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

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1 CSCI 480 Comptr Graphics Lctr 5 Viwing and Projction Shar Transormation Camra Positioning Simpl Paralll Projctions Simpl Prspctiv Projctions [Angl, Ch. 5] Janary 30, 2013 Jrnj Barbic Univrsity o Sothrn Caliornia 1 Rmindr: Ain Transormations Givn a point [x y z], orm homognos coordinats [x y z 1]. Th transormd point is [x y z ]. 2 Transormation Matrics in OpnGL Transormation matrics in OpnGL ar vctors o 16 vals (colmn-major matrics) In glloadmatrix(glloat *m); m = {m 1, m 2,..., m 16 } rprsnts Shar Transormations x-shar scals x proportional to y Lavs y and z vals ixd 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 also dscribd by 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); }

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