Geometry for Computer Graphics (eBook)

Contents 9
Preface 6
1 Geometry 21
1.1 Lines, angles and trigonometry 24
1.1.1 Points and straight lines 24
1.1.2 Angles 24
1.1.3 Trigonometry 25
1.2 Circles 29
1.2.1 Properties of circles 29
1.2.2 Ellipses 30
1.3 Triangles 31
1.3.1 Types of triangle 31
1.3.2 Similar triangles 31
1.3.3 Congruent triangles 32
1.3.4 Theorem of Pythagoras 32
1.3.5 Internal and external angles 33
1.3.6 Sine, cosine and tangent rules 33
1.3.7 Area of a triangle 33
1.3.8 Inscribed and circumscribed circles 34
1.3.9 Centroid of a triangle 35
1.3.10 Spherical trigonometry 35
1.4 Quadrilaterals 36
1.5 Polygons 39
1.5.1 Internal and external angles of a polygon 39
1.5.2 Alternate internal angles of a cyclic polygon 39
1.5.3 Area of a regular polygon 39
1.6 Three-dimensional objects 41
1.6.1 Prisms 41
1.6.2 Pyramids 41
1.6.3 Cylinders 42
1.6.4 Cones 42
1.6.5 Spheres 42
1.6.6 Tori 43
1.6.7 Platonic solids 43
1.7 Coordinate systems 46
1.7.1 Cartesian coordinates in R2 46
1.7.2 Cartesian coordinates in R3 46
1.7.3 Polar coordinates 47
1.7.4 Cylindrical coordinates 47
1.7.5 Spherical coordinates 48
1.8 Vectors 49
1.8.1 Vector between two points 49
1.8.2 Scaling a vector 49
1.8.3 Reversing a vector 49
1.8.4 Unit Cartesian vectors 49
1.8.5 Algebraic notation for a vector 49
1.8.6 Magnitude of a vector 50
1.8.7 Normalizing a vector to a unit length 50
1.8.8 Vector addition/subtraction 50
1.8.9 Compound scalar multiplication 50
1.8.10 Position vector 50
1.8.11 Scalar (dot) product 50
1.8.12 Angle between two vectors 51
1.8.13 Vector (cross) product 51
1.8.14 The commutative law does not hold: axb= --bxa 51
1.8.15 Scalar triple product 51
1.8.16 Vector triple product 52
1.8.17 Vector normal to a triangle 52
1.8.18 Area of a triangle 52
1.9 Quaternions 53
1.9.1 Definition of a quaternion 53
1.9.2 Equal quaternions 53
1.9.3 Quaternion addition and subtraction 53
1.9.4 Quaternion multiplication 53
1.9.5 Magnitude of a quaternion 54
1.9.6 The inverse quaternion 54
1.9.7 Rotating a vector 54
1.9.8 Quaternion as a matrix 54
1.10 Transformations 55
1.10.1 Scaling relative to the origin in R2 55
1.10.2 Scaling relative to a point in R2 55
1.10.3 Translation in R2 55
1.10.4 Rotation about the origin in R2 55
1.10.5 Rotation about a point in R2 56
1.10.6 Shearing along the x-axis in R2 56
1.10.7 Shearing along the y-axis in R2 56
1.10.8 Reflection about the x-axis in R2 56
1.10.9 Reflection about the y-axis in R2 56
1.10.10 Reflection about a line parallel with the x-axis in R2 57
1.10.11 Reflection about a line parallel with the y-axis in R2 57
1.10.12 Translated change of axes in R2 57
1.10.13 Rotated change of axes in R2 57
1.10.14 The identity matrix in R2 57
1.10.15 Scaling relative to the origin in R3 58
1.10.16 Scaling relative to a point in R3 58
1.10.17 Translation in R3 58
1.10.18 Rotation about the x-axis in R3 58
1.10.19 Rotation about the y-axis in R3 59
1.10.20 Rotation about the z-axis in R3 59
1.10.21 Rotation about an arbitrary axis in R3 59
1.10.22 Reflection about the yz-plane in R3 59
1.10.23 Reflection about the zx-plane in R3 59
1.10.24 Reflection about the xy-plane in R3 60
1.10.25 Reflection about a plane parallel with the yz-plane in R3 60
1.10.26 Reflection about a plane parallel with the zx-plane in R3 60
1.10.27 Reflection about a plane parallel with the xy-plane in R3 60
