In our study of computer graphics we will frequently use vectors to solve a va-riety of problems in such diverse areas as geometric modelling, transformations,projections, visibility determination, lighting, shading and texturing, and the de-velopment of curves, surfaces and deformations. Thus it is important to gain athorough understanding of vector algebra.2.1Some Basic Definitions and NotationLet us start by defining the termsscalarandvector. A scalar is a quantity that iscompletely determined by a single numerical value, which consists of a possiblysignedmagnitude(i.e. arealnumber). For instance, quantities such as tempera-ture, time, length, mass, and speed are all represented by scalars. A vector, on theother hand, is a quantity that is determined by adirectionand amagnitude. It isadirectedquantity represented by an arrow. The direction and the length of thisarrow determine the vector’s direction and magnitude, respectively.In everyday language it is common to use quantities such as speed and velocityinterchangeably. This use however is inaccurate as the speed of a vehicle repre-sents the magnitude of its movement alone and gives no indication of its directionof movement, while the velocity of a vehicle represents both its magnitude and itsdirection of movement. A scalar is a single dimensional quantity, while a vectoris a multidimensional quantity.65

66Mathematical and Computer Programming Techniques for Computer GraphicsIn this book we denote vectors in bold italic notation such asv,vorv, whilescalars are denoted by Greek letters or non-bold italic characters such asλorl.A three-dimensional geometric vector can be seen as a translation in three-dimensional Euclidean spaceE3. Given a pointPinE3, we may use a vectorvto move this point to a new positionP, as shown inFig. 2.1. Sometimes wecall the vectorvadisplacement vector, as it displaces pointPto its new positionP. The distance between pointsPandPis called themagnitudeof the vectorvand it is denoted by|v|or the non-bold italic version of the vector namev. Thisdistance is the same for all pointsPinE3. Thus, the original position of pointPis immaterial. For all vectorsvwe say that|v| ≥0.There exists a special vector whose magnitude is zero. This vector translatesevery pointPonto itself, i.e. the position of the point is left unchanged and themagnitude of the vector is zero. We call this vector thezero vectorornull vectorand we denote it by0or0when there is no notational ambiguity (i.e. when it cannot be misinterpreted as a scalar). No direction is associated with the zero vector.Thus,0=0(2.1)It follows that|v|>0for everyv=0(2.2)A vectorvwhose magnitude is equal to one (i.e.|v|=1) is called aunit vector.

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