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Introduction to the Mathematics for Computer Graphics


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Publisher Comments:

John Vince explains a wide range of mathematical techniques and problem-solving strategies associated with computer games, computer animation, virtual reality, CAD, and other areas of computer graphics. Covering all the mathematical techniques required to resolve geometric problems and design computer programs for computer graphic applications, each chapter explores a specific mathematical topic prior to moving forward into the more advanced areas of matrix transforms, 3D curves and surface patches. Problem-solving techniques using vector analysis and geometric algebra are also discussed. All the key areas are covered including: • Numbers • Algebra • Trigonometry • Coordinate geometry • Transforms • Vectors • Curves and surfaces • Barycentric coordinates • Analytic geometry. Plus - and unusually in a student textbook - a chapter on geometric algebra is included. With plenty of worked examples, the book provides a sound understanding of the mathematics required for computer graphics, giving a fascinating insight into the design of computer graphics software, and setting the scene for further reading of more advanced books and technical research papers.


This text covers all mathematical techniques needed to resolve geometric problems and design computer programs for computer graphic applications. It also discusses problem-solving techniques using vector analysis and includes a chapter on geometric algebra.

About the Author

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

Table of Contents

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.- 2D Scaling.- 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 Inside a Triangle.- Area of a Triangle.- Hessian Normal Form.- Intersection of a Circle with a Straight Line.- 3D Geometry.- Equation of a Straight Line.- Point of Intersection of Two Straight Lines.- Equation of a Plane.- Cartesian Form of the Plane Equation.- General Form of the Plane Equation.- Parametric Form of the Plane Equation.- Converting from the Parametric to the General Form.- Plane Equation from Three Points.- Intersecting Planes.- Intersection of Three Planes.- Angle Between  Two Planes.- Angle Between a Line and a Plane.- Intersection of a Line with a Plane.- Summary.- Barycentric Coordinates.- Introduction.- Ceva's Theorem.- Ratios and Proportion.- Mass Points.- Linear Interpolation.- Convex Hull Property.- Areas.- Volumes.- Bezier Curves and Patches.- Summary.- Geometric Algebra.- Introduction.- Symmetric and Antisymmetric Functions.- Trigonometric Foundations.- Vectorial Foundations.- Inner and Outer Products.- The Geometric Product in 2D.- The Geometric Product in 3D.- The Outer Product of 3D Vectors.- Axioms.- Notation.- Grades, Pseudoscalars and Multivectors.- Redefining the Inner and Outer Products.- The Inverse of a Vector.- The Imaginary Properties of the Outer Product.- Duality.- The Relationship between the Vector Product and the Outer Product.- The Relationship Between the Quaternions and Bivectors.- Reflections and Rotations.- 2D Reflections.- 3D Reflections.- 2D Rotations.- Rotors.- Applied Geometric Algebra.- Sine Rule.- Cosine Rule.- A Point Perpendicular to a Point on a Line.- Reflecting a Vector about a Vector.- Orientation of a Point with a Plane.- Summary.- Worked Examples.- Introduction.- Area of a Regular Polygon.- Dihedral Angle if a Dodecahedron.- Conclusion

Product Details

Vince, John
Vince, John A.
Computer Science
Computer graphics
Computer graphics -- Mathematics.
Computer Graphics - General
Computer animation
Computer games
Edition Description:
3rd ed. 2010
Undergraduate Topics in Computer Science
Publication Date:
235 x 155 mm 970 gr

Related Subjects

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Introduction to the Mathematics for Computer Graphics New Trade Paper
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Product details 308 pages Springer - English 9781849960229 Reviews:
"Synopsis" by , This text covers all mathematical techniques needed to resolve geometric problems and design computer programs for computer graphic applications. It also discusses problem-solving techniques using vector analysis and includes a chapter on geometric algebra.
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