Vince Geometry for Computer Graphics

Formulae, Examples and Proofs
1. Auflage 2006
ISBN: 978-1-84628-116-7
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 342 Seiten

A complete overview of the geometry associated with computer graphics that provides everything a reader needs to understand the topic. Includes a summary hundreds of formulae used to solve 2D and 3D geometric problems; worked examples; proofs; mathematical strategies for solving geometric problems; a glossary of terms used in geometry.

Autoren/Hrsg.

Vince, John

Inhaltsverzeichnis

1;Contents;9
2;Preface;6
3;1 Geometry;21
3.1;1.1 Lines, angles and trigonometry;24
3.1.1;1.1.1 Points and straight lines;24
3.1.2;1.1.2 Angles;24
3.1.3;1.1.3 Trigonometry;25
3.2;1.2 Circles;29
3.2.1;1.2.1 Properties of circles;29
3.2.2;1.2.2 Ellipses;30
3.3;1.3 Triangles;31
3.3.1;1.3.1 Types of triangle;31
3.3.2;1.3.2 Similar triangles;31
3.3.3;1.3.3 Congruent triangles;32
3.3.4;1.3.4 Theorem of Pythagoras;32
3.3.5;1.3.5 Internal and external angles;33
3.3.6;1.3.6 Sine, cosine and tangent rules;33
3.3.7;1.3.7 Area of a triangle;33
3.3.8;1.3.8 Inscribed and circumscribed circles;34
3.3.9;1.3.9 Centroid of a triangle;35
3.3.10;1.3.10 Spherical trigonometry;35
3.4;1.4 Quadrilaterals;36
3.5;1.5 Polygons;39
3.5.1;1.5.1 Internal and external angles of a polygon;39
3.5.2;1.5.2 Alternate internal angles of a cyclic polygon;39
3.5.3;1.5.3 Area of a regular polygon;39
3.6;1.6 Three-dimensional objects;41
3.6.1;1.6.1 Prisms;41
3.6.2;1.6.2 Pyramids;41
3.6.3;1.6.3 Cylinders;42
3.6.4;1.6.4 Cones;42
3.6.5;1.6.5 Spheres;42
3.6.6;1.6.6 Tori;43
3.6.7;1.6.7 Platonic solids;43
3.7;1.7 Coordinate systems;46
3.7.1;1.7.1 Cartesian coordinates in R2;46
3.7.2;1.7.2 Cartesian coordinates in R3;46
3.7.3;1.7.3 Polar coordinates;47
3.7.4;1.7.4 Cylindrical coordinates;47
3.7.5;1.7.5 Spherical coordinates;48
3.8;1.8 Vectors;49
3.8.1;1.8.1 Vector between two points;49
3.8.2;1.8.2 Scaling a vector;49
3.8.3;1.8.3 Reversing a vector;49
3.8.4;1.8.4 Unit Cartesian vectors;49
3.8.5;1.8.5 Algebraic notation for a vector;49
3.8.6;1.8.6 Magnitude of a vector;50
3.8.7;1.8.7 Normalizing a vector to a unit length;50
3.8.8;1.8.8 Vector addition/subtraction;50
3.8.9;1.8.9 Compound scalar multiplication;50
3.8.10;1.8.10 Position vector;50
3.8.11;1.8.11 Scalar (dot) product;50
3.8.12;1.8.12 Angle between two vectors;51
3.8.13;1.8.13 Vector (cross) product;51
3.8.14;1.8.14 The commutative law does not hold: axb= --bxa;51
3.8.15;1.8.15 Scalar triple product;51
3.8.16;1.8.16 Vector triple product;52
3.8.17;1.8.17 Vector normal to a triangle;52
3.8.18;1.8.18 Area of a triangle;52
3.9;1.9 Quaternions;53
3.9.1;1.9.1 Definition of a quaternion;53
3.9.2;1.9.2 Equal quaternions;53
3.9.3;1.9.3 Quaternion addition and subtraction;53
3.9.4;1.9.4 Quaternion multiplication;53
3.9.5;1.9.5 Magnitude of a quaternion;54
3.9.6;1.9.6 The inverse quaternion;54
3.9.7;1.9.7 Rotating a vector;54
3.9.8;1.9.8 Quaternion as a matrix;54
3.10;1.10 Transformations;55
3.10.1;1.10.1 Scaling relative to the origin in R2;55
3.10.2;1.10.2 Scaling relative to a point in R2;55
3.10.3;1.10.3 Translation in R2;55
3.10.4;1.10.4 Rotation about the origin in R2;55
3.10.5;1.10.5 Rotation about a point in R2;56
3.10.6;1.10.6 Shearing along the x-axis in R2;56
3.10.7;1.10.7 Shearing along the y-axis in R2;56
3.10.8;1.10.8 Reflection about the x-axis in R2;56
