CS380: Computer Graphics Modeling Transformations. Sung-Eui Yoon ( 윤성의 ) Course URL:

Transcription

1 CS38: Computer Grphics Modeling Trnsformtions Sung-Eui Yoon ( 윤성의 ) Course URL:

2 Clss Ojectives (Ch. 3.5) Know the clssic dt processing steps, rendering pipeline, for rendering primitives Understnd 3D trnsltions nd rottions 2

3 Outline Where re we going? Snek peek t the rendering pipeline Vector lger Modeling trnsformtion Viewing trnsformtion Projections 3

4 The Clssic Rendering Pipeline Oject primitives defined y vertices fed in t the top Pixels come out in the disply t the ottom Commonly hve multiple primitives in vrious stges of rendering 4

5 Modeling Trnsforms 5 Strt with 3D models defined in modeling spces with their own t t t modeling frmes: m 1,m 2,..., m n Modeling trnsformtions orient models within common coordinte frme t clled world spce, w All ojects, light sources, nd the cmer live in world spce Trivil rejection ttempts to eliminte ojects tht cnnot possily e seen An optimiztion

6 Illumintion Illuminte potentilly visile ojects Finl rendered color is determined y oject s orienttion, its mteril properties, nd the light sources in the scene 6

7 Viewing Trnsformtions Mps points from world spce to eye spce: t e = w V Viewing position is trnsformed to the origin Viewing direction is oriented long some xis t 7

8 Clipping nd Projection We specify volume clled viewing frustum Mp the view frustum to the unit cue Clip ojects ginst the view volume, therey eliminting geometry not visile in the imge Project ojects into two-dimensions Trnsform from eye spce to normlized device coordintes 8

9 Rsteriztion nd Disply Trnsform normlized device coordintes to screen spce Rsteriztion converts ojects pixels - Almost every step in the rendering pipeline involves chnge of coordinte systems! - Trnsformtions re centrl to understnding 3D computer grphics 9

10 But, this is rchitecturl overview of recent GPU (Fermi) Unified rchitecture Highly prllel Support CUDA (generl lnguge) Wide memory ndwidth 1

11 11 But, this is rchitecturl overview of recent GPU

12 12 Recent CPU Chips (Intel s Core i7 processors)

13 Vector Alger We lredy sw vector ddition nd multiplictions y sclr Will study three kinds of vector multiplictions Dot product ( ) Cross product ( ) Tensor product ( ) - returns sclr - returns vector - returns mtrix 13

14 14 Dot Product ( ) Returns sclr s Geometric interprettions s: Length of projected onto nd or vice vers Distnce of from the origin in the direction of [ ] [ ] s 1 s, z y x z y x T z y x z y x T = = = = cosθ = θ cosθ

15 15 Cross Product ( ) Return vector tht is perpendiculr to oth nd, oriented ccording to the right-hnd rule The mtrix is clled the skew-symmetric mtrix of c c c z y x x y x z y z = = = [ ] x y y x z x x z y z z y c = c

16 Cross Product ( ) A mnemonic device for rememering the cross-product 16

17 Modeling Trnsformtions Vst mjority of trnsformtions re modeling trnsforms Generlly fll into one of two clsses Trnsforms tht move prts within the model t t m c m Mc = c t 1 1 m1 Trnsforms tht relte locl model s frme to the scene s world frme t t m 1c m 1Mc = w t c 17 Usully, Eucliden trnsforms, 3D rigidody trnsforms, re needed

18 18 Trnsltions Trnslte points y dding offsets to their coordintes The effect of this trnsltion: = = = 1 t 1 t 1 t 1 T where w c m Tc m c m c m Tc m c z y x t t t t t t

19 3D Rottions More complicted thn 2D rottions Rotte ojects long rottion xis Severl pproches Compose three cnonicl rottions out the xes Quternions 19

