Mathematics for computer graphics:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
London
Springer
2014
|
| Edition: | 4. ed. |
| Series: | Undergraduate topics in computer science
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVII, 391 S. Ill., graph. Darst. |
| ISBN: | 9781447162896 |
Staff View
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Record in the Search Index
| _version_ | 1804152237303267328 |
|---|---|
| adam_text | Titel: Mathematics for computer graphics
Autor: Vince, John
Jahr: 2014
Contents
1 Mathematics ............................................................1
1.1 Aims and Objectives of This Book................................1
1.2 Who Should Read This Book?....................................1
1.3 Assumptions Made in This Book..................................1
1.4 How to Use the Book..............................................2
1.5 Is Mathematics Difficult?..........................................2
2 Numbers..................................................................3
2.1 Introduction........................................................3
2.2 Background........................................................3
2.3 Set Notation ......................................................3
2.4 Positional Number System........................................4
2.5 Natural Numbers..................................................4
2.6 Prime Numbers....................................................4
2.7 Integer Numbers..................................................5
2.8 Rational Numbers ................................................5
2.9 Irrational Numbers................................................6
2.10 Real Numbers......................................................6
2.11 The Number Line..................................................6
2.12 Complex Numbers................................................6
2.13 Summary..........................................................9
3 Algebra..................................................................11
3.1 Introduction........................................................11
3.2 Background........................................................11
3.3 Algebraic Laws....................................................12
3.3.1 Associative Law..........................................13
3.3.2 Commutative Law........................................13
3.3.3 Distributive Law ........................................14
3.4 Solving the Roots of a Quadratic Equation........................14
3.5 Indices............................................................15
ix
X Contents
3.5.1 Laws of Indices..........................................16
3.6 Logarithms........................................................16
3.7 Further Notation..................................................18
3.8 Functions..........................................................18
3.8.1 Explicit and Implicit Equations..........................19
3.8.2 Function Notation........................................19
3.8.3 Intervals..................................................20
3.8.4 Function Domains and Ranges..........................21
3.9 Summary..........................................................21
4 Trigonometry............. ..................................23
4.1 Introduction........................................................23
4.2 Background........................................................23
4.3 Units of Angular Measurement....................................23
4.4 The Trigonometric Ratios ........................................24
4.4.1 Domains and Ranges....................................26
4.5 Inverse Trigonometric Ratios......................................27
4.6 Trigonometric Identities..........................................29
4.7 The Sine Rule......................................................29
4.8 The Cosine Rule..................................................30
4.9 Compound Angles................................................30
4.10 Perimeter Relationships ..........................................31
4.11 Summary..........................................................31
5 Coordinate Systems ....................................................33
5.1 Introduction........................................................33
5.2 Background........................................................33
5.3 The Cartesian Plane ..............................................34
5.4 Function Graphs..................................................34
5.5 Geometric Shapes ................................................35
5.5.1 Polygonal Shapes........................................35
5.5.2 Areas of Shapes..........................................36
5.6 Theorem of Pythagoras in 2D....................................37
5.7 3D Cartesian Coordinates ........................................37
5.7.1 Theorem of Pythagoras in 3D............................38
5.7.2 3D Polygons..............................................39
5.7.3 Euler s Rule..............................................39
5.8 Polar Coordinates..................................................39
5.9 Spherical Polar Coordinates......................................40
5.10 Cylindrical Coordinates ..........................................41
5.11 Summary..........................................................42
6 Vectors....................................................................43
6.1 Introduction........................................................43
6.2 Background........................................................43
6.3 2D Vectors........................................................44
Contents xi
6.3.1 Vector Notation..........................................44
6.3.2 Graphical Representation of Vectors....................45
6.3.3 Magnitude of a Vector ..................................46
6.4 3D Vectors........................................................47
6.4.1 Vector Manipulation ....................................48
6.4.2 Scaling a Vector..........................................48
6.4.3 Vector Addition and Subtraction........................49
6.4.4 Position Vectors..........................................49
6.4.5 Unit Vectors..............................................50
6.4.6 Cartesian Vectors........................................51
6.4.7 Vector Products..........................................52
6.4.8 Scalar Product............................................52
6.4.9 The Dot Product in Lighting Calculations..............54
6.4.10 The Scalar Product in Back-Face Detection............55
6.4.11 The Vector Product......................................56
6.4.12 The Right-Hand Rule....................................60
6.5 Deriving a Unit Normal Vector for a Triangle....................61