3.10.9;1.10.9 Reflection about the y-axis in R2;56
3.10.10;1.10.10 Reflection about a line parallel with the x-axis in R2;57
3.10.11;1.10.11 Reflection about a line parallel with the y-axis in R2;57
3.10.12;1.10.12 Translated change of axes in R2;57
3.10.13;1.10.13 Rotated change of axes in R2;57
3.10.14;1.10.14 The identity matrix in R2;57
3.10.15;1.10.15 Scaling relative to the origin in R3;58
3.10.16;1.10.16 Scaling relative to a point in R3;58
3.10.17;1.10.17 Translation in R3;58
3.10.18;1.10.18 Rotation about the x-axis in R3;58
3.10.19;1.10.19 Rotation about the y-axis in R3;59
3.10.20;1.10.20 Rotation about the z-axis in R3;59
3.10.21;1.10.21 Rotation about an arbitrary axis in R3;59
3.10.22;1.10.22 Reflection about the yz-plane in R3;59
3.10.23;1.10.23 Reflection about the zx-plane in R3;59
3.10.24;1.10.24 Reflection about the xy-plane in R3;60
3.10.25;1.10.25 Reflection about a plane parallel with the yz-plane in R3;60
3.10.26;1.10.26 Reflection about a plane parallel with the zx-plane in R3;60
3.10.27;1.10.27 Reflection about a plane parallel with the xy-plane in R3;60
3.10.28;1.10.28 Translated change of axes in R3;60
3.10.29;1.10.29 Rotated change of axes in R3;61
3.10.30;1.10.30 The identity matrix in R3;61
3.11;1.11 Two-dimensional straight lines;62
3.11.1;1.11.1 Normal form of the straight line equation;62
3.11.2;1.11.2 General form of the straight line equation;62
3.11.3;1.11.3 Hessian normal form of the straight line equation;62
3.11.4;1.11.4 Parametric form of the straight line equation;62
3.11.5;1.11.5 Cartesian form of the straight line equation;63
3.11.6;1.11.6 Straight line equation from two points;63
3.11.7;1.11.7 Point of intersection of two straight lines;64
3.11.8;1.11.8 Angle between two straight lines;65
3.11.9;1.11.9 Three points lie on a straight line;65
3.11.10;1.11.10 Parallel and perpendicular straight lines;66
3.11.11;1.11.11 Position and distance of a point on a line perpendicular to the origin;66
3.11.12;1.11.12 Position and distance of the nearest point on a line to a point;67
3.11.13;1.11.13 Position of a point reflected in a line;67
3.11.14;1.11.14 Normal to a line through a point;68
3.11.15;1.11.15 Line equidistant from two points;68
3.11.16;1.11.16 Two-dimensional line segment;69
3.12;1.12 Lines and circles;71
3.12.1;1.12.1 Line intersecting a circle;71
3.12.2;1.12.2 Touching and intersecting circles;71
3.13;1.13 Second degree curves;73
3.13.1;1.13.1 Circle;73
3.13.2;1.13.2 Ellipse;73
3.13.3;1.13.3 Parabola;74
3.13.4;1.13.4 Hyperbola;74
3.14;1.14 Three-dimensional straight lines;75
3.14.1;1.14.1 Straight line equation from two points;75
3.14.2;1.14.2 Intersection of two straight lines;75
3.14.3;1.14.3 The angle between two straight lines;75
3.14.4;1.14.4 Three points lie on a straight line;75
3.14.5;1.14.5 Parallel and perpendicular straight lines;76
3.14.6;1.14.6 Position and distance of a point on a line perpendicular to the origin;76
3.14.7;1.14.7 Position and distance of the nearest point on a line to a point;76
3.14.8;1.14.8 Shortest distance between two skew lines;76
3.14.9;1.14.9 Position of a point reflected in a line;77
3.14.10;1.14.10 Normal to a line through a point;77
3.15;1.15 Planes;78
3.15.1;1.15.1 Cartesian form of the plane equation;78
3.15.2;1.15.2 General form of the plane equation;78
3.15.3;1.15.3 Hessian normal form of the plane equation;78
3.15.4;1.15.4 Parametric form of the plane equation;79