1.10.28 Translated change of axes in R3 60
1.10.29 Rotated change of axes in R3 61
1.10.30 The identity matrix in R3 61
1.11 Two-dimensional straight lines 62
1.11.1 Normal form of the straight line equation 62
1.11.2 General form of the straight line equation 62
1.11.3 Hessian normal form of the straight line equation 62
1.11.4 Parametric form of the straight line equation 62
1.11.5 Cartesian form of the straight line equation 63
1.11.6 Straight line equation from two points 63
1.11.7 Point of intersection of two straight lines 64
1.11.8 Angle between two straight lines 65
1.11.9 Three points lie on a straight line 65
1.11.10 Parallel and perpendicular straight lines 66
1.11.11 Position and distance of a point on a line perpendicular to the origin 66
1.11.12 Position and distance of the nearest point on a line to a point 67
1.11.13 Position of a point reflected in a line 67
1.11.14 Normal to a line through a point 68
1.11.15 Line equidistant from two points 68
1.11.16 Two-dimensional line segment 69
1.12 Lines and circles 71
1.12.1 Line intersecting a circle 71
1.12.2 Touching and intersecting circles 71
1.13 Second degree curves 73
1.13.1 Circle 73
1.13.2 Ellipse 73
1.13.3 Parabola 74
1.13.4 Hyperbola 74
1.14 Three-dimensional straight lines 75
1.14.1 Straight line equation from two points 75
1.14.2 Intersection of two straight lines 75
1.14.3 The angle between two straight lines 75
1.14.4 Three points lie on a straight line 75
1.14.5 Parallel and perpendicular straight lines 76
1.14.6 Position and distance of a point on a line perpendicular to the origin 76
1.14.7 Position and distance of the nearest point on a line to a point 76
1.14.8 Shortest distance between two skew lines 76
1.14.9 Position of a point reflected in a line 77
1.14.10 Normal to a line through a point 77
1.15 Planes 78
1.15.1 Cartesian form of the plane equation 78
1.15.2 General form of the plane equation 78
1.15.3 Hessian normal form of the plane equation 78
1.15.4 Parametric form of the plane equation 79
1.15.5 Converting from the parametric form to the general form 79
1.15.6 Plane equation from three points 79
1.15.7 Plane through a point and normal to a line 80
1.15.8 Plane through two points and parallel to a line 80
1.15.9 Intersection of two planes 80
1.15.10 Intersection of three planes 81
1.15.11 Angle between two planes 81
1.15.12 Angle between a line and a plane 82
1.15.13 Intersection of a line and a plane 82
1.15.14 Position and distance of the nearest point on a plane to a point 82
1.15.15 Reflection of a point in a plane 83
1.15.16 Plane equidistant from two points 83
1.15.17 Reflected ray on a surface 83
1.16 Lines, planes and spheres 84
1.16.1 Line intersecting a sphere 84
1.16.2 Sphere touching a plane 84
1.16.3 Touching spheres 84
1.17 Three-dimensional triangles 86
1.17.1 Point inside a triangle 86
1.17.2 Unknown coordinate value inside a triangle 86
1.18 Parametric curves and patches 87
1.18.1 Parametric curve in R2 87
1.18.2 Parametric curve in R3 87
1.18.3 Planar patch 87
1.18.4 Modulated surface 88
1.18.5 Quadratic Bézier curve 88
1.18.6 Cubic Bézier curve 88
1.18.7 Quadratic Bézier patch 88
1.18.8 Cubic Bézier patch 89
1.19 Second degree surfaces in standard form 90
2 Examples 92
2.1 Trigonometry 94
2.2 Circles 97
2.3 Triangles 98
2.3.1 Checking for similar triangles 98
2.3.2 Checking for congruent triangles 98