20 Geometry of Rottion Nturl sis for rottion of vector out specified xis: 2

21 21 Geometry of Rottion

22 Tensor Product ( ) 22 Cretes mtrix tht when pplied to vector c return scled y the project of onto c

23 Tensor Product ( ) Useful when = The mtrix is clled the symmetric mtrix of We shll denote this A 23 = ( ) c = ( c)

24 24 Snity Check Consider rottion y out the x-xis You cn check it in ny computer grphics ook, ut you don t need to memorize it θ θ θ θ sin 1 1 ) cos (1 1 cos ), 1 ( = Rotte = 1 cos sin sin cos 1 θ θ θ θ

25 25 Rottion using Affine Trnsformtion â x x [ ] 1 ˆ t s o x [ ] 1 ˆ t s o R x x θ s t Assume tht these sis vectors re normlized

26 Quternion Developed y W. Hmilton in 1843 Bsed on complex numers Two populr nottions for quternion, q w + xi + yj + zk, where i 2 = j 2 = k 2 = ijk = -1 [w, v], where w is sclr nd v is vector Conversion from the xis, v, nd ngle, t q = [cos (t/2), sin (t/2) v] Cn represent rottion Exmple: rotte y degree long x xis: q x = [cos (/2), sin(/2) (1,, )] 26

27 Bsic Quternion Opertions Addition q + q = [w + w, v + v ] Multipliction qq = [ww -v v, v x v + wv +w v] Conjugte q* = [w, -v] Norm N(q) = w 2 + x 2 + y 2 + z 2 Inverse q -1 = q* / N(q) 27

28 Bsic Quternion Opertions q is unit quternion if N(q)= 1 Then q -1 = q* Identity [1, (,, )] for multipliction [, (,, )] for ddition 28

29 Rottions using Quternions Suppose tht you wnt to rotte vector/point v with q Then, the rotted v v = q r q -1, where r = [, v]) Compositing rottions R = R2 R1 (rottion R1 followed y rottion R2) 29

30 Quternion to Rottion Mtrix Q = w + xi + yj + zk R m = 1-2y 2-2z 2 2xy-2wz 2xz+2wy 2xy+2wz 1-2x 2-2z 2 2yz-2wx 2xz-2wy 2yz+2wx 1-2x 2-2y 2 We cn lso convert rottion mtrix to quternion 3

31 Advntge of Quternions More efficient wy to generte ritrry rottions Less storge thn 4 x 4 mtrix Esier for smooth rottion Numericlly more stle thn 4x4 mtrix (e.g., no drifting issue) More redle 31

32 Clss Ojectives were: Know the clssic dt processing steps, rendering pipeline, for rendering primitives Understnd 3D trnsltions nd rottions 32

33 33 PA2: Simple Animtion & Trnsformtion

34 OpenGL: Disply Lists Disply lists A group of OpenGL commnds stored for lter executions Cn e optimized in the grphics hrdwre Thus, cn show higher performnce Ver. 4.3: Vertex Arry Oject is much etter Immedite mode Cuses commnds to e executed immeditely 34

35 An Exmple 35 void drwcow() { if (frme == ) { cow = new WveFrontOBJ( "cow.oj" ); cowid = glgenlists(1); glnewlist(cowid, GL_COMPILE); cow->drw(); glendlist(); }.. glclllist(cowid);.. }

36 API for Disply Lists Gluint glgenlists (rnge) - generte continuous set of empty disply lists void glnewlist (list, mode) & glendlist () : specify the eginning nd end of disply list void glclllists (list) : execute the specified disply list 36

37 OpenGL: Getting Informtion from OpenGL void min( int rgc, chr* rgv[] ) { int rv,gv,v; glgetintegerv(gl_red_bits,&rv); glgetintegerv(gl_green_bits,&gv); glgetintegerv(gl_blue_bits,&v); printf( "Pixel colors = %d : %d : %d\n", rv, gv, v );. } void disply () {.. glgetdoulev(gl_modelview_matrix, cow2wld.mtrix());.. } 37

38 Homework Wtch SIGGRAPH Videos Go over the next lecture slides 38

39 Next Time Viewing trnsformtions 39