Computer Graphics Through Key Mathematics

ISBN-13: 9781852334222 / Angielski / Miękka / 2001 / 343 str.

H. Jones; Huw Jones

Opis

Computer Graphics through Key Mathematics introduces the mathematics that support computer graphics on a 'need to know' basis. Its approach means you don't have to do advanced mathematical manipulation in order to understand the capabilities, scope and limitations of the computer graphics systems that create impressive images. The book is written in a clear, easy-to-understand way and is aimed at all those who have missed out on an extended mathematical education but who are studying or working in areas where computer graphics or 3D design plays an vital part. All those who have no formal training but who want to understand the foundations of computer graphics systems should read this book, as should mathematicians who want to understand how their subject is used in computer image synthesis.

Spis treści

1 The Processes of Computer Graphics.- Object Model Building.- Depiction of Models.- Conclusion.- 2 Numbers, Counting and Measuring.- Natural Numbers.- Integers.- Rational Numbers.- Real Numbers.- Complex Numbers.- Representations of Number.- The Computer Representation of Number.- Boolean Algebra.- Summary.- 3 Coordinates and Dimension: Representations of Space and Colour.- Cartesian Coordinates.- Defining Space by Equations and Inequalities.- Angles.- Trigonometry and Polar Coordinates.- Dimension.- Coordinate Systems in Three Dimensions.- Colour and its Representation.- Summary.- 4 Functions and Transformations: Ways of Manipulating Space.- Functions as Mappings.- Graphs of Functions.- Transformations in 2D.- Transformations in 3D.- Combining Affine Transformations.- Inversion of Affine Transformations.- Inversion of Functions.- Shape Transformation by Function Change.- Conclusions.- 5 Form from Function: Analysis of Shapes.- The Straight Line.- Drawing General Function Graphs.- Graphs of Polynomials.- Calculus: Differentiation.- Calculus: Integration.- Series Expansions.- Calculus and Animation.- The Exponential Function.- The Conic Sections.- Some Standard 3D Forms and their Equations.- Summary.- 6 Matrices: Tools for Manipulating Space.- Matrices in Computer Graphics.- Definition and Notation.- Forms of Matrices.- Operations on Matrices: Addition.- Operations on Matrices: Multiplication.- The Identity Matrix.- Matrices and Equations.- The Inverse of a Square Matrix.- Matrices, Transformations and Homogeneous Coordinates: Two Dimensions.- Matrices, Transformations and Homogeneous Coordinates: Three Dimensions.- Inverse of a Transformation Matrix.- Perspective Projection.- Computer Implementation of Matrix Methods.- Summary.- 7 Vectors: Descriptions of Spatial Relationships.- Definition of a Vector.- Notation.- Addition of Vectors: The Parallelogram and Triangle Laws.- Multiplication of a Vector by a Scalar.- Examples of Vector Quantities.- Vectors in 2D Cartesian Spaces.- Vectors in 3D Cartesian Spaces.- Multiplication of Vectors: The Scalar or Dot Product.- Multiplication of Vectors: The Vector or Cross Product.- Representation of Lines Using Vectors.- Classification of Points against Planes Using Vectors.- Representation of Planes in Standard Form.- Intersection of a Line with a Plane.- Inclusion of a Point in a Triangle.- Reflected and Refracted Rays.- Distance between Two Skew Lines.- Intersection of Two Planes.- Summary.- 8 Geometric Modelling and Fractals: Building Descriptions of Objects.- Data Structures.- Geometric Modelling Systems.- Voxel Modelling Methods.- Constructive Solid Geometry (CSG).- Boundary Representation (B-Rep).- Isosurface Modelling.- Fractals.- Fractal Dimension.- Fractals Based in the Complex Plane: Julia and Mandelbrot Sets.- Fractals in Simulation of Natural Phenomena.- Summary.- 9 Splines: Generation of Curves and Surfaces.- Reasons for Splines.- Interpolation.- Bézier Splines for Curve Drawing.- Animation Control Using Cubic Bézier Curves.- Drawing Bézier Curves.- Interpolating Splines for Curve Generation.- Animation Control Using Interpolating Splines.- B-Splines.- Non-Uniform Rational B-Splines: NURBS.- Circles and Other Conic Sections.- Surface Construction Using Bézier Patches.- Surface Generation Based on Other Forms of Curve.- Depiction of Surface Patches.- Summary.- 10 Drawing and Rendering: How to Create Pictures.- What is a 3D Drawing?.- Methods for Rendering.- Hidden Surface Removal.- Flat or Lambert Shading.- Scan Line Methods.- Gouraud Shading.- Phong Shading.- Exact Object Rendering.- Shadows.- Specular Highlights.- Textures.- Ray Tracing.- Radiosity.- Anti-Aliasing.- Summary.- Suggestions for Further Reading.- Mathematical and Scientific Minds.- Understanding of Mathematics.- Computer Graphics in General.- Geometric Modelling.- Fractals and Related Issues.- Journals.- Names.