【正文】 ts of algebra to prepare the reader for the algebraic manipulations used in later chapters. Although algebra can be a very abstract mathematical tool, here we only need to explore those practical features relevant to its application to puter graphics. NotationThe word ‘a(chǎn)lgebra ’ es from the Arabic aljabr w’almuqabal, meaning ‘restoration and reduction’. Today’s algebraic notation has evolved over thousands of years during which di?erent civilizations have developed ways of annotating mathematical and logical problems. In retrospect, it does seem strange that centuries passed before the ‘equals’ sign (=) was invented and concepts such as ‘zero’ (ce 876) were introduced, especially as they now seem so important. But we are not at the end of this evolution, because new forms of annotation and manipulation will continue to emerge as new mathematical ideas are invented.One fundamental concept of algebra is the idea of giving a name to an unknown quantity. For example, m is often used to represent the slope of a 2D line, and c is the line’s ycoordinate where it intersects the y axis. Ren180。e Descartes (1596–1650) formalized the idea of using letters from the beginning of the alphabet (a, b, c, etc.) to represent arbitrary numbers, and letters at the end of the alphabet (p, q, r, s, t,... x, y, z) to identify variables representing quantities such as pressure (p), temperature (t), and coordinates (x, y, z).With the aid of the basic arithmetic operators +, ?, , 247。 we can develop expressions that describe the behaviour of a physical process or a speci?c putation. For example, the expression ax + by ?d equals zero for a straight line. The variables x and y are the coordinates of any point on the line and the values of a, b, d determine the position and orientation of the line. There is an implied multiplication between ax and by, which would be expressed as a ?x and b?y if we were using a programming = sign permits the line equation to be expressed as a selfevident statement: 0 = ax + by ? d. Such a statement implies that the expressions on the left and righthand sides of the = sign are ‘equal’ or ‘balanced’. So whatever is done to one side must also be done to the other in order to maintain equality or balance. For example, if we add d to both sides, the straightline equation bees d = ax + by. Similarly, we could double or treble both expressions, divide them by 4, or add 6, without disturbing the underlying relationship.Algebraic expressions also contain a wide variety of other notation, such as√x square root of x√n x n th root of xxn x to the power nsin(x) sine of x cos(x) cosine of x tan(x) tangent of x log(x) logarithm of xln(x) natural logarithm of xParentheses are used to isolate part of an expression in order to select a subexpression that is manipulated in a particular way. For example, the parentheses in c(a + b)+ d ensure that the variables a and b are added together before being multiplied by c and ?nally added to d. Algebraic LawsThere are three basic laws that are fundamental to manipulating algebraic expressions: associative, mutative and distributive. In the following descriptions, the term binary operation represents the arithmetic operations +, ?,or , which are always associated with a pair of numbers or variables. Associative LawThe associative law in algebra states that when three or more elements are linked together through a binary operation, the result is independent of how each pair of elements is grouped. The associative law of addition isa + (b + c)= (a + b)+ c (). 1 + (2 + 3) = (1 + 2) + 3and the associative law of multiplication isa (b c)= (a b) c (). 1 (2 3) = (1 2) 3Note that substraction is not associative:a ? (b ? c) ?= (a ? b) ? c (). 1 ? (2 ? 3) ?= (1 ? 2) ? 3 Commutative LawThe mutative law in algebra states that when two elements are linked through some binary operation, the result is independent of the order of the elements. The mutative law of addition isa + b = b + a (). 1 + 2 = 2 + 1and the mutative law of multiplication isa b = b a (). 2 3=3 2Note that subtraction is not mutative:a ? b ?= b ? a (). 2 ? 3 ?=3 ? 2 Distributive LawThe distributive law in algebra describes an operation which when performed on a bination of elements is the same as performing the operation on the individual elements. The distributive law does not work in all cases of arithmetic. For example, multiplication over addition holds:a (b + c)= ab + ac (). 3 (4 + 5) = 3 4+3 5whereas addition over multiplication does not:a + (b c) ?= (a + b) (a + c) (). 3 + (4 5) ?= (3 + 4) (3 + 5)Although most of these laws seem to be natural for numbers, they do not necessarily apply to all mathematical constructs. For instance, the vector product, which multiplies two vectors together, is not mutative. Solving the Roots of a Quadratic EquationTo put the above laws and notation into practice, let’s take a simple example to illustrate the logical steps in solving a problem. The task involves solving the roots of a quadratic equation, . those values of x that make the equation equal the quadratic equation where a ?= 02.22√ 2第 61 頁(yè) 共 62 頁(yè) IndicesA notation for repeated multiplication is with the use of indices. For instance, in the above example with a quadratic equation x2 is used to represent x x. This notation leads to a variety of situations where laws are required to explain how the result is to be puted. Laws of IndicesThe laws of indices can be expressed aswhich are easily veri?ed using some simple examples. Examples1: 23 22 =8 4= 32 = 252: 24 247。 22 = 16 247。 4=4= 223: (22 )3 = 64= 26From the above laws, it is evident that LogarithmsTwo people are associated with the invention of logarithms: John Napier (1550–1617) and Joost Bu168。rgi (1552–1632). Both men were