It presents in a unified manner an introduction to the mathematical theory underlying computer graphic applications. It covers topics of keen interest to students. Mathematical elements for computer graphics David F. Rogers, James Alan Adams INTRODUCTION TO COMPUTER GRAPHIC TECHNOIiOGY l. 1. Mathematical Elements for Computer Graphics. Front Cover. David F. Rogers, James Alan Adams Rogers, United States Naval Academy, Annapolis, MD.
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Mathematical Elements for Computer Graphics By David Rogers - Download Format - PDF | Direct Download | Size - MB. Language. Mathematical elements for computer graphics by David F. Rogers, , McGraw -Hill edition, in English - 2nd ed. Mathematical elements for computer graphics / David F. Rogers, J. Alan Adams. Author. Rogers, David F., Other Authors. Adams, J. Alan (James Alan).
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Sign In. Admns, Janes Alan , jowt autl-or. Tt tle ; R6 A"' Al. Sealing Algorittm. Mt bears directly oo the develO Jtent and iltplerrentatioo of truly productive applications. By the. AlJrost everyone can expect to be affected by this rapidly expanding techoology. For this reason the reader will find rrore extensive discussions of rotatioo, translatioo, perspective , and curve and surface descriptloo than of clippi. A discussion of existing techniques for reptesent.
United' s APT , t. JJtplish a sncoth rotation in soft'Ihus this is. I t cncx: Representing pictures to be presented. Interacting wi. TMY be as s. The specif1. Mtes to specify a nurrber sud'l as 60 , , i. In harogcncous roord. Although in many grilliUcs applications t. In ge. CUpping i. In the scissoring t. Winq space than is required. Only those lines or l. In two dim:: An additional requ. Ue1ent for II'OSt pictures is the presentation of alpha-.
If dlaracters am qenc. In fnct this is rocessa. Perhaps the best kncMn interactive device l. I-OSitioned over a l. Function switches, sJu. The analog tablet is the ItDSt versatile and accurate devloe for cx:: JX inting function , the indication occurs in the data base and rot in the displ.
Except in unusual cnvl. A storage tube display is shown in Fig. Because of this the display of dynamic ITOtions is not possible. Xllcs display is of the call igraFhic or li. Ular to the effect "ttich results fran runni.. A ref, A raster scan CRT grafh,ics display uses a starxlard television monitor for In the raster scan display the picture is cnrposed of a. F1gure sho-"9 the qenernl. Figum Conceptl:.
UTbcr of rrethods of cl assifying couputcr graphics devims. I f points are plotted close enough t:. Still another rret: In essence this rrethocl requires detex: Appendices A and C contain the architecture o f a software sche: Graphics , r-ldinm..
Or Industry: All of these transformations can be ac: If a ar. This effect is c11 led shear and is stnm in F1g. Bofore m pletirq our discussion or the transfoll'Mtion of points, consider the effect lD.
A straight line can be tlcfined by bov rosilion v To see this, oonsider a line beb. Aoon E and f'. To shari that these l.. We no" rrult. However, the total effect of a 2 x 2 matrix transformation l. S easier to see, considering the effects of rotation, r eflection, and scaling separately. In order to illustrate these. If the r.. It is n:: Sinre mtri. First ootice fran Eq. The effect of the t.. It is a l so possibl e to easily det: It can be shown that the area of any t.
Figure stXJWS that p: X int nntrix nust equal the llUiber of. Using this matrix in SJ. Alternately W"e may consider the third elerre. For notatiooal pw: Previously the transforrratioo was perfo01ed suc:: M lies is in th;t: Jh the origin. Csing si. With three ca1pooents it follo. In harogenoous a: Onal spaoe.
For ex. In a gearet. Ml in Fig. Ming representation of the sane set of equations can be used to rreet this rcquirenent; X y 1 r 1 2 0 e [0 0 1] 1 -3 0. Conslrlc- tloiO parallel lines defined by. In this rranner horrogeneous ooordi.. Jera tic.
Based on our previous experience we imrali. The other tenn9 are determined by considering In a rotation about the. The fact that three-d. Figure 'Ihl:: An algoritbn which will produce a. The alqont. J-5, whid1 sho,. It 1s seen th The idea of a vcct.. Jgh perspective views are often used by artists and architocts because. Lcns can be obtaint:: In perspective gearctry no bolO lines arc parallel. Xretric projection transfotmations onto the appropriate zero plane always contain a ooll.
Such project. In order to develop the oonditicns for diJretric and isaretric projections,. Qle rrotln: Consider the unit. Using the values for a dilretric projectioo, the t: A clinet.. Perhaps rrore cam at is the isoretric projectioo. S 0. This result is well knc: USing the angular values for an isoretrlc projection the transform: Cblique projections such as cavalier and cabi. Jonality of the coordinate system.