2.3.3 Solving the angles and sides of a triangle 99
2.3.4 Calculating the area of a triangle 100
2.3.5 The center and radius of the inscribed and circumscribed circles for a triangle 101
2.4 Quadrilaterals 103
2.5 Polygons 105
2.6 Three-dimensional objects 107
2.6.1 Cone, cylinder and sphere 107
2.6.2 Conical frustum, spherical segment and torus 107
2.6.3 Tetrahedron 108
2.7 Coordinate systems 109
2.7.1 Cartesian coordinates in R2 109
2.7.2 Cartesian coordinates in R3 109
2.7.3 Polar coordinates 109
2.7.4 Cylindrical coordinates 110
2.7.5 Spherical coordinates 111
2.8 Vectors 113
2.8.1 Vector between two points 113
2.8.2 Scaling a vector 113
2.8.3 Reversing a vector 113
2.8.4 Magnitude of a vector 113
2.8.5 Normalizing a vector to a unit length 113
2.8.6 Vector addition/subtraction 113
2.8.7 Position vector 114
2.8.8 Scalar (dot) product 114
2.8.9 Angle between two vectors 114
2.8.10 Vector (cross) product 114
2.8.11 Scalar triple product 115
2.8.12 Vector normal to a triangle 115
2.8.13 Area of a triangle 115
2.9 Quaternions 116
2.9.1 Quaternion addition and subtraction 116
2.9.2 Quaternion multiplication 116
2.9.3 Magnitude of a quaternion 116
2.9.4 The inverse quaternion 116
2.9.5 Rotating a vector 116
2.9.6 Quaternion as a matrix 117
2.10 Transformations 118
2.10.1 Scaling relative to the origin in R2 118
2.10.2 Scaling relative to a point in R2 118
2.10.3 Translation in R2 118
2.10.4 Rotation about the origin in R2 119
2.10.5 Rotation about a point in R2 119
2.10.6 Shearing along the x-axis in R2 119
2.10.7 Shearing along the y-axis in R2 120
2.10.8 Reflection about the x-axis in R2 120
2.10.9 Reflection about the y-axis in R2 120
2.10.10 Reflection about a line parallel with the x-axis in R2 121
2.10.11 Reflection about a line parallel with the y-axis in R2 121
2.10.12 Translated change of axes in R2 121
2.10.13 Rotated change of axes in R2 122
2.10.14 The identity matrix in R2 122
2.10.15 Scaling relative to the origin in R3 122
2.10.16 Scaling relative to a point in R3 123
2.10.17 Translation in R3 123
2.10.18 Rotation about the x-axis in R3 123
2.10.19 Rotation about the y-axis in R3 124
2.10.20 Rotation about the z-axis in R3 124
2.10.21 Rotation about an arbitrary axis in R3 124
2.10.22 Reflection about the yz-plane in R3 125
2.10.23 Reflection about the zx-plane in R3 125
2.10.24 Reflection about the xy-plane in R3 125
2.10.25 Reflection about a plane parallel with the yz-plane in R3 126
2.10.26 Reflection about a plane parallel with the zx-plane in R3 126
2.10.27 Reflection about a plane parallel with the xy-plane in R3 126
2.10.28 Translated axes in R3 127
2.10.29 Rotated axes in R3 127
2.10.30 The identity matrix in R3 127
2.11 Two-dimensional straight lines 128
2.11.1 Convert the normal form of the line equation to its general form and the Hessian normal form 128
2.11.2 Derive the unit normal vector and perpendicular from the origin to the line for the line equation 3x+4y+6=0 128
2.11.3 Derive the straight-line equation from two points 129
2.11.4 Point of intersection of two straight lines 130
2.11.5 Calculate the angle between two straight lines 131
2.11.6 Test if three points lie on a straight line 132
2.11.7 Test for parallel and perpendicular lines 133
2.11.8 Find the position and distance of the nearest point on a line to the origin 134
2.11.9 Find the position and distance of the nearest point on a line to a point 135
