CSCI-4972/6963 Advanced Computer Graphics

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1 Luo Jr. CSCI-497/696 Advaned Computer Graphis Professor Barb Cutler MRC 9A Piar Animation Studios, 986 Topis for the Semester Mesh Simplifiation Meshes representation simplifiation subdivision surfaes generation volumetri modeling Simulation partile sstems rigid bod, deformation, loth, wind/water flows ollision detetion weathering Rendering ra traing appearane models shadows loal vs. global illumination radiosit, photon mapping, subsurfae sattering, et. proedural modeling teture snthesis hardware & more Hoppe Progressive Meshes SIGGRAPH Mesh Generation & Volumetri Modeling Modeling Subdivision Surfaes Hoppe et al., Pieewise Smooth Surfae Reonstrution 994 Cutler et al., Simplifiation and Improvement of Geri s Game Piar 997 Tetrahedral Models for Simulation 4 5 6

2 Partile Sstems Phsial Simulation Rigid Bod Dnamis f ( t ) Collision Detetion Frature b Deformation p ( t ( ) ) t f ( t ) p b ( t ) v ( t ) p b ( t ) Müller et al., Stable Real-Time Deformations Star Trek: The Wrath of Khan f ( t ) Fluid Dnamis Ra Casting For ever piel onstrut a ra from the ee For ever objet in the sene Find intersetion with the ra Keep the losest Foster & Mataas, 996 Visual Simulation of Smoke Fedkiw, Stam & Jensen SIGGRAPH 9 Ra Traing Subsurfae Sattering Shade (interation of light and material) Seondar ras (shadows, refletion, refration) An Improved Illumination Model for Shaded Displa Whitted 98 Jensen et al., A Pratial Model for Subsurfae Light Transport Surfae

3 Appearane Models Sllabus & Course Website θ r θ i Wojieh Matusik Whih version should I register for? CSCI 696 units of graduate redit CSCI units of undergraduate redit same letures, assignments, quies, & grading riteria φ i φ r Other Questions? Henrik Wann Jensen 4 Introdutions Who are ou? name ear/degree graphis bakground (if an) researh/job interests wh ou are taking this lass something fun, interesting, or unusual about ourself Outline Orthographi & Perspetive Projetions Eample: Iterated Funtion Sstems (IFS) 5 6 What is a Transformation? Simple Transformations Maps points (, ) in one oordinate sstem to points (', ') in another oordinate sstem ' a + b + ' d + e + f For eample, Iterated Funtion Sstem (IFS): Can be ombined Are these operations invertible? Yes, eept sale 7 8

4 Transformations are used to: Position objets in a sene Change the shape of objets Create multiple opies of objets Projetion for virtual ameras Desribe animations Rigid-Bod / Eulidean Transforms Preserves distanes Preserves angles Rigid / Eulidean Identit 9 Similitudes / Similarit Transforms Linear Transformations Preserves angles Similitudes Rigid / Eulidean Similitudes Rigid / Eulidean Linear Identit Isotropi Saling Identit Isotropi Saling Saling Refletion Shear Linear Transformations Affine Transformations L(p + q) L(p) + L(q) L(ap) a L(p) preserves parallel lines Affine Similitudes Rigid / Eulidean Linear Similitudes Rigid / Eulidean Linear Identit Isotropi Saling Saling Refletion Shear Identit Isotropi Saling Saling Refletion Shear 4 4

5 Projetive Transformations preserves lines Projetive Affine Similitudes Linear Rigid / Eulidean Saling Identit Isotropi Saling Refletion Shear General (Free-Form) Transformation Does not preserve lines Not as pervasive, omputationall more involved Perspetive 5 Sederberg and Parr, Siggraph Outline Orthographi & Perspetive Projetions Eample: Iterated Funtion Sstems (IFS) How are Transforms Represented? ' ' ' a + b + ' d + e + f a b d e + f p' M p + t 7 8 Homogeneous Coordinates Add an etra dimension in D, we use matries In D, we use 4 4 matries Eah point has an etra value, w ' ' ' w' p' a e i m b f j n g k o M p d h l p w 9 in homogeneous oordinates ' ' Affine formulation a b d e ' a + b + ' d + e + f + f p' M p + t Homogeneous formulation ' ' p' a b d e f M p 5

