Projections - 3D Viewing. Overview Lecture 4. Projection - 3D viewing. Projections. Projections Parallel Perspective

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1 Ovrviw Lctur 4 Projctions - 3D Viwing Projctions Paralll Prspctiv 3D Viw Volum 3D Viwing Transformation Camra Modl - Assignmnt 2 OFF fils 3D mor compl than 2D On mor dimnsion Displa dvic still 2D Analog to taking a photograph Projctions Paralll Projction Orthographic Top Front Sid Aonomtri Isomtric Obliqu Cabint Cavalir Prspctiv Projction On point Two point Thr point Camra modl Dtrmind b whr ou plac th projction plan rlativ to principal as, and what angl th projctors mak with th projction plan. Paralll projctions ar usd in nginring and architctur drawings, bcaus th can b usd for masurmnts. Prspctiv projction imitats our s or camra and looks mor natural.

2 Projctions Paralll Projction If objct positions ar transformd to th projction plan along paralll lins. Dtrmind b Dirction of Projction Prspctiv Projction If objct positions ar transformd to th projction plan along lins that convrg to cntr of projction or prp. prp projction rfrnc point Paralll Projction Prsrv rlativ proportions of objcts. Orthographic: Dirction of projction is normal to th projction plan. projction Obliqu: Vp plan Th projction plan and th dirction of Orthographic projction ar not Projction prpndicular to ach othr. Vp Obliqu Projction

3 Paralll Orthographic Projction Top (Plan Viw) Lngth and angls can b Front Elvation masurd accuratl from th drawings. Sid Elvation Rar Elvation 3D natur difficult to s. Commonl usd in nginring and architctural drawings. Aonomtric Orthographic Can displa mor than on fac of an objct. Th projction plan is not normal to a principal ais. Uniform forshortning. Mor lik prspctiv. Paralllism of lins ar prsrvd, but not angls. Isomtric, dimtric, trimtric. Isomtric Orthographic Dimtric and Trimtric Orthographic Th projction plan intrscts ach coordinat ais at th sam distanc. Th projction plan maks qual angls (120 ) with ach principal ais. Allowing masurmnts along th as to b mad to th sam scal. Dimtric: Angls btwn two of th principal as ar qual. Nd two scal ratios. Trimtric: Angls diffrnt btwn th thr principal as. Nd thr scal ratios.

4 Dimtric, Trimtric and Isomtric Paralll Obliqu Projction Dirction of projction is not normal to th projction plan. Th projction plan is normal to a principal ais, so th projction of th fac of th objct paralll to this plan allows masurmnt of angls and distancs. Othr facs allow th masurmnt of distancs along principal as, but not angls. Obliqu vs. Orthographic Paralll Obliqu Projction Orthographic Projction Obliqu Projction ( p, p ) (,,) α L Window Window (,) φ Viw Plan

5 Cavalir: Paralll Obliqu Projction Th dirction of projction maks 45 angl with th projction plan. Dpth = width and hight. No forshortning Cabint: Th dirction of projction maks an angl of arctan(2) = 63,4 with th projction plan. Forshortning of a half mor 3D ralistic. Summar of Paralll Projctions Prspctiv Projction Prspctiv projction is catgorid b thir numbr of vanishing points and thrfor b th numbr of as th projction plan cuts. Do not prsrv rlativ proportions of objcts. Vanishing point Prspctiv Projctions ais vanishing point Two point Prspctiv Projction ais vanishing point On point Prspctiv Projction 3 point 2 point 1 point

6 3D Viwing Coordinat Sstm Viw Rfrnc Point Origin. Viw-Plan Normal N Positiv dirction for th viwing v ais and th orintation of th viw plan. Viw-Up Vctor V Th up dirction for th viw. Positiv dirction for v. V not paralll to N. U Vctor Prpndicular to both V and N. w w v v 3D Viwing Transformation w v N Viwing Piplin: World Coord. Viwing Coord. Dvic Coord. Nd to stablish a viw rfrnc coordinat sstm. Camra Modl From Point - F: Th position of th camra. At Point - A: Whr th camra is aimd. Up vctor - U: Dfins th up dirction. Viw angl - v: Fild of viw. Viwing Transformation Transform 3D world coordinats ( w, w, w ) into 3D coordinats (,, ). Transform 3D coordinats (,, ) into 2D normal dvic coordinats ( ndc, ndc ). F nds up in th origin of th coordinat sstm. A nds up on th positiv -ais. UP vctor nds up in th positiv Y-Z plan.

7 Camra Modl World to E Transformation Lin-of-sight vctor Lin of sight F Right-handd world coordinat sstm P = ( Pw F) V F Lin of sight () Lft-handd coordinat sstm A Z-ais of coord. c = A F A F should b mappd to World to E Transformation Vctor prpndicular to U and A-F Vctor prpndicular to c and a World to E Transformation X-ais of coord. a = ( A F ) U ( A F ) U should b mappd to b = Y-ais of coord. (( A F ) U ) ( A F ) (( A F ) U ) ( A F ) should b mappd to 0 1 0

8 World to E Transformation Convrt From E to NDC 3D to 2D Combining all thr conditions a b V c 1 = v tan = 2 S D V is orthogonal, so V t 1 = V s = D s = D Convrt From E to NDC 3D to 2D ndc ndc = Vc = Vc Vwidth + 2 tan v 2 Vhight + 2 tan v 2 Rstriction on F, A, U, v F and A ma not b th sam point not abl to dfin lin-of-sight U cannot b a null vctor nd a uniqu up dirction U cannot b paralll to lin-of-sight nd a uniqu rotational position v must b 0 < v < 180

9 Zoom Enlarg an imag b rducing th angl v. Incrasing th viw angl maks th imag smallr. Viwing angls btwn 40 and 60 giv th most ralistic viw. OFF Fil Format OFF #hadr Nvrtics Nfacs Ndgs X[0] Y[0] Z[0] : : : X[Nv-1] Y[Nv-1] Z[Nv-1] NV V[0] V[1]..V[NV-1] COLORSPEC W will not us COLORSPEC, rad and discard. OFF Fil Format Eampl cub.off OFF