2.11.10 Find the reflection of a point in a line passing through the origin 136
2.11.11 Find the reflection of a point in a line 137
2.11.12 Find the normal to a line through a point 138
2.11.13 Find the line equidistant from two points 139
2.11.14 Creating the parametric line equation for a line segment 140
2.11.15 Intersecting two line segments 140
2.12 Lines and circles 142
2.12.1 Line intersecting a circle 142
2.12.2 Touching and intersecting circles 145
2.13 Second degree curves 147
2.13.1 Circle 147
2.13.2 Ellipse 147
2.13.3 Parabola 147
2.13.4 Hyperbola 148
2.14 Three-dimensional straight lines 149
2.14.1 Derive the straight-line equation from two points 149
2.14.2 Intersection of two straight lines 149
2.14.3 Calculate the angle between two straight lines 150
2.14.4 Test if three points lie on a straight line 150
2.14.5 Test for parallel and perpendicular straight lines 151
2.14.6 Find the position and distance of the nearest point on a line to the origin 151
2.14.7 Find the position and distance of the nearest point on a line to a point 151
2.14.8 Find the reflection of a point in a line 152
2.14.9 Find the normal to a line through a point 152
2.14.10 Find the shortest distance between two skew lines 153
2.15 Planes 154
2.15.1 Cartesian form of the plane equation 154
2.15.2 General form of the plane equation 154
2.15.3 Hessian normal form of the plane equation 154
2.15.4 Parametric form of the plane equation 155
2.15.5 Converting a plane equation from parametric form to general form 155
2.15.6 Plane equation from three points 156
2.15.7 Plane through a point and normal to a line 157
2.15.8 Plane through two points and parallel to a line 157
2.15.9 Intersection of two planes 158
2.15.10 Intersection of three planes 160
2.15.11 Angle between two planes 162
2.15.12 Angle between a line and a plane 162
2.15.13 Intersection of a line and a plane 163
2.15.14 Position and distance of the nearest point on a plane to a point 163
2.15.15 Reflection of a point in a plane 164
2.15.16 Plane equidistant from two points 164
2.15.17 Reflected ray on a surface 165
2.16 Lines, planes and spheres 167
2.16.1 Line intersecting a sphere 167
2.16.2 Sphere touching a plane 168
2.16.3 Touching spheres 169
2.17 Three-dimensional triangles 170
2.17.1 Coordinates of a point inside a triangle 170
2.17.2 Unknown coordinate value inside a triangle 171
2.18 Parametric curves and patches 173
2.18.1 Parametric curves in R2 173
2.18.2 Parametric curves in R3 177
2.18.3 Planar patch 181
2.18.4 Parametric surfaces in R3 182
2.18.5 Quadratic Bézier curve 184
2.18.6 Cubic Bézier curve 184
2.18.7 Quadratic Bézier patch 185
2.18.8 Cubic Bézier patch 186
2.19 Second degree surfaces in standard form 187
3 Proofs 188
3.1 Trigonometry 190
3.1.1 Trigonometric functions and identities 190
3.1.2 Cofunction identities 190
3.1.3 Pythagorean identities 190
3.1.4 Useful trigonometric values 191
3.1.5 Compound angle identities 192
3.1.6 Double-angle identities 194
3.1.7 Multiple-angle identities 194
3.1.8 Functions of the half-angle 195
3.1.9 Functions of the half-angle using the perimeter of a triangle 196
3.1.10 Functions converting to the half-angle tangent form 197
3.1.11 Relationships between sums of functions 199
3.1.12 Inverse trigonometric functions 201
3.2 Circles 202
3.2.1 Proof: Angles subtended by the same arc 202