6.6 Areas..............................................................61
6.6.1 Calculating 2D Areas....................................62
6.7 Summary..........................................................63
7 Transforms..............................................................65
7.1 Introduction........................................................65
7.2 Background........................................................65
7.3 2D Transforms....................................................66
7.3.1 Translation ..............................................66
7.3.2 Scaling ..................................................66
7.3.3 Reflection................................................66
7.4 Matrices............................................................67
7.4.1 Systems of Notation......................................70
7.4.2 The Determinant of a Matrix............................70
7.5 Homogeneous Coordinates........................................71
7.5.1 2D Translation ..........................................72
7.5.2 2D Scaling ..............................................72
7.5.3 2D Reflections ..........................................73
7.5.4 2D Shearing..............................................75
7.5.5 2D Rotation..............................................76
7.5.6 2D Scaling ..............................................78
7.5.7 2D Reflection............................................79
7.5.8 2D Rotation About an Arbitrary Point..................79
7.6 3D Transforms....................................................80
7.6.1 3D Translation ..........................................80
7.6.2 3D Scaling ..............................................81
7.6.3 3D Rotation..............................................81
7.6.4 Gimbal Lock............................................85
x¡¡ Contents
7.6.5 Rotating About an Axis..................................86
7.6.6 3D Reflections ..........................................87
7.7 Change of Axes....................................................87
7.7.1 2D Change of Axes......................................88
7.7.2 Direction Cosines........................................89
7.7.3 3D Change of Axes......................................90
7.8 Positioning the Virtual Camera....................................90
7.8.1 Direction Cosines........................................91
7.8.2 Euler Angles ............................................93
7.9 Rotating a Point About an Arbitrary Axis........................96
7.9.1 Matrices..................................................96
7.9.2 Quaternions..............................................103
7.9.3 Adding and Subtracting Quaternions....................104
7.9.4 Multiplying Quaternions................................104
7.9.5 Pure Quaternion..........................................105
7.9.6 The Inverse Quaternion..................................105
7.9.7 Unit Quaternion..........................................106
7.9.8 Rotating Points About an Axis..........................106
7.9.9 Roll, Pitch and Yaw Quaternions........................109
7.9.10 Quaternions in Matrix Form ............................Ill
7.9.11 Frames of Reference....................................112
7.10 Transforming Vectors..............................................113
7.11 Determinants......................................................114
7.12 Perspective Projection............................................118
7.13 Summary..........................................................120
8 Interpolation............................................................121
8.1 Introduction........................................................121
8.2 Background........................................................121
8.3 Linear Interpolation ..............................................121
8.4 Non-linear Interpolation..........................................124
8.4.1 Trigonometric Interpolation..............................124
8.4.2 Cubic Interpolation......................................125
8.5 Interpolating Vectors..............................................130
8.6 Interpolating Quaternions ........................................133
8.7 Summary..........................................................134
9 Curves and Patches......................................................135
9.1 Introduction........................................................135
9.2 Background........................................................135
9.3 The Circle..........................................................135
9.4 The Ellipse........................................................136
9.5 Bézier Curves......................................................136
9.5.1 Bernstein Polynomials..................................136
9.5.2 Quadratic Bézier Curves................................140
9.5.3 Cubic Bernstein Polynomials............................141
Contents xiii
9.6 A Recursive Bézier Formula......................................143
9.7 Bézier Curves Using Matrices....................................144
9.7.1 Linear Interpolation......................................145
9.8 B-Splines..........................................................147
9.8.1 Uniform B-Splines......................................148
9.8.2 Continuity................................................150
9.8.3 Non-uniform B-Splines..................................151
9.8.4 Non-uniform Rational B-Splines........................151
9.9 Surface Patches....................................................152
9.9.1 Planar Surface Patch....................................152
9.9.2 Quadratic Bézier Surface Patch..........................153
9.9.3 Cubic Bézier Surface Patch..............................155
9.10 Summary..........................................................157
10 Analytic Geometry......................................................159
10.1 Introduction........................................................159
10.2 Background........................................................159
10.2.1 Angles....................................................159
10.2.2 Intercept Theorems......................................160
10.2.3 Golden Section..........................................161
10.2.4 Triangles ................................................161
10.2.5 Centre of Gravity of a Triangle..........................162
10.2.6 Isosceles Triangle........................................162
10.2.7 Equilateral Triangle......................................162
10.2.8 Right Triangle............................................162
10.2.9 Theorem of Thaïes......................................163
10.2.10 Theorem of Pythagoras..................................163