JTple orthograiiUc projection onto a zero plane perpendl. An alg: To include the 2-coordinate infoOM. This is what causes the "unnatural view in Fig. Vectors o! D- 'rransforrred 1na. As we have seen, i f the viewing point is on a line normli to the center of.
A cooplete set of transfo: Notice the symret. An algorithm which will genexate a general perspective vie.. Equatloo represents four equations: Asslll'C that the rreasured position of a point in one perspective projectioo is 0.
Note that i f no solutioo result. An al gorithm which utilizes this technique is given in Apperrlix C. As a third way of CXX'lsidenng Eq. Others, such as hicklen-l.. Arother useful display technique is the use of stereo pairs to create the illusi. Jn a stereoytafhic projectioo, a separate perspective view JTUSt be created for each eye. For a hunan with average eyesight, the strongest stereo effect occurs at a d.
One final pl;'E! To create a stereo pah, the rr.
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An algor1tbn to generate a stereo pair for an ob C'Ct wtu. Ahuja, o. Gecrtetry for COnst. Et-1 Syst. Forrest, A. Ons, and ViSUil. I EEE, vol. OCJ and designing curves. Either a p.. A oonpararretric expressJ. Jever, a p:: Jlynanial of.
For t. A general secord-deqree, urpll. An even s.
It is necessary to change the coordinate system or approx. FUrther, when points on an axisdependent nonparanetric curve are calculated with equal i.
For a curve of one pararreter, the p.: If the parameter is t and the curve is t: Since a point on a para: Jy a single value of the. Due to its axis-independent nature, a parametric curve is easily transforned into a curve of the sarre shape but wi.
A ooopa. The gtadrant of the circle shown in Fig. Equal increments of x were used to obtain the FOints on the arc. It also produces fOQr results since the arc lengths are unequal.
Figure 4-lb is obtained fran the hr1sic parametric form given by 4-U P t [oose sine] 'lhis produces good output since the arc l engths are equal, but the calculatial is inefficient because trigJa'I'Ctric functioos require several operations to catpJted points on a curve and thus give a better. COruc sections fonn useful planar curves for rrany applicatialS.
Jw the intersection of a plane and right circular oone defines the various conic sections, i. The general definitial of a cadc section can be given as:. Since we can always rotate the curve such that.
A parabola occurs if a ': If g - 0, the hyperbola degenerates into a pajr of intersecting straight lines. NoW that in the. Jtion was not tnifonn. The shape of ships and airplanes were adequately defined in this manner. COOsider the ooterminaticn of a circle through three paints given by [1 1 , [2 2] , and [3 2] respectively see also Sec.
O Thus, the center of the cucular arc which r: For this case, we require. Sired charact. Dlsider again the problem of pa. If the points are far renoved fJ: To avoid this, translate the origin or the original coordinate system to the first point and define a new ooordi.
A pararretric rcpresentaticn of conic sections is of interest bx: Vlded a tp: S obvicus equal-angle increments.
Mathematical Elements for Computer Graphics by J. Alan Adams and David F. Rogers (1989, Paperback)
N points. An alternate rrothod might be to use equal perirretcr lengths. To ove. IIred to be inclined at an angle i to the horizcntal , as sJ1o.. In th: Consider an origin-centered parabola opening to the right,.
Thus, the arrount of the parabola to be displayed must be limited by ch: If this is done then CD. In the algorithm a fixerl ntri: A rrore general method defining a C"..! A"l alternate para11etric representation of a hyperbola which yields the polyqt: I f we coosidcr the branch of the hyperoola in the first and fourth quadrants and wish to plot the portion of the hyperbola for a.
To draw the arc of the circle, we first translate diroctly to the center of the circl e, i. Hith '1hc locaLion of tl1c OUltcr of the circle and.
AT I 00 The snoothness or fcumcss of a curve is generally a quahtative judgrrent. Assure that such a curve is to be. It requires the solutioo to elliptic integrals whose wtegratioo cx: The required nunber of circular arcs depends upon the separation distanoe between the dat.. Detai 1s of this technique for generating snooth curves by circular arc interpolat-ion are given in the cited reference. The output. G, , Li.
Gottfried, B. Berlin , OCt. Y,, J'ldll. Snith, L. Snoothness," Central lnstl. Alternately a space curve rray be expressed in a nonp. Notice that tho intc. Sfios tho rrent applies when sol vi.
After each interval in the pararreter t , the variables x and y return to their initial values, but z increa. If the physical spline is considered to be a thin elastic bcilrn, then Eulers equati01 cf Ref. La, and R x is the radius of cw: In general the mathenatical spline.