3.15.5;1.15.5 Converting from the parametric form to the general form;79
3.15.6;1.15.6 Plane equation from three points;79
3.15.7;1.15.7 Plane through a point and normal to a line;80
3.15.8;1.15.8 Plane through two points and parallel to a line;80
3.15.9;1.15.9 Intersection of two planes;80
3.15.10;1.15.10 Intersection of three planes;81
3.15.11;1.15.11 Angle between two planes;81
3.15.12;1.15.12 Angle between a line and a plane;82
3.15.13;1.15.13 Intersection of a line and a plane;82
3.15.14;1.15.14 Position and distance of the nearest point on a plane to a point;82
3.15.15;1.15.15 Reflection of a point in a plane;83
3.15.16;1.15.16 Plane equidistant from two points;83
3.15.17;1.15.17 Reflected ray on a surface;83
3.16;1.16 Lines, planes and spheres;84
3.16.1;1.16.1 Line intersecting a sphere;84
3.16.2;1.16.2 Sphere touching a plane;84
3.16.3;1.16.3 Touching spheres;84
3.17;1.17 Three-dimensional triangles;86
3.17.1;1.17.1 Point inside a triangle;86
3.17.2;1.17.2 Unknown coordinate value inside a triangle;86
3.18;1.18 Parametric curves and patches;87
3.18.1;1.18.1 Parametric curve in R2;87
3.18.2;1.18.2 Parametric curve in R3;87
3.18.3;1.18.3 Planar patch;87
3.18.4;1.18.4 Modulated surface;88
3.18.5;1.18.5 Quadratic Bézier curve;88
3.18.6;1.18.6 Cubic Bézier curve;88
3.18.7;1.18.7 Quadratic Bézier patch;88
3.18.8;1.18.8 Cubic Bézier patch;89
3.19;1.19 Second degree surfaces in standard form;90
4;2 Examples;92
4.1;2.1 Trigonometry;94
4.2;2.2 Circles;97
4.3;2.3 Triangles;98
4.3.1;2.3.1 Checking for similar triangles;98
4.3.2;2.3.2 Checking for congruent triangles;98
4.3.3;2.3.3 Solving the angles and sides of a triangle;99
4.3.4;2.3.4 Calculating the area of a triangle;100
4.3.5;2.3.5 The center and radius of the inscribed and circumscribed circles for a triangle;101
4.4;2.4 Quadrilaterals;103
4.5;2.5 Polygons;105
4.6;2.6 Three-dimensional objects;107
4.6.1;2.6.1 Cone, cylinder and sphere;107
4.6.2;2.6.2 Conical frustum, spherical segment and torus;107
4.6.3;2.6.3 Tetrahedron;108
4.7;2.7 Coordinate systems;109
4.7.1;2.7.1 Cartesian coordinates in R2;109
4.7.2;2.7.2 Cartesian coordinates in R3;109
4.7.3;2.7.3 Polar coordinates;109
4.7.4;2.7.4 Cylindrical coordinates;110
4.7.5;2.7.5 Spherical coordinates;111
4.8;2.8 Vectors;113
4.8.1;2.8.1 Vector between two points;113
4.8.2;2.8.2 Scaling a vector;113
4.8.3;2.8.3 Reversing a vector;113
4.8.4;2.8.4 Magnitude of a vector;113
4.8.5;2.8.5 Normalizing a vector to a unit length;113
4.8.6;2.8.6 Vector addition/subtraction;113
4.8.7;2.8.7 Position vector;114
4.8.8;2.8.8 Scalar (dot) product;114
4.8.9;2.8.9 Angle between two vectors;114
4.8.10;2.8.10 Vector (cross) product;114
4.8.11;2.8.11 Scalar triple product;115
4.8.12;2.8.12 Vector normal to a triangle;115
4.8.13;2.8.13 Area of a triangle;115
4.9;2.9 Quaternions;116
4.9.1;2.9.1 Quaternion addition and subtraction;116
4.9.2;2.9.2 Quaternion multiplication;116
4.9.3;2.9.3 Magnitude of a quaternion;116
4.9.4;2.9.4 The inverse quaternion;116
4.9.5;2.9.5 Rotating a vector;116
4.9.6;2.9.6 Quaternion as a matrix;117
4.10;2.10 Transformations;118
4.10.1;2.10.1 Scaling relative to the origin in R2;118
4.10.2;2.10.2 Scaling relative to a point in R2;118
4.10.3;2.10.3 Translation in R2;118
4.10.4;2.10.4 Rotation about the origin in R2;119
4.10.5;2.10.5 Rotation about a point in R2;119
4.10.6;2.10.6 Shearing along the x-axis in R2;119
