hearn,baker - computer graphics - c version 2nd ed

Three dimensional cartesan reference frames figure a

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' . Three-Dimensional Cartes~an Reference Frames -- Figure A-6(a) shows the conventional orientation for the coordinate axes in a Figure A-.5 three-dimensional Cartesian reference system. This is called a right-handed sys- An angle esubtended by a tem because the right-hand thumb points in the positive z direction when we circular arc of lengths and imagine grasping the z axis with the fingers curling from the positive x axis to the radius r. positive y axis (through 90•‹), as illustrated in Fig. A-6(b). Most computer graph- ics packages require object descriptions and manipulations to be specified in right-handed Cartesian coordinates. For discussions t h u g h o u t this book (in- cluding the appendix), we assume that all Cartesian reference frames are right- handed. Another possible arrangement of Cartesian axes is the left-handed system shown in Fig. A-7. For t h s system, the left-hand thumb points in the positive z direction when we imagne grasping the z axis so that the fingers of the left hand curl from the positive x axis to the positive y axis through 90". This orientation of axes is sometimes conven~ent for describing depth of obpcts relative to a display screen. If screen locations are described in the xy plane of a left-handed system with the coordinate origin in the lower-left screen corner, positive z values indi- cate positions behind the screen, as in Fig. A-7(ah Larger values along the posi- tive z axis are then interpreted as being farther from the viewer. Three-Dimensional Curbilinear Coordinate Systems Any non-Cartesian reference frame is referred to as a curvilinear coordinate sys-p tern. The choice of coordinate system for a particular graphics application de- pends on a number of factors, such as symmetry, ease of computation, and visu- - - - . . . . F i p m A-6 Coordinate representation of a point P at position (x, y, z ) in a right-handed Cartesian reference system.
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Figure A-7 Left-handed Cartesian coordinate system superimposed on the surface of a video monitor. figure A -8 * . ,' x, axis A general cu~linear coordinate '*ye- reference frame. alization advantages. Figure A-8 shows a general curvilinear coordinate reference frame formed with three coordinate surfaces, where each surface has one coordl- nate held constant. For instance, the x,x2 surface is defined with x, held constant. Coordinate axes in any reference frame are the intersection curves of the coordl- nate surfaces. If the coordinate surfaces intersect at right angles, we have an or- thogonaI curvilinear coordinate system. Nonorthogonal reference frames are usefd for specialized spaces, such as visualizations of motions governed by the laws of general relativity, but in general, they are used less frequently in graphics applications than orthogonal systems.
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