Piecewise splines of lo.. COrresponding tangent vectors at these given points 1 are indicated by the derivatives with respect to the parazreter t. WithU1 the cubic segncnt the pararreter t varies betw-een t: Substituting F. Fran Eq. In the general case Eqs. Then Elq. One approach is to set the rrax. Jm parazret.
IL"W ,. Parametnc cub1. Other CJi cxn: Recallmq Eqs. Equation s! Cubic splines will have CXJntinuous first and secc. S-3 and S Ccrrparisons of the effects of the varicus erd conditions is shown Fig. I"Ve has relaxed boundaJ: Notice that the di. We wish to detez: It should be noted that it l. S not necessary to mvcrt the rratrix to obtain the internal derivatives. An algont: L"te fl. For e: S CXXltrolle3 b'J varytng the tangent vector ragnitudes at the data points!
Arother rethod for ir"provinq curve sn: Details are descnbcd m the c. It hAs been ased. Each parabola qces through three points and each. INhere as shown in F. A parabola can be oorpletely specified by two 3 5 end p;: OOrdinate systern, whereas the blendin: J parabolas P r and 3. For t: According to Ove! Parabolic blc OO and a re11 s. When an artist, stylist, or designer sketches, he or she uses short, over.
In sare applicaticns it rray be desirable to sketch a shape by using parabolic blend. Assure that the positioo vectol: Equation rray now be used to b l em the bNo parabolas to yield the desired curve between P4 and P 5: An alternate method o f curve description has been described by Bezier.
A Bczior curve is associated with the "vertices" of a polygon which uniquely define the curve shape.
ICh greater intuitive feel. This qreaUy increases flexibility and owr: The curve points are then give. Equatioos and along with Eq. Another characteristic of the interpolatioo fln:: Bezicr curves, as sha. For this c;. Assune equal increrents: Although tt 1s not necessary to calSider curve derivatives in generating.
Continuity carlitions beb. FOr the particular case of two adjacent 1 n n-1 cubic Bezicr curves cf Fig. Note that continuity of slope c: A polygoo with six vertices will always produce a fift: Sinoe any point on a Beder curve is a resul. Practically, this elim.
Fer e. The theory for a-splines was first suggested by Sc: An additional variable rrust be uc: Recall that the pa. Fbr cx. S integer values for the. M, if the 0. The 8-spl. The function P t is a p: Due to the flexibility of B-spl. Figure sho. Jeen the four vertices, the fourth-order curve corresponds to the Be'.
Id-order curve produces a looser curve bet Figure 5-U shows the effect of rrultiple vertices in the defini. In Fig. The first curve is of order seven , equal to the m.: The second curve is of order five. Notice that a "knuckle" occurs at the double vertex since the slope and curvature are discontinoous. A duplicate vertex is required to create a knu:: A triple vertex creates a kmx: This ability is a mmLll"1 requirement in stup desl. Figure shcJo Jwn in Fig. S found in tho Bezier.
For t 0. Algoritl'lm9 which will generate the required B-spl.
Engineering Office, DeoerTber Aided Des. Adams, J. Alan , "A Ccnparison of l-: Forru;t, A. Cox, M. Surfaces and surface representatim play a cntical role in rost design and manufacturing prcx: JiCI'ically controlled ooch.
Here we arc not oonoe. U controlled rmchine tool tapes can be obtained. Various techniques of anal. SUt'face normals can also be u91 to indicate the shape and orientation of a surface patch.
In thls sectiCI'l we restrict ourselves to a four-sided plane surface. The derivatives of the point functions x and y as defined by Fqs. Similar calculations can be made for an expressiat for a planar surfc'! First, we ass We will assme that a surface may be represented in a. In order to efficiently represent a surface, sane notaticnal dctcils are.
A poinL on the surface rre. Note that the latter representation does not necessarily define a single fixed point. We will consider that a surface patch can be built up ftorn knoNn data:. In particular , tn. Ne wish to find the equation for the lofted surface bebo. Examination of F. It can be readily stn. A rrore general COOns surface is discussed in Sec. Ney blend the boundacy curves to produce the internal shape of the surface.
Ne l"'f: M turn our at tention to sane practical aspects. One of the nore useful patch descriptions uses para-. Before prooeeding, sare notatiooal details are islportant. For oc: In particular that.
Alternately Fq. Examination of Fi: For a given surface patch all thP elennnts in the P-. Sinoe for the surfao: For eY. P u,O: P-rratrix, i. Figure 6-lOa. Also note that increnenting the y-cx: J'lg the y-cx: Jres used.
