Mathematics for Computer Graphics (eBook)

John Vince has been writing books for 25 years. Previous publications with Springer include: Geometric Algebra: An Algebraic System for Computer Games and Animation Springer, 2009, ISBN 978-1-84882-378-5 Vector Analysis for Computer Graphics Springer, 2007, ISBN 978-1-84628-803-6 Mathematics for Computer Graphics Springer, 2006, ISBN 1-84628-034-6 Introduction to Virtual Reality Springer, 2004, ISBN 1-85233-739-7 More information can be found at http://www.johnvince.co.uk/

Mathematics.- Introduction.- Is Mathematics Difficult?.- Who Should Read this book?.- Aims and Objectives of this Book.- Assumptions Made in this Book.- How to use the Book.- Numbers.- Introduction.- Natural Numbers.- Prime Numbers.- Integers.- Rational Numbers.- Irrational Numbers.- Real Numbers.- The Number Line.- Complex Numbers.- Summary.- Algebra.- Introduction.- Notation.- Algebraic Laws.- Associative Law.- Commutative Law.- Distributive Law.- Solving the Roots of a Quadratic Equation.- Indices.- Laws of Indices.- Examples.- Logarithms.- Further Notation.- Summary.- Trigonometry.- Introduction.- The Trigonometric Ratios.- Example.- Inverse Trigonometric Ratios.- Trigonometric Relationships.- The Sine Rule.- The Cosine Rule.- Compound Angles.- Perimeter Relationships.- Summary.- Cartesian Coordinates.- Introduction.- The Cartesian xy-plane.- Function Graphs.- Geometric Shapes.- Polygonal Shapes.- Areas of Shapes.- Theorem of Pythagoras in 2D.- 3D Coordinates.- Theorem of Pythagoras in 3D.- 3D polygons.- Euler’s Rule.- Summary.- Vectors.- Introduction.- 2D Vectors.- Vector Notation.- Graphical Representation of Vectors.- Magnitude of a Vector.- 3D Vectors.- Vector Manipulation.- Multiplying a Vector by a Scalar.- Vector Addition and Subtraction.- Position Vectors.- Unit Vectors.- Cartesian Vectors.- Vector Multiplication.- Scalar Product.- Example of the Scalar Product.- The Dot Product in Lightening Calculations.- The Scalar Product in Back-Face Detection.- The Vector Product.- The Right-Hand Rule.- Deriving a Unit Normal Vector for a Triangle.- Areas.- Calculating 2D Areas.- Summary.- Transforms.- Introduction.- 2D Transforms.- Translation.- Scaling.- Reflection.- Matrices.- Systems of Notation.- The Determinant of a Matrix.- Homogeneous Coordinates.- 2D Translation.- 2D Scaling.- 2D Reflections.- 2D Shearing.- 2D Rotation.- 2DScaling.- 2D Reflection.- 2D Rotation about an Arbitrary Point.- 3D Transforms.- 3D Translation.- 3D Scaling.- 3D Rotation.- Gimbal Lock.- Rotating about an Axis.- 3D Reflections.- Change of Axes.- 2D Change of Axes.- Direct Cosines.- 3D Change of Axes.- Positioning on the Virtual Camera.- Direction Cosines.- Euler Angles.- Rotating a point about an Arbitrary Axis.- Matrices.- Quaternions.- Adding and Subtracting Quaternions.- Multiplying Quaternions.- Pure Quaternion.- The Inverse Quaternion.- Unit Quaternion.- Rotating Points about an Axis.- Roll, Pitch and Yaw Quaternions.- Quaternions in Matrix Form.- Frames of Reference.- Transforming Vectors.- Determinants.- Perspective Projection.- Summary.- Interpolation.- Introduction.- Linear Interpolation.- Non-Linear Interpolation.- Trigonometric Interpolation.- Cubic Interpolation.- Interpolating Vectors.- Interpolating Quaternions.- Summary.- Curves and Patches.- Introduction.- The Circle.- The Ellipse.- Bezier Curves.- Bernstein Polynomials.- Quadratic Bezier Curves.- Cubic Bernstein Polynominals.- A Recursive Bezier Formula.- Bezier Curves Using Matrices.- Linear Interpolation.- B-Splines.- Continuity.- Non-Uniform B-Splines.- Non-Uniform Rational B-Splines.- Surface Patches.- Planar Surface Patch.- Quadratic Bezier Surface Patch.- Cubic Bezier Surface Patch.- Summary.- Analytical Geometry.- Introduction.- Review of Geometry.- Angles.- Intercept Theorems.- Golden Section.- Triangles.- Centre of Gravity of a Triangle.- Isosceles Triangle.- Equilateral Triangle.- Right Triangle.- Theorem of Thales.- Theorem of Pythagoras.- Quadrilaterals.- Trapezoid.- Parallelogram.- Rhombus.- Regular Polygon (n-gon).- Circle.- 2D Analytical Geometry.- Equation of a Straight Line.- The Hessian Normal Form.- Space Partitioning.- The Hessian Normal Form From Two Points.- Intersection Points.- Intersection Point of Two Straight Lines.- Intersection Point of Two Line Segments.- Point Insi