4.10.7;2.10.7 Shearing along the y-axis in R2;120
4.10.8;2.10.8 Reflection about the x-axis in R2;120
4.10.9;2.10.9 Reflection about the y-axis in R2;120
4.10.10;2.10.10 Reflection about a line parallel with the x-axis in R2;121
4.10.11;2.10.11 Reflection about a line parallel with the y-axis in R2;121
4.10.12;2.10.12 Translated change of axes in R2;121
4.10.13;2.10.13 Rotated change of axes in R2;122
4.10.14;2.10.14 The identity matrix in R2;122
4.10.15;2.10.15 Scaling relative to the origin in R3;122
4.10.16;2.10.16 Scaling relative to a point in R3;123
4.10.17;2.10.17 Translation in R3;123
4.10.18;2.10.18 Rotation about the x-axis in R3;123
4.10.19;2.10.19 Rotation about the y-axis in R3;124
4.10.20;2.10.20 Rotation about the z-axis in R3;124
4.10.21;2.10.21 Rotation about an arbitrary axis in R3;124
4.10.22;2.10.22 Reflection about the yz-plane in R3;125
4.10.23;2.10.23 Reflection about the zx-plane in R3;125
4.10.24;2.10.24 Reflection about the xy-plane in R3;125
4.10.25;2.10.25 Reflection about a plane parallel with the yz-plane in R3;126
4.10.26;2.10.26 Reflection about a plane parallel with the zx-plane in R3;126
4.10.27;2.10.27 Reflection about a plane parallel with the xy-plane in R3;126
4.10.28;2.10.28 Translated axes in R3;127
4.10.29;2.10.29 Rotated axes in R3;127
4.10.30;2.10.30 The identity matrix in R3;127
4.11;2.11 Two-dimensional straight lines;128
4.11.1;2.11.1 Convert the normal form of the line equation to its general form and the Hessian normal form;128
4.11.2;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
4.11.3;2.11.3 Derive the straight-line equation from two points;129
4.11.4;2.11.4 Point of intersection of two straight lines;130
4.11.5;2.11.5 Calculate the angle between two straight lines;131
4.11.6;2.11.6 Test if three points lie on a straight line;132
4.11.7;2.11.7 Test for parallel and perpendicular lines;133
4.11.8;2.11.8 Find the position and distance of the nearest point on a line to the origin;134
4.11.9;2.11.9 Find the position and distance of the nearest point on a line to a point;135
4.11.10;2.11.10 Find the reflection of a point in a line passing through the origin;136
4.11.11;2.11.11 Find the reflection of a point in a line;137
4.11.12;2.11.12 Find the normal to a line through a point;138
4.11.13;2.11.13 Find the line equidistant from two points;139
4.11.14;2.11.14 Creating the parametric line equation for a line segment;140
4.11.15;2.11.15 Intersecting two line segments;140
4.12;2.12 Lines and circles;142
4.12.1;2.12.1 Line intersecting a circle;142
4.12.2;2.12.2 Touching and intersecting circles;145
4.13;2.13 Second degree curves;147
4.13.1;2.13.1 Circle;147
4.13.2;2.13.2 Ellipse;147
4.13.3;2.13.3 Parabola;147
4.13.4;2.13.4 Hyperbola;148
4.14;2.14 Three-dimensional straight lines;149
4.14.1;2.14.1 Derive the straight-line equation from two points;149
4.14.2;2.14.2 Intersection of two straight lines;149
4.14.3;2.14.3 Calculate the angle between two straight lines;150
4.14.4;2.14.4 Test if three points lie on a straight line;150
4.14.5;2.14.5 Test for parallel and perpendicular straight lines;151
4.14.6;2.14.6 Find the position and distance of the nearest point on a line to the origin;151
4.14.7;2.14.7 Find the position and distance of the nearest point on a line to a point;151
4.14.8;2.14.8 Find the reflection of a point in a line;152
4.14.9;2.14.9 Find the normal to a line through a point;152