This on occasion may lead to. A mnber of sc: In the Bezier fornulation , only the oorner. The bol. It is saret. In particular the surface. Ven by. Serre of the difficulties associated with Eq.
Fbr a a-spline surface the corresponding Cartesian prodlX: Jve the sarre degree of mult iplicity in a given direction.
Further di. Sts a uore qeneral repre-. The r subscr ipt indicates that the general blen: The bicubic surfaces discussed in Sec. Forrest Ref. If both the condi. Yhen m n.
Figure il l ustrates the difference bet: Jcen the. S,w are also consideruc:. To actually inplerrent these techniques in a production or desi gn environ-. Paris, France, ; Bezier, P. Peau-s, G. COons, S. Project tW: Thesis, Carrbridqe University, NoveTber Bezier, P. Desiqn," Academic Press , New ork, A concoptual view of a c: It consists of routines to translate, rotate, clip etc. USt interl! All graphics devirns, of course, reoeivc info: Here cursor is used in a general sense to rrean the e.
J;bsolute and relative cx:: CUxved line node is used to activate hard. Jt is not considered. OO interactive device control func: Tho above algorit: WARE graphic primitive every t. If yes - CQ'ltinue, i f no - tum cursor off - c: E'F 0. If yes - CXlntinue, if no -.
COII'and serves to erase the screen for storage tube CRl' grapuc devices , to indicate a new plot for pen and ink devices, am to rotify a refresh grapuc devioe that the picture is curplete and a new frmre nust be started. Fran the user ' s point of view the control of interactive devices rrust be. Jrately 0. Thus, a 34 is the element in the thixd reM and fourth oolum. A natrix of m l"CMS and n oohrm. An exanple of a 3 x 3 identity rutri.
Jc is: Consider one matrix of order n1 x m1 and a semnd rmtrix of order For n'lll. In algebra, where single variabl es are ccnsidered , if ax:. Y 1 then x "' a- 1J',. The algont: J'lates is g!. It is based on sec. MAT T: T 2,2: ALir; AL.
EXT I. LET TZ: T 3,3: N T2 Lt. LET T2: COS T2 1. UT f"Ok I: Reflections about other planes can be cbtained by a cacb.
UT FOR 1: N HAT V: JIS w'. PFGJ lif. LET T 1,1: C PPJJex: Timl 'P: ES ' 't: Ef' Vf X, Y, Z-:? CTlOI 'C: TORS Lt. E FOR I: S 4,1J ' Y2-S 4 ,J2. Oti- 2 'X ,: A I,J. D 3,1 LET T 2, 1: D 6,1 LET T 3,1: An algorithm. Alll PROGf! AtJ Dlfo! N LET X2: X5 N MAT X: ZER t-. OF CE. IPSE Al.. CTH Gf Sffil - t-: LET C1: SIM 11 U. COS P U. P Ll1 C3: LET X1: CR Til l t.
TilE lt-. X AND Y. NFJoi At. NT Lf. U ATE ' lY:: LET T3: T31 SE 1 CCR! T3- CAl. E Ptilt. ZER 2 ,2 JI. AT F': ZER 2. DS TAf. SliO LlT A: X,Y , R,Tl. OF' cEt. GfiEES 'f: C lliE. EtlT ' ll. Jwn data r;oints with various end bounda. IC ' C2: MltiZEliO El. LET L J: FOR K: LET F 3 ,K: If' Cl: LET S K, 1: LET M 1 , t, ATF t.
Z COf. X CF' EO. EXT I t".
DlM N 25 ,? LET G: C ' X-COY. THE FO! NO rR It: LfT U 1, 1: TS 'Y ,: AflfiAY Q t. Z ' 2-cot'POUEt. T FOP U: O THEN Direction oosine , 55 Directnx, 95 Discriminate, 96 Display buffer, 12 Display oontro11er, Cover photos: Photos 3, 4, 5 show vanous v1ews of the shear stress as a tunct1on of the pressure gradi ent, parameterized with surface temperature.
Photos 1 and 2 show the surface heat transfer rates as a function of the pressure gradient, parametenzed with surface temperature Ref. Rogers, J. Alan Adams Uploaded by Souradeep Bhattacharja. Best book for learning the basics of computer graphics. Flag for inappropriate content. For Later. Related titles. Advanced Machining Processes by Prof. Vijay Kumar Jain. Mathematical elements for Computer Graphics - David F. Mortenson, Michael E.J' O o r where 9 is a scaler.
A plastic right. Let the o. Wns can be rore useful. Public Private login e. However, at the present tiJre we are not interested in. Prem Chander. Timl 'P: Sealing Algorittm. As we have seen, i f the viewing point is on a line normli to the center of.
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