Curves & Surfaces. Last Time? Today. Readings for Today (pick one) Limitations of Polygonal Meshes. Today. Adjacency Data Structures

Transcription

1 Las Time? Adjacency Daa Srucures Geomeric & opologic informaion Dynamic allocaion Efficiency of access Curves & Surfaces Mesh Simplificaion edge collapse/verex spli geomorphs progressive ransmission view-dependen refinemen Readings for (pick one) "Geomery Images", Gu, Gorler, & Hoppe, SIGGRAPH 2002 Limiaions of Polygonal Models "Teddy: A Skeching Inerface for 3D Freeform Design", Igarashi e al., SIGGRAPH 1999 Exending o Surfaces Tensor Produc Limiaions of Polygonal Meshes Limiaions of Polygonal Models Planar faces (& silhouees) Fixed resoluion Deformaion is difficul No naural parameerizaion (for exure mapping) Exending o Surfaces Tensor Produc 1

2 Color & Normal Inerpolaion Wha is Gouraud Shading? I s easy in OpenGL o specify differen colors and/or normals a he verices of riangles: Why is his useful? Insead of shading wih he normal of he riangle, we ll shade he verices wih he average normal and inerpolae he shaded color across each face This gives he illusion of a smooh surface wih smoohly varying normals ray racing scan conversion fla shading scan conversion gouraud shading How do we compue Average Normals? Is i expensive?? Phong Normal Inerpolaion (No Phong Shading) Gouraud no always good enough Inerpolae he average verex normals across he face and compue per-pixel shading Sill low, fixed resoluion (missing fine deails) Sill have polygonal silhouees Inersecion deph is planar (e.g. ray racing visualizaion) Collisions problems for simulaion Solid Texuring problems... Normals should be re-normalized (ensure lengh=1) Before shaders, per-pixel shading was no possible in hardware (Gouraud shading is acually a decen subsiue!) Some Non-Polygonal Modeling Tools Coninuiy definiions: C0 coninuous curve/surface has no breaks/gaps/holes G1 coninuous Exrusion Surface of Revoluion angen a join has same direcion C1 coninuous curve/surface derivaive is coninuous angen a join has same direcion and magniude Cn coninuous Spline Surfaces/Paches Quadrics and oher implici polynomials curve/surface hrough nh derivaive is coninuous imporan for shading Shape Opimizaion Using Reflecion Lines, Tosun e al.,

3 Quesions? Limiaions of Polygonal Models Exending o Surfaces Tensor Produc Definiion: Wha's a Spline? Inerpolaion Curves / Splines Smooh curve defined by some conrol poins Moving he conrol poins changes he curve Inerpolaion (approximaion) (approximaion) Inerpolaion Curves Curve is consrained o pass hrough all conrol poins Given poins P 0, P 1,... P n, find lowes degree polynomial which passes hrough he poins x() = a n-1 n a a 1 + a 0 y() = b n-1 n b b 1 + b 0 Linear Inerpolaion Simples "curve" beween wo poins Q() = Spline Basis Funcions a.k.a. Blending Funcions 3

4 Inerpolaion vs. Approximaion Curves Inerpolaion vs. Approximaion Curves Inerpolaion Curve over consrained los of (undesirable?) oscillaions Inerpolaion curve mus pass hrough conrol poins Approximaion curve is influenced by conrol poins Approximaion Curve more reasonable? Quesions? Limiaions of Polygonal Models Exending o Surfaces Tensor Produc Cubic Curve Cubic Curve 4 conrol poins Curve passes hrough firs & las conrol poin Curve is angen a P 1 o (P 2 -P 1 ) and a P 4 o (P 4 -P 3 ) de Caseljau's algorihm for consrucing curves A curve is bounded by he convex hull of is conrol poins. 4

5 Cubic Curve Connecing Cubic Curves Asymmeric: Curve goes hrough some conrol poins bu misses ohers Bernsein Polynomials How can we guaranee C 0 coninuiy? How can we guaranee G 1 coninuiy? How can we guaranee C 1 coninuiy? Can guaranee higher C 2 or higher coninuiy Connecing Cubic Curves Where is his curve C 0 coninuous? G 1 coninuous? C 1 coninuous? Wha s he relaionship beween: he # of conrol poins, and he # of cubic subcurves? Higher-Order Curves > 4 conrol poins Bernsein Polynomials as he basis funcions Every conrol poin affecs he enire curve No simply a local effec More difficul o conrol for modeling Quesions? Limiaions of Polygonal Models Exending o Surfaces Tensor Produc 5

6 Cubic s 4 conrol poins Locally cubic Curve is no consrained o pass hrough any conrol poins Cubic s Ieraive mehod for consrucing s A curve is also bounded by he convex hull of is conrol poins. Shirley, Fundamenals of Compuer Graphics Cubic s Connecing Cubic Curves Can be chained ogeher Beer conrol locally (windowing) Connecing Cubic Curves Curve Conrol Poins Wha s he relaionship beween he # of conrol poins, and he # of cubic subcurves? Defaul wih Disconinuiy Repea inerior conrol poin which passes hrough end poins Repea end poins 6

7 is no he same as is no he same as Relaionship o he conrol poins is differen Convering beween & Convering beween & original conrol poins as new conrol poins o mach Using he basis funcions: new conrol poins o mach original conrol poins as NURBS (generalized s) : uniform cubic NURBS: Non-Uniform Raional non-uniform = differen spacing beween he blending funcions, a.k.a. knos raional = raio of polynomials (insead of cubic) Nea Bezier Spline Trick A Bezier curve wih 4 conrol poins: P 0 P 1 P 2 P 3 Can be spli ino 2 new Bezier curves: P 0 P 1 P 2 P 3 P 3 P 4 P 5 P 3 A curve is bounded by he convex hull of is conrol poins. 7

8 Spline Surface via Tensor Produc Limiaions of Polygonal Models Of wo vecors: Exending o Surfaces Tensor Produc Similarly, we can define a surface as he ensor produc of wo curves... Farin, Curves and Surfaces for Compuer Aided Geomeric Design Bilinear Pach Bilinear Pach Smooh version of quadrilaeral wih non-planar verices... Bu will his help us model smooh surfaces? Do we have conrol of he derivaive a he edges? Ruled Surfaces in Ar & Archiecure hp:// Chiras Iulia Asri Isabella Maiss Sheiners Anoni Gaudi Children s School Barcelona hp:// Readings for Friday (pick one) Hoppe e al., Piecewise Smooh Surface Reconsrucion SIGGRAPH 1994 DeRose, Kass, & Truong, "Subdivision Surfaces in Characer Animaion", SIGGRAPH 1998 Pos a commen or quesion on he LMS discussion by 10am on Tuesday 8

9 Homework 1: Quesions/Commens? 9