4.14.10;2.14.10 Find the shortest distance between two skew lines;153
4.15;2.15 Planes;154
4.15.1;2.15.1 Cartesian form of the plane equation;154
4.15.2;2.15.2 General form of the plane equation;154
4.15.3;2.15.3 Hessian normal form of the plane equation;154
4.15.4;2.15.4 Parametric form of the plane equation;155
4.15.5;2.15.5 Converting a plane equation from parametric form to general form;155
4.15.6;2.15.6 Plane equation from three points;156
4.15.7;2.15.7 Plane through a point and normal to a line;157
4.15.8;2.15.8 Plane through two points and parallel to a line;157
4.15.9;2.15.9 Intersection of two planes;158
4.15.10;2.15.10 Intersection of three planes;160
4.15.11;2.15.11 Angle between two planes;162
4.15.12;2.15.12 Angle between a line and a plane;162
4.15.13;2.15.13 Intersection of a line and a plane;163
4.15.14;2.15.14 Position and distance of the nearest point on a plane to a point;163
4.15.15;2.15.15 Reflection of a point in a plane;164
4.15.16;2.15.16 Plane equidistant from two points;164
4.15.17;2.15.17 Reflected ray on a surface;165
4.16;2.16 Lines, planes and spheres;167
4.16.1;2.16.1 Line intersecting a sphere;167
4.16.2;2.16.2 Sphere touching a plane;168
4.16.3;2.16.3 Touching spheres;169
4.17;2.17 Three-dimensional triangles;170
4.17.1;2.17.1 Coordinates of a point inside a triangle;170
4.17.2;2.17.2 Unknown coordinate value inside a triangle;171
4.18;2.18 Parametric curves and patches;173
4.18.1;2.18.1 Parametric curves in R2;173
4.18.2;2.18.2 Parametric curves in R3;177
4.18.3;2.18.3 Planar patch;181
4.18.4;2.18.4 Parametric surfaces in R3;182
4.18.5;2.18.5 Quadratic Bézier curve;184
4.18.6;2.18.6 Cubic Bézier curve;184
4.18.7;2.18.7 Quadratic Bézier patch;185
4.18.8;2.18.8 Cubic Bézier patch;186
4.19;2.19 Second degree surfaces in standard form;187
5;3 Proofs;188
5.1;3.1 Trigonometry;190
5.1.1;3.1.1 Trigonometric functions and identities;190
5.1.2;3.1.2 Cofunction identities;190
5.1.3;3.1.3 Pythagorean identities;190
5.1.4;3.1.4 Useful trigonometric values;191
5.1.5;3.1.5 Compound angle identities;192
5.1.6;3.1.6 Double-angle identities;194
5.1.7;3.1.7 Multiple-angle identities;194
5.1.8;3.1.8 Functions of the half-angle;195
5.1.9;3.1.9 Functions of the half-angle using the perimeter of a triangle;196
5.1.10;3.1.10 Functions converting to the half-angle tangent form;197
5.1.11;3.1.11 Relationships between sums of functions;199
5.1.12;3.1.12 Inverse trigonometric functions;201
5.2;3.2 Circles;202
5.2.1;3.2.1 Proof: Angles subtended by the same arc;202
5.2.2;3.2.2 Proof: Alternate segment theorem;202
5.2.3;3.2.3 Proof: Area of a circle, sector and segment;203
5.2.4;3.2.4 Proof: Chord theorem;205
5.2.5;3.2.5 Proof: Secant theorem;205
5.2.6;3.2.6 Proof: Secant–tangent theorem;205
5.2.7;3.2.7 Proof: Area of an ellipse;206
5.3;3.3 Triangles;208
5.3.1;3.3.1 Proof: Theorem of Pythagoras;208
5.3.2;3.3.2 Proofs: Properties of triangles;208
5.3.3;3.3.3 Proof: Altitude theorem;211
5.3.4;3.3.4 Proof: Area of a triangle;212
5.3.5;3.3.5 Proof: Internal and external angles of a triangle;215
5.3.6;3.3.6 Proof: The medians of a triangle are concurrent at its centroid;215
5.3.7;3.3.7 Proof: Radius and center of the inscribed circle for a triangle;217
5.3.8;3.3.8 Proof: Radius and center of the circumscribed circle for a triangle;220
5.4;3.4 Quadrilaterals;226
5.4.1;3.4.1 Proof: Properties of quadrilaterals;226
