphere |
240 |
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12.12 Calculate if a Sphere Touches a Plane |
244 |
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12.13 Summary |
245 |
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13 Conclusion |
246 |
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References |
247 |
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Index |
248 |
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