TRANSFORMATIONS OF SURFACE NORMAL VECTORS with applications to three dimensional computer graphics

Transcription

1 TRANSFORMATIONS OF SURFACE NORMAL VECTORS wih applicaions o hree dimensional compuer graphics Ken Turkowski Advanced Technology Group Graphics Sofware Apple Compuer, Inc. Absrac: Given an affine 4x4 modeling ransformaion marix, we derive he marix ha represens he ransformaion of a surface s normal vecors. This is similar o he modeling marix only when any scaling is isoropic. We furher derive resuls for ransformaions of ligh direcion vecors and shading compuaions in clipping space. 6 July 990 Apple Technical Repor No.

2 Turkowski Transformaions of Surface Normal Vecors 6 July 990 Transformaions of Surface Normal Vecors Ken Turkowski 6 July 990 Why is a Normal Vecor no jus a Difference Beween Two Poins? In(fig. ), below, on he lef, we illusrae a recangle, and is normals N i ha have been modeled as sraigh line segmens. On he righ is an affine ransformaion of he recangle and is normals, where he endpoins of he sraigh line segmens represening he normals have been ransformed in he same way as oher poins ha make up he recangle. Noe ha hese so-called normal vecors are no longer perpendicular o he surface. (fig. ) Is his he ype of behavior ha we expec from normal vecors? I hink no. This leads us o believe ha normals behave differenly under ransformaions han he surfaces ha hey correspond o. The ransformaions are relaed, hough, by he requiremen ha he surfaces and heir normals remain orhogonal. We use his propery o derive he normal ransformaion marix from he he poin ransformaion marix. Transformaion of Normal Vecors under Affine Modeling Transformaions Given a planar surface, a angen vecor can be expressed as he difference beweeo poins on he surface: = p p 0 (eq. ) The normal o he surface can be calculaed from: n = for and wo non-colinear angen vecors and. (eq. ) By consrucion, he normal is orhogonal o he angen vecors, so ha: Apple Compuer, Inc. Page

3 Turkowski Transformaions of Surface Normal Vecors 6 July 990 n = n = 0 (eq. 3) When a surface is ransformed by he modeling marix M: p = pm (eq. 4) where m 0 m 0 0 m 0 m m 0 M = m 0 m 30 m 3 m 3 (eq. 5) and p are a se of poins ha define he surface, hen he angen vecors are ransformed by he marix M : = M (eq. 6) where M = m 0 m 0 m 0 m m m (eq. 7) is he submarix of M ha excludes ranslaions, as can be verified by applying (eq. ), (eq. 5) and (eq. 4). In order o have he relaion beween angen and normal vecors hold, we apply (eq. 3): n = M M n = ( M ) ( nm ) = n = 0 (eq. 8) or ha n = nn = nm (eq. 9) which implies ha he normal vecor is ransformed by he ranspose of he inverse of he angen s ransformaion marix. This is a well-known heorem of ensor algebra, where is called a conravarian ensor of rank, and n is a covarian ensor of rank. Simplifying he Compuaions Unless he marix M is uniary (i.e. lengh and angle preserving), he ransformaion of a uni normal is generally no iself a uni normal. In his ligh, we do no need o compue he full inverse, when he adjoiill do, since a pos-normalizaion sep is necessary anyway. In erms of he angen ransforma- Apple Compuer, Inc. Page

4 Turkowski Transformaions of Surface Normal Vecors 6 July 990 ion marix M or he modeling marix M, he adjoin is: adj ( M ) = m m m m 0 m m 0 m m 0 m 0 m m m 0 m 0 m 0 m 0 m m 0 m m m 0 m m m 0 m 0 (eq. 0) so a ransformaion N ha preserves he relaionship beween he normal and he surface angen under he modeling ransformaion M is: m m m m m 0 m 0 m m N = m 0 m m 0 m 0 m 0 m m m 0 m 0 m m 0 m 0 m m m 0 m 0 (eq. ) Non-Planar Surfaces We can generalize hese resuls o non-planar surfaces, as long as he geomery of he surface is described by a se of poins. In paricular, he definiion of he surface normal can be replaced by n = s, (eq. ) he gradien of a surface s a a given poin. This makes (eq. 9) and (eq. ) useful for Bézier surface paches. Uniary Transformaions In he case where M is a uniary marix (i.e. he modeling marix M is composed of only roaions and ranslaions), is inverse is simply he ranspose; since he ranspose of he ranspose of a marix is he original marix iself, normals are ransformed by he same marix as he angens, and hence: N = M (eq. 3) Noe ha his does no hold rue when he marix includes skewing or oher anisoropic scaling. Inverse Transformaions of he Lighing Vecor When compuing he shading in a simple hree dimensional graphics sysem, i is someimes advanageous o ransform a direcional ligh vecor from he world space ino he modeling space in order o perform he do produc wih he surface normals of an objec in is canonical modeling space. Iorld space, he shading compuaion akes he form: d = = (eq. 4) Apple Compuer, Inc. Page 3