5.4.2;3.4.2 Proof: The opposite sides and angles of a parallelogram are equal;229
5.4.3;3.4.3 Proof: The diagonals of a parallelogram bisect each other;229
5.4.4;3.4.4 Proof: The diagonals of a square are equal, intersect at right angles and bisect the opposite angles;230
5.4.5;3.4.5 Proof: Area of a parallelogram;231
5.4.6;3.4.6 Proof: Area of a quadrilateral;231
5.4.7;3.4.7 Proof: Area of a general quadrilateral using Heron’s formula;233
5.4.8;3.4.8 Proof: Area of a trapezoid;235
5.4.9;3.4.9 Proof: Radius and center of the circumscribed circle for a rectangle;236
5.5;3.5 Polygons;237
5.5.1;3.5.1 Proof: The internal angles of a polygon;237
5.5.2;3.5.2 Proof: The external angles of a polygon;237
5.5.3;3.5.3 Proof: Alternate internal angles of a cyclic polygon;238
5.5.4;3.5.4 Proof: Area of a regular polygon;239
5.5.5;3.5.5 Proof: Area of a polygon;240
5.5.6;3.5.6 Proof: Properties of regular polygons;241
5.6;3.6 Three-dimensional objects;243
5.6.1;3.6.1 Proof: Volume of a prism;243
5.6.2;3.6.2 Proof: Surface area of a rectangular pyramid;244
5.6.3;3.6.3 Proof: Volume of a rectangular pyramid;245
5.6.4;3.6.4 Volume of a rectangular pyramidal frustum;246
5.6.5;3.6.5 Proof: Volume of a triangular pyramid;246
5.6.6;3.6.6 Proof: Surface area of a right cone;247
5.6.7;3.6.7 Proof: Surface area of a right conical frustum;247
5.6.8;3.6.8 Proof: Volume of a cone;248
5.6.9;3.6.9 Proof: Volume of a right conical frustum;249
5.6.10;3.6.10 Proof: Surface area of a sphere;249
5.6.11;3.6.11 Proof: Volume of a sphere;250
5.6.12;3.6.12 Proof: Area and volume of a torus;252
5.6.13;3.6.13 Proof: Radii of the spheres associated with the Platonic solids;252
5.6.14;3.6.14 Proof: Inner and outer radii for the Platonic solids;257
5.6.15;3.6.15 Proof: Dihedral angles for the Platonic solids;261
5.6.16;3.6.16 Proof: Surface area and volume of the Platonic solids;265
5.7;3.7 Coordinate systems;268
5.7.1;3.7.1 Cartesian coordinates;268
5.7.2;3.7.2 Polar coordinates;268
5.7.3;3.7.3 Cylindrical coordinates;269
5.7.4;3.7.4 Spherical coordinates;269
5.8;3.8 Vectors;271
5.8.1;3.8.1 Proof: Magnitude of a vector;271
5.8.2;3.8.2 Proof: Normalizing a vector to a unit length;271
5.8.3;3.8.3 Proof: Scalar (dot) product;271
5.8.4;3.8.4 Proof: Commutative law of the scalar product;272
5.8.5;3.8.5 Proof: Associative law of the scalar product;272
5.8.6;3.8.6 Proof: Angle between two vectors;272
5.8.7;3.8.7 Proof: Vector (cross) product;273
5.8.8;3.8.8 Proof: The non-commutative law of the vector product;273
5.8.9;3.8.9 Proof: The associative law of the vector product;274
5.8.10;3.8.10 Proof: Scalar triple product;274
5.9;3.9 Quaternions;275
5.9.1;3.9.1 Definition of a quaternion;275
5.10;3.10 Transformations;279
5.10.1;3.10.1 Proof: Scaling in R2;279
5.10.2;3.10.2 Proof: Translation in R2;280
5.10.3;3.10.3 Proof: Rotation in R2;280
5.10.4;3.10.4 Proof: Shearing in R2;281
5.10.5;3.10.5 Proof: Reflection in R2;282
5.10.6;3.10.6 Proof: Change of axes in R2;283
5.10.7;3.10.7 Proof: Identity matrix in R2;284
5.10.8;3.10.8 Proof: Scaling in R3;284
5.10.9;3.10.9 Proof: Translation in R3;285
5.10.10;3.10.10 Proof: Rotation in R3;285
5.10.11;3.10.11 Proof: Reflection in R3;287
5.10.12;3.10.12 Proof: Change of axes in R3;289
5.10.13;3.10.13 Proof: Identity matrix in R3;290
5.11;3.11 Two-dimensional straight lines;291
5.11.1;3.11.1 Proof: Cartesian form of the line equation;291