10.2.11 Quadrilateral ............................................164
10.2.12 Trapezoid................................................164
10.2.13 Parallelogram............................................164
10.2.14 Rhombus ................................................165
10.2.15 Regular Polygon ........................................165
10.2.16 Circle....................................................165
10.3 2D Analytic Geometry............................................167
10.3.1 Equation of a Straight Line..............................167
10.3.2 The Hessian Normal Form..............................168
10.3.3 Space Partitioning........................................169
10.3.4 The Hessian Normal Form from Two Points............170
10.4 Intersection Points................................................171
10.4.1 Intersecting Straight Lines..............................171
10.4.2 Intersecting Line Segments..............................172
10.5 Point Inside a Triangle............................................174
10.5.1 Area of a Triangle........................................174
10.5.2 Hessian Normal Form....................................176
10.6 Intersection of a Circle with a Straight Line......................177
xiv Contents
10.7 3D Geometry......................................................179
10.7.1 Equation of a Straight Line..............................179
10.7.2 Intersecting Two Straight Lines..........................180
10.8 Equation of a Plane................................................183
10.8.1 Cartesian Form of the Plane Equation..................183
10.8.2 General Form of the Plane Equation....................186
10.8.3 Parametric Form of the Plane Equation..................186
10.8.4 Converting from the Parametric to the General Form . 187
10.8.5 Plane Equation from Three Points......................189
10.9 Intersecting Planes................................................191
10.9.1 Intersection of Three Planes ............................194
10.9.2 Angle Between Two Planes..............................196
10.9.3 Angle Between a Line and a Plane......................198
10.9.4 Intersection of a Line with a Plane......................199
10.10 Summary..........................................................201
11 Barycentric Coordinates................................................203
11.1 Introduction........................................................203
11.2 Background........................................................203
11.3 Ceva s Theorem ..................................................203
11.4 Ratios and Proportion ............................................205
11.5 Mass Points........................................................206
11.6 Linear Interpolation ..............................................211
11.7 Convex Hull Property ............................................218
11.8 Areas..............................................................218
11.9 Volumes............................................................226
11.10 Bézier Curves and Patches........................................228
11.11 Summary..........................................................229
12 Geometric Algebra......................................................231
12.1 Introduction........................................................231
12.2 Background........................................................231
12.3 Symmetric and Antisymmetric Functions........................231
12.4 Trigonometric Foundations........................................233
12.5 Vectorial Foundations ............................................234
12.6 Inner and Outer Products..........................................235
12.7 The Geometric Product in 2D....................................236
12.8 The Geometric Product in 3D....................................238
12.9 The Outer Product of Three 3D Vectors..........................240
12.10 Axioms............................................................241
12.11 Notation ..........................................................242
12.12 Grades, Pseudoscalars and Multivectors..........................243
12.13 Redefining the Inner and Outer Products..........................244
12.14 The Inverse of a Vector............................................244
12.15 The Imaginary Properties of the Outer Product..................246
12.16 Duality............................................................248
Contents xv
12.17 The Relationship Between the Vector Product and the Outer
Product............................................................249
12.18 The Relationship between Quaternions and Bivectors............249
12.19 Reflections and Rotations ........................................250
12.19.1 2D Reflections ..........................................250
12.19.2 3D Reflections ..........................................251
12.19.3 2D Rotations............................................252
12.20 Rotors..............................................................254
12.21 Applied Geometric Algebra......................................257
12.22 Summary..........................................................263
13 Calculus: Derivatives....................................................265
13.1 Introduction........................................................265
13.2 Background........................................................265
13.3 Small Numerical Quantities......................................265
13.4 Equations and Limits..............................................267
13.4.1 Quadratic Function......................................267
13.4.2 Cubic Equation..........................................268
13.4.3 Functions and Limits....................................270
13.4.4 Graphical Interpretation of the Derivative..............272
13.4.5 Derivatives and Differentials............................273
13.4.6 Integration and Antiderivatives..........................273
13.5 Function Types....................................................275
13.6 Differentiating Groups of Functions............... 276
13.6.1 Sums of Functions ......................................276
13.6.2 Function of a Function................. 278
13.6.3 Function Products.................... 281
13.6.4 Function Quotients................... 286
13.7 Differentiating Implicit Functions................ 287
13.8 Differentiating Exponential and Logarithmic Functions..... 291
13.8.1 Exponential Functions ................. 291
13.8.2 Logarithmic Functions................. 293
13.9 Differentiating Trigonometric Functions............. 295
13.9.1 Differentiating tan........................................295
13.9.2 Differentiating csc........................................296