6 Homogeneous Coordinates Most of the time w, and we an ignore it ' ' ' a e i b f j g k d h l Homogeneous Visualiation Divide b w to normalie (homogenie) W? Point at infinit (diretion) If we multipl a homogeneous oordinate b an affine matri, w is unhanged (,, ) (,, ) (7,, ) (4,, ) (4, 5, ) (8,, ) w w Translate (t, t, t) Translate(,,) Sale (s, s, s) Sale(s,s,s) p' Wh bother with the etra dimension? Beause now translations an be enoded in the matri! p p' Isotropi (uniform) saling: s s s p q q' ' ' ' t t t ' ' ' s s s 4 About ais ZRotate(θ) p' θ p About (k, k, k), a unit vetor on an arbitrar ais (Rodrigues Formula) Rotate(k, θ) θ k ' ' ' os θ sin θ -sin θ os θ ' ' ' kk(-)+ kk(-)+ks kk(-)-ks kk(-)-ks kk(-)+ kk(-)-ks kk(-)+ks kk(-)-ks kk(-)+ where os θ & s sin θ 5 6 6

7 Storage Often, w is not stored (alwas ) Needs areful handling of diretion vs. point Mathematiall, the simplest is to enode diretions with w In terms of storage, using a -omponent arra for both diretion and points is more effiient Whih requires to have speial operation routines for points vs. diretions Outline Orthographi & Perspetive Projetions Eample: Iterated Funtion Sstems (IFS) 7 8 How are transforms ombined? Sale then Translate Non-ommutative Composition Sale then Translate: p' T ( S p ) TS p (,) (,) Sale(,) (,) (,) Translate(,) (,) (5,) (,) (,) Sale(,) (,) (,) Translate(,) (,) (5,) Use matri multipliation: p' T ( S p ) TS p TS Caution: matri multipliation is NOT ommutative! 9 Translate then Sale: p' S ( T p ) ST p Translate(,) (4,) Sale(,) (,) (6,) (,) (,) (8,4) 4 Non-ommutative Composition Sale then Translate: p' T ( S p ) TS p TS Translate then Sale: p' S ( T p ) ST p ST 6 Outline Orthographi & Perspetive Projetions Eample: Iterated Funtion Sstems (IFS) 4 4 7

8 Orthographi vs. Perspetive Orthographi Simple Orthographi Projetion Projet all points along the ais to the plane Perspetive 4 44 Simple Perspetive Projetion Projet all points along the ais to the d plane, eepoint at the origin: Alternate Perspetive Projetion Projet all points along the ais to the plane, eepoint at the (,,-d): homogenie homogenie * d / * d / d / d /d 45 * d / ( + d) * d / ( + d) ( + d)/ d /d 46 In the limit, as d Outline this perspetive projetion matri... /d...is simpl an orthographi projetion Orthographi & Perspetive Projetions Eample: Iterated Funtion Sstems (IFS)

9 Iterated Funtion Sstems (IFS) Capture self-similarit Contration (redue distanes) An attrator is a fied point A U fi (A) Eample: Sierpinski Triangle Desribed b a set of n affine transformations In this ase, n translate & sale b Eample: Sierpinski Triangle Another IFS: The Dragon for lots of random input points (, ) for j to num_iters randoml pik one of the transformations ( k+, k+ ) f i ( k, k ) displa ( k, k ) Inreasing the number of iterations 5 5 D IFS in OpenGL GL_POINTS GL_QUADS Assignment : OpenGL Warmup Get familiar with: C++ environment OpenGL Transformations simple Vetor & Matri lasses Have Fun! 5 Will not be graded (but ou should still do it and submit it!) 54 9

10 Outline Orthographi & Perspetive Projetions Eample: Iterated Funtion Sstems (IFS) OpenGL Basis: GL_POINTS gldisable(gl_lighting); glbegin(gl_points); glcolorf(.,.,.); glvertef( ); glend(); lighting should be disabled OpenGL Basis: GL_QUADS glenable(gl_lighting); glbegin(gl_quads); glnormalf( ); glcolorf(.,.,.); glvertef( ); glvertef( ); glvertef( ); glvertef( ); glend(); lighting should be enabled... an appropriate normal should be speified OpenGL Basis: Transformations Useful ommands: glmatrimode(gl_modelview); glpushmatri(); glpopmatri(); glmultmatrif( ); From OpenGL Referene Manual Questions? For Net Time: Read Hugues Hoppe Progressive Meshes SIGGRAPH 996 Post a omment or question on the ourse WebCT/LMS disussion b am on Frida /5 Image b Henrik Wann Jensen 59 6