5.11.2;3.11.2 Proof: Hessian normal form (after Otto Hesse (1811–1874));292
5.11.3;3.11.3 Proof: Equation of a line from two points;292
5.11.4;3.11.4 Proof: Point of intersection of two straight lines;294
5.11.5;3.11.5 Proof: Angle between two straight lines;295
5.11.6;3.11.6 Proof: Three points lie on a straight line;296
5.11.7;3.11.7 Proof: Parallel and perpendicular straight lines;297
5.11.8;3.11.8 Proof: Shortest distance to a line;298
5.11.9;3.11.9 Proof: Position and distance of a point on a line perpendicular to the origin;298
5.11.10;3.11.10 Proof: Position and distance of the nearest point on a line to a point;299
5.11.11;3.11.11 Proof: Position of a point reflected in a line;300
5.11.12;3.11.12 Proof: Normal to a line through a point;302
5.11.13;3.11.13 Proof: Line equidistant from two points;303
5.11.14;3.11.14 Proof: Equation of a two-dimensional line segment;304
5.11.15;3.11.15 Proof: Point of intersection of two two-dimensional line segments;305
5.12;3.12 Lines and circles;307
5.12.1;3.12.1 Proof: Line and a circle;307
5.12.2;3.12.2 Proof: Touching and intersecting circles;309
5.13;3.13 Second degree curves;312
5.13.1;3.13.1 Circle;312
5.13.2;3.13.2 Ellipse;312
5.13.3;3.13.3 Parabola;314
5.13.4;3.13.4 Hyperbola;315
5.14;3.14 Three-dimensional straight lines;316
5.14.1;3.14.1 Proof: Straight-line equation from two points;316
5.14.2;3.14.2 Proof: Intersection of two straight lines;316
5.14.3;3.14.3 Proof: Angle between two straight lines;317
5.14.4;3.14.4 Proof: Three points lie on a straight line;317
5.14.5;3.14.5 Proof: Parallel and perpendicular straight lines;317
5.14.6;3.14.6 Proof: Position and distance of a point on a line perpendicular to the origin;318
5.14.7;3.14.7 Proof: Position and distance of the nearest point on a line to a point;318
5.14.8;3.14.8 Proof: Position of a point reflected in a line;319
5.14.9;3.14.9 Proof: Normal to a line through a point;320
5.14.10;3.14.10 Proof: Shortest distance between two skew lines;321
5.15;3.15 Planes;322
5.15.1;3.15.1 Proof: Equation to a plane;322
5.15.2;3.15.2 Proof: Plane equation from three points;325
5.15.3;3.15.3 Proof: Plane through a point and normal to a line;327
5.15.4;3.15.4 Proof: Plane through two points and parallel to a line;327
5.15.5;3.15.5 Proof: Intersection of two planes;327
5.15.6;3.15.6 Proof: Intersection of three planes;329
5.15.7;3.15.7 Proof: Angle between two planes;330
5.15.8;3.15.8 Proof: Angle between a line and a plane;330
5.15.9;3.15.9 Proof: Intersection of a line and a plane;330
5.15.10;3.15.10 Proof: Position and distance of the nearest point on a plane to a point;331
5.15.11;3.15.11 Proof: Reflection of a point in a plane;332
5.15.12;3.15.12 Proof: Plane equidistant from two points;332
5.15.13;3.15.13 Proof: Reflected ray on a surface;333
5.16;3.16 Lines, planes and spheres;334
5.16.1;3.16.1 Proof: Line intersecting a sphere;334
5.16.2;3.16.2 Proof: Sphere touching a plane;335
5.16.3;3.16.3 Proof: Touching spheres;335
5.17;3.17 Three-dimensional triangles;337
5.17.1;3.17.1 Proof: Point inside a triangle;337
5.17.2;3.17.2 Proof: Unknown coordinate value inside a triangle;337
5.18;3.18 Parametric curves and patches;338
5.18.1;3.18.1 Proof: Planar surface patch;338
5.18.2;3.18.2 Proof: Bézier curves in R2 andR3;338
5.18.3;3.18.3 Proof: Bézier surface patch in R3;340
6;4 Glossary;343
7;5 Bibliography;351
8;Index;353