Reminder: Affine Transformations. Viewing and Projection. Shear Transformations. Transformation Matrices in OpenGL. Specification via Ratios

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1 CSCI 420 Comptr Graphics Lctr 5 Viwing and Projction Jrnj Barbic Univrsity o Sothrn Caliornia Shar Transormation Camra Positioning Simpl Paralll Projctions Simpl Prspctiv Projctions [Angl, Ch. 5] Rmindr: Ain Transormations Givn a point [x y z], orm homognos coordinats [x y z 1]. Th transormd point is [x y z ]. 1 2 Transormation Matrics in OpnGL Transormation matrics in OpnGL ar vctors o 16 vals (colmn-major matrics) In glloadmatrix(glloat *m); Shar Transormations x-shar scals x proportional to y Lavs y and z vals ixd m = {m 1, m 2,..., m 16 } rprsnts Som books transpos all matrics! 3 4 Spciication via Shar Angl Spciication via Ratios cot(θ) = (x -x) / y x = x + y cot(θ) y = y z = z y (x,y) (x,y ) x -x θ y θ = shar angl For xampl, shar in both x and z dirction Lav y ixd Slop α or x-shar, γ or z-shar Solv Yilds θ x 5 6 1

2 Composing Transormations Lt p = A q, and q = B s. Thn p = (A B) s. s q p B A Composing Transormations Fact: Evry ain transormation is a composition o rotations, scalings, and translations So, how do w compos ths to orm an x-shar? Exrcis! AB matrix mltiplication 7 8 Otlin Shar Transormation Camra Positioning Simpl Paralll Projctions Simpl Prspctiv Projctions Transorm Camra = Transorm Scn Camra position is idntiid with a ram Eithr mov and rotat th objcts Or mov and rotat th camra Initially, camra at origin, pointing in ngativ z-dirction 9 10 Th Look-At Fnction Convnint way to position camra gllookat(x, y, z, x, y, z, x, y, z); = y point = ocs point = p vctor OpnGL cod void display() { glclar (GL_COLOR_BUFFER_BIT GL_DEPTH_BUFFER_BIT); glmatrixmod (GL_MODELVIEW); glloadidntity(); gllookat ( x, y, z, x, y, z, x, y, z ); gltranslat(x, y, z);... rndrbnny(); 11 gltswapbrs(); } 12 2

3 Implmnting th Look-At Fnction Plan: 1. Transorm world ram to camra ram Compos a rotation R with translation T W = T R 2. Invrt W to obtain viwing transormation V V = W -1 = (T R) -1 = R -1 T -1 Driv R, thn T, thn R -1 T -1 World Fram to Camra Fram I Camra points in ngativ z dirction n = ( ) / is nit normal to Thror, R maps [0 0-1] T to [n x n y n z ] T n World Fram to Camra Fram II R maps [0,1,0] T to projction o onto This projction v qals: α = ( n) / n = n v 0 = α n v = v 0 / v 0 World Fram to Camra Fram III St w to b orthogonal to n and v w = n x v (w, v, -n) is right-handd V 0 α n w v n Smmary o Rotation gllookat( x, y, z, x, y, z, x, y, z ); n = ( ) / v = ( ( n) n) / ( n) n w = n x v Rotation mst map: (1,0,0) to w (0,1,0) to v (0,0,-1) to n World Fram to Camra Fram IV Translation o origin to = [ x y z 1] T

4 Camra Fram to Rndring Fram V = W -1 = (T R) -1 = R -1 T -1 R is rotation, so R -1 = R T Ptting it Togthr Calclat V = R -1 T -1 T is translation, so T -1 ngats displacmnt This is dirnt rom book [Angl, Ch ] Thr,, v, n ar right-handd (hr:, v, -n) Othr Viwing Fnctions Roll (abot z), pitch (abot x), yaw (abot y) Otlin Shar Transormation Camra Positioning Simpl Paralll Projctions Simpl Prspctiv Projctions Assignmnt 2 poss a rlatd problm Projction Matrics Rcall gomtric piplin Projction taks 3D to 2D Projctions ar not invrtibl Projctions ar dscribd by a 4x4 matrix Homognos coordinats crcial Paralll and prspctiv projctions Paralll Projction Projct 3D objct to 2D via paralll lins Th lins ar not ncssarily orthogonal to projction plan 23 sorc: Wikipdia 24 4

5 Paralll Projction Problm: objcts ar away do not appar smallr Can lad to impossibl objcts : Orthographic Projction A spcial kind o paralll projction: projctors prpndiclar to projction plan Simpl, bt not ralistic Usd in blprints (mltiviw projctions) Pnros stairs sorc: Wikipdia Orthographic Projction Matrix Projct onto z = 0 x p = x, y p = y, z p = 0 In homognos coordinats Prspctiv Prspctiv charactrizd by orshortning Mor distant objcts appar smallr Paralll lins appar to convrg Rdimntary prspctiv in cav drawings: Lascax, Franc sorc: Wikipdia Discovry o Prspctiv Fondation in gomtry (Eclid) Mral rom Pompii, Italy Middl Ags Art in th srvic o rligion Prspctiv abandond or orgottn Ottonian manscript, ca

6 Rnaissanc Rdiscovry, systmatic stdy o prspctiv Filippo Brnllschi Flornc, 1415 Projction (Viwing) in OpnGL Rmmbr: camra is pointing in th ngativ z dirction Orthographic Viwing in OpnGL glortho(xmin, xmax, ymin, ymax, nar, ar) Prspctiv Viwing in OpnGL Two intracs: glfrstm and glprspctiv glfrstm(xmin, xmax, ymin, ymax, nar, ar); z min = nar, z max = ar z min = nar, z max = ar Fild o Viw Intrac glprspctiv(ovy, aspctratio, nar, ar); nar and ar as bor aspctratio = w / h Fovy spciis ild o viw as hight (y) angl OpnGL cod void rshap(int x, int y) { glviwport(0, 0, x, y); glmatrixmod(gl_projection); glloadidntity(); glprspctiv(60.0, 1.0 * x / y, 0.01, 10.0); glmatrixmod(gl_modelview); }

7 Prspctiv Viwing Mathmatically Exploiting th 4 th Dimnsion Prspctiv projction is not ain: d has no soltion or M d = ocal lngth y/z = y p /d so y p = y/(z/d) = y d / z Not that y p is non-linar in th dpth z! Ida: xploit homognos coordinats or arbitrary w Prspctiv Projction Matrix Us mltipl o point Projction Algorithm Inpt: 3D point (x,y,z) to projct Solv with 1. Form [x y z 1] T 2. Mltiply M with [x y z 1] T ; obtaining [X Y Z W] T 3. Prorm prspctiv division: X / W, Y / W, Z / W Otpt: (X / W, Y / W, Z / W) (last coordinat will b d) Prspctiv Division Normaliz [x y z w] T to [(x/w) (y/w) (z/w) 1] T Prorm prspctiv division atr projction Projction in OpnGL is mor complx (inclds clipping) 41 7