Z-buffering, Interpolation and More W-buffering Doug Rogers NVIDIA Corporation
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1 -buerig, Iterpolatio a More -buerig Doug Roger NVIDIA Corporatio roger@viia.om Itroutio Covertig ooriate rom moel pae to ree pae i a erie o operatio that mut be learly uertoo a implemete or viibility problem may our. Thi paper the proper alulatio ue or proper perpetive projetio. Ater a poit i rotate a tralate ito eye-pae, a perpetive traormatio our that overt eye-pae ooriate ito ree pae ooriate. The ree pae ooriate (x a y are ue to reer triagle a the ooriate i ue to etermie viibility. There are everal way to alulate x a y. Origi x Oe way i to plae the amera at the origi a the projetio plae uit i rot o it. Uig imilar triagle, x a y a be alulate by: x y ; x ; y Eq. To alulate the ree pae ooriate ( rom eye-pae, a perpetive traormatio i applie. Thi reult i a o-liear itributio o eye-pae value. Thi hyperboli harateriti alloate mot o the preiio loe to the oberver. From Jim Bli' Corer p5, the projetio matrix M i alulate by: ω iel o view; ear lippig plae; ar lippig plae i( ω o ω M Note: you a multiply the top let etry i matrix M with the ree apet ratio i eire. ou houl alo ivie by the matrix M by (ee the ote about og at the bottom o thi paper or etail Moroe Street Sata Clara, CA 955 T F
2 3535 Moroe Street Sata Clara, CA 955 T F For a poit a (,,,. i eye-pae, the traorme poitio i (',',',': ' ' Sree Cooriate The traorme poitio (',',' are projete ito ree pae ooriate by iviig by ': Set x y Equatio yiel or the ear lippig plae ( a or the ar lippig plae (. The value i the value that get ale by the available -buer reolutio a tore i the -buer. Compare the x a y alulatio to equatio. A a example: 3 i about halway ito the -buer; the itae rage (, 3] ue the other hal. Thi i ue to the oliear, hyperboli ature o Eq.
3 Equatio i part o the geeralie ormula rom Newma a Sproull, 98: β α + ; where α a β are uer eie otat. I our implemetatio: α ; β he >>, α a β, o Thi i a hart o v. eye-pae. ou a ee that at itae. we have ue hal o the - buer rage, a at 5. we have ue about 85% o the -buer rage. Liear Value ( buer I you o ot wat to ariie o muh rage up loe (hal the -buer or rage to i thi ae, ( you a make your -buer value liear by uig a ormula uh a:. Eq 3 Uig the above example, itae yiel Eye Liear ( buer buer Sree v. Eye ( ( he alulatig value thi way, the value are evely itribute your over the rage o the -buer. Problem may arie, however, whe you have polygo that: overlap. Iteretio lie will be le aurate Moroe Street Sata Clara, CA 955 T F
4 are parallel a very loe. iterpolatio may aue more ightig. are very large. The iterpolatio will be more proe to error. I you o't have thee oitio or o't are about the error whe thi happe, you a make the liear. Liear i ot a upporte D3D eature, o you have to o the traorm yourel. Thi matrix will yiel liear value: M R R ; R ( ; ame a liear equatio 3. Corret Hyperboli Liear (iterpolatio error Iteretig Cube The above example i two ube that iteret. I the liear value example, you a learly ee the iterpolatio error. -buerig Thi problem o itributio ha bee aree i Diret3D 6.. Calulatig liear eye-pae value a uig thi or hie urae removal i alle -buerig. -buerig i implemete i the TNT harware a a be a ueul alterative to -buerig. -buerig houl be relatively eay to a to your appliatio. ou a hek the apability bit or thi harware eature. I ptricap.wratercap, or a evie, D3DPRASTERCAPS_BUFFER iiate that -buerig i upporte. Eable -buerig with: SetReerState(D3DRENDERSTATE_ENABLE, D3DB_USE; 3535 Moroe Street Sata Clara, CA 955 T F
5 For more iormatio about -Buerig, ee my -buerig paper. Iterpolatio Iterpolatio i perorme with -buer by liearly iterpolatig ree value beaue thee value are alreay hyperboli. -buerig i iterpolate uig /, the iverte to yiel the perpetive orret -buer value. -buer e alulate the plae equatio rom the triagle, the liearly iterpolate aro the urae. The plae equatio or the ree pae urae i: A x + B y + C + D (Eq.4. The otat A, B, C a D or the plae are alulate by: A (y - y ( - - ( - (y - y B C D ( - (x - x - (x - x ( - (x - x (y - y - (y - y (x - x -(A x + B y + C The x i, y i ooriate are lampe to ubpixel ooriate o the TNT. For a give x, y, we a etermie the aoiate value by olvig equatio 4 or : A x B y D C Diviig by C, our iterpolatio value E, F, a G are alulate by: A B D E ; F ; G C C C So our iterpolate ree i: i E xi + F yi + G For eah uit o poitive horiotal movemet, we a E to thi equatio. For eah uit o poitive vertial movemet, we a F. -buer -buerig i ompute i the ame way, but we are iterpolatig itea o. e mut perorm our iterpolatio the ivert to obtai the value that i plae i the -buer. Coeptually, the iterpolate value i ompute by alig the vertex value to maximie the preiio o the -buer, the iverte. i iterpolate the the reult i the iverte agai to get the ial value that i plae i the -buer Moroe Street Sata Clara, CA 955 T F
6 Sale the value * ale _ ator Ivert or eah vertex. ar w. ' Sie the evie river reeive the reiproal o homogeeou (RH, we a ombie tep oe a two a jut ale to the -buer ie. ar w RH vr ar ; where vr ar, ar i the ar lippig plae. ale _ ator See my -buerig paper or how ale_ator a ar are iitialie. 3 Calulate plae equatio, but ue iverte a ale value itea o ree value. A (y - y (w - w - (w - w (y - y B C D (w - w (x - x - (x - x (w - w (x - x (y - y - (y - y (x - x -(A x + B y + C w e obtai the iterpolatio otat, E, F a G i the ame way a we o or -buerig above. The iterpolate ivere value i alulate by: wi E x F y G i + i + 4 The iterpolate ivere value i iverte, givig whih i tore i the -buer Moroe Street Sata Clara, CA 955 T F
7 3535 Moroe Street Sata Clara, CA 955 T F A Note About Fog: The D3D oumetatio iiate that the etry mut be oe or og eet to be applie properly whe uig bae og: So our projetio matrix: M beome the equivalet M the reue to ; the M See my paper "Implemetig Fog i Diret3D" or more iormatio about Fog.
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