5 Turkowski Transformaions of Surface Normal Vecors 6 July 990 Applying he normal ransformaion rule (eq. 9), we ge he equivalen relaion in modeling space: d = ( n m M ) = n m ( M ) = n m ( M ) (eq. 5) where he subscrip w indicaes world space, and he subscrip m indicaed modeling space. Then he ligh vecor iorld space is ransformed by he inverse 3x3 modeling submarix o bring i ino modeling space. However, his analysis assumes ha he modeling and lighing vecors are already normalized appropriaely for world space. In he general case, we have he equaion: d = i w ( ) n w = n M m = n i M m w (eq. 6) where he norm in he denominaor is given by: N = = n m M ( ) = n m M M n m ( ) = n m G n m ( ) ( ) (eq. 7) The marix G = M M (eq. 8) is called he firs fundamenal marix [Faux 80], and represens he fac ha he modeling space is in general non-euclidean, i.e. he disance beween poins as measured in he modeling space is no simply he square roo of he sum of he squares of he componenwise differences beween he poins. Such a space is called a Riemannian space, and comes abou because he modeling submarix is no in general isoropic (i.e. scaling is no equal in all dimensions). In oher words, if he meric iorld space is o be regarded as Euclidean (as i should for shading calculaions), hen he meric in modeling space will no, in general, be Euclidean. If he ligh vecor is normalized iorld space, hen he shading equaion hen becomes: M ( ) d = n m M ( n m Gn m ) (eq. 9) These calculaions in Riemannian space are more complicaed han he corresponding calculaions in Euclidean space. Since he ransformaion from modeling o world space (eq. 9) occurs a each normal, Apple Compuer, Inc. Page 4

6 Turkowski Transformaions of Surface Normal Vecors 6 July 990 whereas he Riemannian normalizaion (eq. 9) occurs a each pixel, i is compuaionally less expensive o do he shading iorld space wih he Euclidean norm: d = ( ) (eq. 0) Non-affine Transformaions The resuls of (eq. 9) are rue regardless of he form of he modeling marix M. This allows us o apply i o non-affine (projecive) ransformaions as well, i.e. modeling marices ihich he righmos column has non-zero enries. Of course, (eq. 0) and (eq. ) are no quie as simple, and he resulan normals are 4-vecors. Unforunaely, hough, orhogonaliy is no invarian under projecion. Shading Compuaions in Clipping Space We can compue diffuse shading in homogeneous coordinaes by mainaining he do produc: d = nn nn i = nn nn i = nn nn ( VC ) VC ( ) [ i = n ( NVC )] nn normalized surface normal i( VC) ligh vecor in clipping space (eq. ) where he vecors n and i have heir fourh componen se o zero since hey have only a direcion, no a posiion. The marices V and C ransform from modeling o viewing, and viewing o clipping spaces, respecively. The normalizaion of he inerpolaed vecor is a problem, hough, since i is compued in world space, no clipping space, so his is no a pracical echnique for smooh shading by inerpolaing normal vecors in 4-space. Back-Face Culling I can significanly enhance he hroughpu of a renderer by eliminaing back-facing polygons from consideraion during hidden surface removal and scan conversion. Back-facing means ha he polygon is no visible from he eye under he desired perspecive viewing ransformaion. Le us look a he siuaion in several spaces: modeling, world, viewing, and clipping. Back-face Culling in Modeling Space A poin on he plane of he polygon can be characerized by he equaion: where p m n m = 0 (eq. ) Apple Compuer, Inc. Page 5

7 Turkowski Transformaions of Surface Normal Vecors 6 July 990 p m = [ x y z ], n m = [ n x n y n z d ] (eq. 3) are given in modeling space. The polygon is visible if he eye is on he proper side of he plane: ( e w M )n m < 0 (eq. 4) where e w is given iorld space. Of course, i isn necessary o inver he enire 4x4 M marix, since i can be compued from he inverse of a 3x3 insead: M = M m 0 M 0 (eq. 5) where he M marix is pariioned as follows: M = M 0 m 0 Back-face Culling in World Space To perform he visibiliy es iorld space, we invoke associaiviy of marix muliplicaion on (eq. 4): e w M ( n m ) = e w n m M ( ) < 0 (eq. 7) Back-face Culling in Viewing Space In viewing space, he eye is a he origin and we ge: or e v = [ ] e v n m ( MV) 0 n m ( MV) 0 0 < 0 < 0 (eq. 8) (eq. 9) (eq. 30) We define a pariion for he viewing marix similar o ha of he modeling marix: Apple Compuer, Inc. Page 6

8 Turkowski Transformaions of Surface Normal Vecors 6 July 990 V = V 0 v 0 (eq. 3) The concaenaioih he modeling marix M is hen: MV = M 0 m 0 V 0 = v 0 M V 0 m 0 V + v 0 (eq. 3) Back-face Culling in Clipping Space The concaenaioih he clipping marix produces: e c n m ( MVC) < 0 (eq. 33) Conclusions We have derived he normal ransformaion (eq. 9) corresponding o an affine poin ransformaion of a surface. A simple way of calculaing his marix is given in(eq. ). The compuaions in a 3D graphics sysem wih infinie ligh sources can be simplified wih (eq. 5). Back-face culling is done by aking he do produc of he plane equaioih he eye poin. I seems as if he compuaions are simples in modeling space. Apple Compuer, Inc. Page 7

9 Turkowski Transformaions of Surface Normal Vecors 6 July 990 References Faux 80 Faux, I.D., and Pra, M.J. Compuaional Geomery for Design and Manufacure. Chincheser, Wes Sussex, England: Ellis Horwood Limied, 980. Apple Compuer, Inc. Page 8