13.9.3 Differentiating sec........................................296
13.9.4 Differentiating cot........................................298
13.9.5 Differentiating arcsin, arccos and arctan........ 298
13.9.6 Differentiating arccsc, arcsec and arccot........ 299
13.10 Differentiating Hyperbolic Functions.............. 300
13.10.1 Differentiating sinh, cosh and tanh......................301
13.11 Higher Derivatives........................ 302
13.12 Higher Derivatives of a Polynomial............... 303
13.13 Identifying a Local Maximum or Minimum........... 305
13.14 Partial Derivatives ........................ 307
xvi Contents
13.14.1 Visualising Partial Derivatives..........................311
13.14.2 Mixed Partial Derivatives................................312
13.15 Chain Rule........................................................314
13.16 Total Derivative....................................................316
13.17 Summary..........................................................317
14 Calculus: Integration....................................................319
14.1 Introduction........................................................319
14.2 Indefinite Integral..................................................319
14.3 Integration Techniques............................................320
14.3.1 Continuous Functions....................................320
14.3.2 Difficult Functions ......................................320
14.3.3 Trigonometrie Identities ................................321
14.3.4 Exponent Notation......................................324
14.3.5 Completing the Square..................................325
14.3.6 The Integrand Contains a Derivative....................326
14.3.7 Converting the Integrand into a Series of Fractions . . 329
14.3.8 Integration by Parts......................................330
14.3.9 Integration by Substitution..............................337
14.3.10 Partial Fractions..........................................341
14.4 Area Under a Graph ..............................................344
14.5 Calculating Areas..................................................345
14.6 Positive and Negative Areas......................................352
14.7 Area Between Two Functions....................................354
14.8 Areas with the y-Axis ............................................355
14.9 Area with Parametric Functions..................................356
14.10 Bernhard Riemann................................................358
14.10.1 Domains and Intervals ..................................358
14.10.2 The Riemann Sum ......................................359
14.11 Summary..........................................................360
15 Worked Examples ......................................................361
15.1 Introduction........................................................361
15.2 Area of Regular Polygon..........................................361
15.3 Area of Any Polygon..............................................362
15.4 Dihedral Angle of a Dodecahedron ..............................363
15.5 Vector Normal to a Triangle......................................364
15.6 Area of a Triangle Using Vectors..................................365
15.7 General Form of the Line Equation from Two Points............365
15.8 Angle Between Two Straight Lines ..............................366
15.9 Test if Three Points Lie on a Straight Line........................367
15.10 Position and Distance of the Nearest Point on a Line to a Point . 367
15.11 Position of a Point Reflected in a Line............................370
15.12 Intersection of a Line and a Sphere................................372
15.13 Sphere Touching a Plane..........................................376
15.14 Summary..........................................................378
Contents xvii
16 Conclusion ............................... 379
Appendix A Limit of (sin$)/0 ...................... 381
Appendix B Integrating cos 6...................... 385
Index..................................... 387
|
| any_adam_object | 1 |
| author | Vince, John 1941- |
| author_GND | (DE-588)120106604 |
| author_facet | Vince, John 1941- |
| author_role | aut |
| author_sort | Vince, John 1941- |
| author_variant | j v jv |
| building | Verbundindex |
| bvnumber | BV041888791 |
| classification_rvk | SK 110 ST 320 |
| ctrlnum | (OCoLC)870171517 (DE-599)BVBBV041888791 |
| discipline | Informatik Mathematik |
| edition | 4. ed. |
| format | Book |
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| genre | (DE-588)4151278-9 Einführung gnd-content |
| genre_facet | Einführung |
| id | DE-604.BV041888791 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T01:07:31Z |
| institution | BVB |
| isbn | 9781447162896 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027332754 |
| oclc_num | 870171517 |
| open_access_boolean | |
| owner | DE-83 |
| owner_facet | DE-83 |
| physical | XVII, 391 S. Ill., graph. Darst. |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | Springer |
| record_format | marc |
| series2 | Undergraduate topics in computer science |
| spelling | Vince, John 1941- Verfasser (DE-588)120106604 aut Mathematics for computer graphics John Vince 4. ed. London Springer 2014 XVII, 391 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Undergraduate topics in computer science Informatik (DE-588)4026894-9 gnd rswk-swf Computergrafik (DE-588)4010450-3 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Computergrafik (DE-588)4010450-3 s Mathematik (DE-588)4037944-9 s DE-604 Informatik (DE-588)4026894-9 s 1\p DE-604 Erscheint auch als Online-Ausgabe 978-1-4471-6290-2 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027332754&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Vince, John 1941- Mathematics for computer graphics Informatik (DE-588)4026894-9 gnd Computergrafik (DE-588)4010450-3 gnd Mathematik (DE-588)4037944-9 gnd |
| subject_GND | (DE-588)4026894-9 (DE-588)4010450-3 (DE-588)4037944-9 (DE-588)4151278-9 |
| title | Mathematics for computer graphics |
| title_auth | Mathematics for computer graphics |
| title_exact_search | Mathematics for computer graphics |
| title_full | Mathematics for computer graphics John Vince |
| title_fullStr | Mathematics for computer graphics John Vince |
| title_full_unstemmed | Mathematics for computer graphics John Vince |
| title_short | Mathematics for computer graphics |
| title_sort | mathematics for computer graphics |
| topic | Informatik (DE-588)4026894-9 gnd Computergrafik (DE-588)4010450-3 gnd Mathematik (DE-588)4037944-9 gnd |
| topic_facet | Informatik Computergrafik Mathematik Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027332754&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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