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Mathematics for Computer Graphics Applications introduces the mathematics that is the foundation of today’s advanced computer graphics applications. Computer graphics, as you know, is at the leading edge of the revolution in how we use computers and automation to enhance our lives. It has dramatically changed the practice of engineering and science, medicine, and the media. The wonderful special effects that entertain and inform us at the movies, on TV, or while surfing cyberspace are possible only through powerful computer graphics applications. From the creatures of Jurassic Park and the imaginary worlds of Myst and Star Wars, to the dance of virtual molecules on a chemist’s computer display screen, these effects are possible only because an equally powerful and wonderful world of mathematics is at work behind the scenes. Some of the mathematics is traditional but with a new job to do, and some is brand new.
So, this is a textbook with a purpose, and that is to provide you with a strong base for more advanced studies in mathematics, computer science, physics, and engineering, and in more specialized subjects such as algebraic and computational geometry, geometric modeling, and CAD/CAM. I hope I have achieved that purpose in a way that is both useful and enjoyable. Here is an outline of what you will find in each chapter:
Chapter 1 Vectors Vectors have a strong intuitive appeal because of their innate geometric character. This chapter builds upon this intuition to introduce the algebra and geometry of vectors, upon which much of the text depends.
Chapter 2 Matrix Methods Matrix algebra provides a concise and powerful way to formulate and solve very large and complex sets of equations. Subjects covered include definitions, matrix equivalence, matrix arithmetic, partitioning, determinants, inversion, and eigenvectors.
Chapter 3 Transformations explores geometric transformations and their invariant properties. It introduces systems of equations that produce linear transformations and develops the algebra and geometry of translations, rotations, reflection, and scaling. Vectors and matrix algebra are put to good use here in ways that clearly demonstrate their effectiveness. The mathematics of transformations is indispensable to animation and special effects, as well as many areas of physics and engineering.
Chapter 4 Symmetry and Groups This chapter uses geometry and intuition to define symmetry and introduces easy-to-understand but powerful analytic methods to further explore this subject. It defines a group and the more specialized symmetry group. It concludes with a look at the rotational symmetry of polyhedra.
Chapter 5 Limit and Continuity Limit processes are at work in computer graphics applications to ensure the display of smooth, continuous-looking curves and surfaces. And continuity is now an important consideration in the construction and rendering of shapes for CAD/CAM and similar applications. This chapter takes a look at some old and some new ways to learn about limit and continuity, including the method of exhaustion, sequences and series, functions, limit theorems, continuity and continuous functions.
Chapter 6 Topology What makes a sphere different from a torus? Why are left and right not reliable directions on a Möbius strip? This chapter answers questions like these. It shows that topology is the study of continuity and connectivity and how these characteristics are preserved when geometric figures are deformed. Topological equivalence, topology of a closed path, piecewise flat surfaces, closed curved surfaces, orientation, and curvature are among the topics covered here.
Chapter 7 Halfspaces Halfspaces are always fun to work with because we can combine very simple halfspaces to create complex shapes that are otherwise extremely difficult, if not impossible, to represent mathematically. This chapter explores both two- and three-dimensional halfspaces. It first reviews elementary set theory and how to interpret it geometrically in Venn diagrams and then explains the Boolean operators union, intersection and difference and how to use them to create more complex shapes.
Chapter 8 Points, Chapter 9 Lines, and Chapter 10 Planes discuss these three misleadingly simple geometric objects and show how they are the basic building blocks for other more complex geometric objects.
Chapter 11 Polygons The use of polygons in computer graphics is widespread. For example, it is easy to subdivide and approximate the surfaces of solids with planes bounded by them. Topics covered here include definitions of various types of polygons, their geometric properties, convex hulls, construction of the regular polygons, and symmetry.
Chapter 12 Polyhedra defines convex, concave, and stellar polyhedra, with particular attention to the five regular polyhedra, or Platonic solids. It defines Euler’s Formula and shows how to use it to prove that only five regular polyhedra are possible in a space of three dimensions. Other topics include definitions of the various families of polyhedra, nets, the convex hull of a polyhedron, the connectivity matrix, halfspace representations of polyhedra, model data structures, and maps.
Chapter 13 Constructive Solid Geometry Traditional geometry does not tell us how to create even the simplest shapes we see all around us. Constructive solid geometry is a way to describe these shapes as combinations of even simpler shapes. The chapter revisits and applies some elementary set theory, halfspaces, and Boolean algebra. Binary trees are introduced, and their properties are developed.
Chapter 14 Curves explores the mathematical definition of a curve as a set of parametric equations, a form that is very useful to CAD/CAM, geometric modeling, and other computer graphic applications. Parametric equations are the basis for Bézier, NURBS, and Hermite curves. Both plane and space curves are introduced, followed by discussions of the tangent vector, blending functions, conic curves, reparameterization, and continuity.
Chapter 15 The Bézier Curve This curve is not only an important part of almost every computer-graphics illustration program, it is a standard tool of animation. The chapter begins by describing a surprisingly simple geometric construction of a Bézier curve, followed by a derivation of its algebraic definition, basis functions, control points, and how to join two curves end-to-end to form a single, more complex curve.
Chapter 16 Surfaces develops the parametric equations of surfaces, a natural extension of the mathematics of curves discussed in Chapter 14. Topics include the surface patch, plane and cylindrical surfaces, the bicubic surface, the Bézier surface, and the surface normal.
Chapter 17 Computer Graphics Display Geometry introduces some of the basic geometry and mathematics of computer graphics, including display coordinate systems, windows, line and polygon clipping, polyhedra edge visibility, and silhouettes.
Chapter 18 Display and Scene Transformations discusses orthographic and perspective transformations and explores some scene transformations: orbit, pan, and aim. Chapters 17 and 18 draw heavily on concepts introduced earlier in the text and lay the foundation for more advanced studies in computer graphics applications.
If you have questions about this book's contents and use, please contact me via e-mail at mortenson@olympus.net.
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