3.2.2 Proof: Alternate segment theorem 202
3.2.3 Proof: Area of a circle, sector and segment 203
3.2.4 Proof: Chord theorem 205
3.2.5 Proof: Secant theorem 205
3.2.6 Proof: Secant–tangent theorem 205
3.2.7 Proof: Area of an ellipse 206
3.3 Triangles 208
3.3.1 Proof: Theorem of Pythagoras 208
3.3.2 Proofs: Properties of triangles 208
3.3.3 Proof: Altitude theorem 211
3.3.4 Proof: Area of a triangle 212
3.3.5 Proof: Internal and external angles of a triangle 215
3.3.6 Proof: The medians of a triangle are concurrent at its centroid 215
3.3.7 Proof: Radius and center of the inscribed circle for a triangle 217
3.3.8 Proof: Radius and center of the circumscribed circle for a triangle 220
3.4 Quadrilaterals 226
3.4.1 Proof: Properties of quadrilaterals 226
3.4.2 Proof: The opposite sides and angles of a parallelogram are equal 229
3.4.3 Proof: The diagonals of a parallelogram bisect each other 229
3.4.4 Proof: The diagonals of a square are equal, intersect at right angles and bisect the opposite angles 230
3.4.5 Proof: Area of a parallelogram 231
3.4.6 Proof: Area of a quadrilateral 231
3.4.7 Proof: Area of a general quadrilateral using Heron’s formula 233
3.4.8 Proof: Area of a trapezoid 235
3.4.9 Proof: Radius and center of the circumscribed circle for a rectangle 236
3.5 Polygons 237
3.5.1 Proof: The internal angles of a polygon 237
3.5.2 Proof: The external angles of a polygon 237
3.5.3 Proof: Alternate internal angles of a cyclic polygon 238
3.5.4 Proof: Area of a regular polygon 239
3.5.5 Proof: Area of a polygon 240
3.5.6 Proof: Properties of regular polygons 241
3.6 Three-dimensional objects 243
3.6.1 Proof: Volume of a prism 243
3.6.2 Proof: Surface area of a rectangular pyramid 244
3.6.3 Proof: Volume of a rectangular pyramid 245
3.6.4 Volume of a rectangular pyramidal frustum 246
3.6.5 Proof: Volume of a triangular pyramid 246
3.6.6 Proof: Surface area of a right cone 247
3.6.7 Proof: Surface area of a right conical frustum 247
3.6.8 Proof: Volume of a cone 248
3.6.9 Proof: Volume of a right conical frustum 249
3.6.10 Proof: Surface area of a sphere 249
3.6.11 Proof: Volume of a sphere 250
3.6.12 Proof: Area and volume of a torus 252
3.6.13 Proof: Radii of the spheres associated with the Platonic solids 252
3.6.14 Proof: Inner and outer radii for the Platonic solids 257
3.6.15 Proof: Dihedral angles for the Platonic solids 261
3.6.16 Proof: Surface area and volume of the Platonic solids 265
3.7 Coordinate systems 268
3.7.1 Cartesian coordinates 268
3.7.2 Polar coordinates 268
3.7.3 Cylindrical coordinates 269
3.7.4 Spherical coordinates 269
3.8 Vectors 271
3.8.1 Proof: Magnitude of a vector 271
3.8.2 Proof: Normalizing a vector to a unit length 271
3.8.3 Proof: Scalar (dot) product 271
3.8.4 Proof: Commutative law of the scalar product 272
3.8.5 Proof: Associative law of the scalar product 272
3.8.6 Proof: Angle between two vectors 272
3.8.7 Proof: Vector (cross) product 273
3.8.8 Proof: The non-commutative law of the vector product 273
3.8.9 Proof: The associative law of the vector product 274
3.8.10 Proof: Scalar triple product 274
3.9 Quaternions 275
3.9.1 Definition of a quaternion 275
3.10 Transformations 279
3.10.1 Proof: Scaling in R2 279
3.10.2 Proof: Translation in R2 280
3.10.3 Proof: Rotation in R2 280
3.10.4 Proof: Shearing in R2 281
3.10.5 Proof: Reflection in R2 282
