Mathematical and Computer Programming Techniques for Computer Graphics (eBook)

Vector Algebra Survival Kit Some Basic Definitions and Notation Multiplication of a Vector by a Scalar Vector Addition Position Vectors and Free Vectors The Vector Equation of a Line Linear Dependence / Independence of Vectors Vector Bases The Components of a Vector Multiplication of a Vector by a Scalar Vector Addition Vector Equality Orthogonal, Orthonormal and Right-Handed Vector Bases Cartesian Bases and Cartesian Coordinates The Length of a Vector The Scalar Product of Vectors The Scalar Product Expresses in Terms of its Components Properties and Applications of the Scalar Product The Direction Ratios and Direction Cosines of a Vector The Vector Product of Two Vectors The Vector Product Expressed in Terms of its Components Properties of the Vector Product Triple Produces of Vectors The Components of a Vector Relative to a Non-orthogonal Basis The Decomposition of a Vector According to a Basis The Vector Equation of the Line Revisited The Vector Equation of the Place Some Applications of Vector Algebra in Analytical Geometry Summary of Vector Algebra Axioms and Rules A Simple Vector Algebra C Library Matrix Algebra Survival Kit The Definition of a Matrix Square Matrices Diagonal Matrices The Identity Matrix The Zero or Null Matrix The Transpose of a Matrix Symmetric and Antisymmetric Matrices Triangular Matrices Scalar Matrices Equality of Matrices Matrix Operations The Minor of a Matrix The Determinant of a Matrix The Computational Rules of Determinants The Cofactor of an Element of a Matrix and the Cofactor Matrix The Ajoint Matrix or Adjugate Matrix The Reciprocal or Inverse of a Matrix A Theorem on Invertible Matrices and their Determinants Axioms and Rules of Matrix Inversion Solving a System of Linear Simultaneous Equations Orthogonal Matrices Two Theorems on Vector by Matrix Multiplication The Row / Column Reversal Matrix Summary of Matrix Algebra Axioms and Rules A Simple Matrix Algebra C Library Vector Spaces or Linear Spaces The Definition of a Scalar Field The Definition of a Vector Space Linear Combinations of Vectors Linear Dependence and Linear Independence of Vectors Spans and Bases of a Vector Space Transformations between Bases Transformations between Orthonormal Bases An Alternative Notation for Change of Basis Transformations Two-Dimensional Transformations The Definition of a 2D Transformation The Concatenation of Transformations 2D Graphics Transformations 2D Primitive Transformations 2D Composite Transformations The Sign of the Angles in Transformations Some Important Observations The Matrix Representation of 2D Transformations The Matrix Representation of Primitive Transformations Some Transformation Matrix Properties The Concatenation of Transformation Matrices Local Frame and Global Frame Transformations Transformations of the Frame of Reference or Coordinate System The Viewing Transformation Homogeneous Coordinates A Simple C Library for 2D Transformations Two-Dimensional Clipping Clipping a 2D Point to a Rectangular Clipping Boundary Clipping a 2D Line Segment to a Rectangular Clipping Boundary The Cohen and Sutherland 2D Line-Clipping Algorithm 2D Polygon Clipping References Three-Dimensional Transformations Primitive 3D Transformations The Global and Local Frames of Reference A

"1 Set Theory Survival Kit (p. 3-4)

Unlike many other branches of mathematics, where the formulation of ideas and concepts occurs gradually over time and is developed by many mathematicians before it is formalised into a single theory, the formulation of set theory is almost the single-handed creation of one mathematician, namely Georg Cantor. Georg Ferdinand Ludwig Philipp Cantor (1845–1918) was born in Russia to a Danish father and a Russian mother and spent most of his life in Germany.

Between the years 1879 and 1884 Cantor published a six-part treatise on set theory (where he introduced some of the fundamental notions of this theory) followed by the publication of a two-part treatise between the years 1895 and 1897 (where he clarified and systematised what he had introduced in his first cycle of publications). Between the years 1897 and 1902 a number of paradoxes in Cantor’s set theory began to emerge. These paradoxes were discovered by Cantor himself and, among others, by the Italian mathematician Cesare Burali-Forti (1861–1931), the German mathematician Ernst Friedrich Ferdinand Zermelo (1871–1953) and the British mathematician Bertrand Arthur William Russell (1872–1970).

In 1908, Zermelo was the first to attempt to introduce an axiomatic approach to the study of set theory. Since then, many mathematicians proved influential in the further development of set theory. Among these are the German mathematician Adolf Abraham Halevi Fraenkel (1891–1965), the Hungarian mathematician and computer scientist John von Neumann (1903–1957), the Swiss mathematician Paul Isaac Bernays (1888–1977) and the Czech mathematician Kurt G¨odel (1906– 1978).

Since its introduction, set theory has proved to be of great importance to the modern formulation of many topics of pure mathematics. In current mathematical practice, such topics as numbers, relations, intervals, functions and transformations are defined in terms of sets. In our study of computer graphics we will frequently use sets to explain a number of other mathematical concepts. Thus, it is important to gain a good understanding of sets and set theory.

1.1 Some Basic Notations and Definitions

1.1.1 Sets and Elements

The concept of the set is one of the basic concepts of mathematics and is fundamental to most branches of modern mathematics. Thus, we start our discussion by defining the terms set and element or member. A set is any well-defined list, collection or class of objects, in which the order and multiplicity of these objects has no significance and is ignored. These objects are called the elements or members of the set. The phrase well-defined means that there is a clear and unambiguous way of defining the elements of a set, i.e. of determining if a given element is a member of a given set. Sets may be finite or infinite depending on the number of their elements. Set theory is the branch of mathematics that concerns the study of sets and their properties."