3.10.6 Proof: Change of axes in R2 283
3.10.7 Proof: Identity matrix in R2 284
3.10.8 Proof: Scaling in R3 284
3.10.9 Proof: Translation in R3 285
3.10.10 Proof: Rotation in R3 285
3.10.11 Proof: Reflection in R3 287
3.10.12 Proof: Change of axes in R3 289
3.10.13 Proof: Identity matrix in R3 290
3.11 Two-dimensional straight lines 291
3.11.1 Proof: Cartesian form of the line equation 291
3.11.2 Proof: Hessian normal form (after Otto Hesse (1811–1874)) 292
3.11.3 Proof: Equation of a line from two points 292
3.11.4 Proof: Point of intersection of two straight lines 294
3.11.5 Proof: Angle between two straight lines 295
3.11.6 Proof: Three points lie on a straight line 296
3.11.7 Proof: Parallel and perpendicular straight lines 297
3.11.8 Proof: Shortest distance to a line 298
3.11.9 Proof: Position and distance of a point on a line perpendicular to the origin 298
3.11.10 Proof: Position and distance of the nearest point on a line to a point 299
3.11.11 Proof: Position of a point reflected in a line 300
3.11.12 Proof: Normal to a line through a point 302
3.11.13 Proof: Line equidistant from two points 303
3.11.14 Proof: Equation of a two-dimensional line segment 304
3.11.15 Proof: Point of intersection of two two-dimensional line segments 305
3.12 Lines and circles 307
3.12.1 Proof: Line and a circle 307
3.12.2 Proof: Touching and intersecting circles 309
3.13 Second degree curves 312
3.13.1 Circle 312
3.13.2 Ellipse 312
3.13.3 Parabola 314
3.13.4 Hyperbola 315
3.14 Three-dimensional straight lines 316
3.14.1 Proof: Straight-line equation from two points 316
3.14.2 Proof: Intersection of two straight lines 316
3.14.3 Proof: Angle between two straight lines 317
3.14.4 Proof: Three points lie on a straight line 317
3.14.5 Proof: Parallel and perpendicular straight lines 317
3.14.6 Proof: Position and distance of a point on a line perpendicular to the origin 318
3.14.7 Proof: Position and distance of the nearest point on a line to a point 318
3.14.8 Proof: Position of a point reflected in a line 319
3.14.9 Proof: Normal to a line through a point 320
3.14.10 Proof: Shortest distance between two skew lines 321
3.15 Planes 322
3.15.1 Proof: Equation to a plane 322
3.15.2 Proof: Plane equation from three points 325
3.15.3 Proof: Plane through a point and normal to a line 327
3.15.4 Proof: Plane through two points and parallel to a line 327
3.15.5 Proof: Intersection of two planes 327
3.15.6 Proof: Intersection of three planes 329
3.15.7 Proof: Angle between two planes 330
3.15.8 Proof: Angle between a line and a plane 330
3.15.9 Proof: Intersection of a line and a plane 330
3.15.10 Proof: Position and distance of the nearest point on a plane to a point 331
3.15.11 Proof: Reflection of a point in a plane 332
3.15.12 Proof: Plane equidistant from two points 332
3.15.13 Proof: Reflected ray on a surface 333
3.16 Lines, planes and spheres 334
3.16.1 Proof: Line intersecting a sphere 334
3.16.2 Proof: Sphere touching a plane 335
3.16.3 Proof: Touching spheres 335
3.17 Three-dimensional triangles 337
3.17.1 Proof: Point inside a triangle 337
3.17.2 Proof: Unknown coordinate value inside a triangle 337
3.18 Parametric curves and patches 338
3.18.1 Proof: Planar surface patch 338
3.18.2 Proof: Bézier curves in R2 andR3 338
3.18.3 Proof: Bézier surface patch in R3 340
4 Glossary 343
5 Bibliography 351
Index 353