Las Time: Curves & Surfaces Expeced value and variance Mone-Carlo in graphics Imporance sampling Sraified sampling Pah Tracing Irradiance Cache Phoon Mapping Quesions? Today Moivaion Limiaions of Polygonal Models Gouraud Shading & Phong Normal Inerpolaion Some Modeling Tools & Definiions Curves Surfaces / Paches Subdivision Surfaces Limiaions of Polygonal Meshes Can We Disguise he Faces? Planar faces (& silhouees) Fixed resoluion Deformaion is difficul No naural parameerizaion (for exure mapping)
Gouraud Shading Insead of shading wih he normal of he riangle, shade he verices wih he average normal and inerpolae he color across each face Illusion of a smooh surface wih smoohly varying normals Phong Normal Inerpolaion (No Phong Shading) Inerpolae he average verex normals across he face and compue per-pixel shading Mus be renormalized 1K faces 1K smooh 10K faces 10K smooh Beer, bu no always good enough Sill low, fixed resoluion (missing fine deails) Sill have polygonal silhouees Inersecion deph is planar (e.g. ray visualizaion) Collisions problems for simulaion Solid Texuring problems... Some Non-Polygonal Modeling Tools Exrusion Spline Surfaces/Paches Surface of Revoluion Quadrics and oher implici polynomials Coninuiy definiions: C 0 coninuous curve/surface has no breaks/gaps/holes G 1 coninuous angen a join has same direcion C 1 coninuous curve/surface derivaive is coninuous angen a join has same direcion and magniude C n coninuous curve/surface hrough n h derivaive is coninuous imporan for shading
Quesions? Today Moivaion Curves Wha's a Spline? Linear Inerpolaion Inerpolaion Curves vs. Approximaion Curves Bézier (NURBS) Surfaces / Paches Subdivision Surfaces Definiion: Wha's a Spline? Inerpolaion Curves / Splines Smooh curve defined by some conrol poins Moving he conrol poins changes he curve Inerpolaion Bézier (approximaion) (approximaion) www.abm.org Linear Inerpolaion Simples "curve" beween wo poins Q() = Spline Basis Funcions a.k.a. Blending Funcions 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-1 +... + a 2 2 + a 1 + a 0 y() = b n-1 n-1 +... + b 2 2 + b 1 + b 0
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? Cubic Bézier Curve 4 conrol poins Curve passes hrough firs & las conrol poin Curve is angen a P 0 o (P 0 -P 1 ) and a P 4 o (P 4 -P 3 ) Cubic Bézier Curve de Caseljau's algorihm for consrucing Bézier curves A Bézier curve is bounded by he convex hull of is conrol poins. Cubic Bézier Curve Connecing Cubic Bézier Curves Bernsein Polynomials Asymmeric: Curve goes hrough some conrol poins bu misses ohers 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 Bézier 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 Bézier subcurves? Higher-Order Bézier 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 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 MIT EECS 6.837, Durand of and Compuer Culer Graphics Cubic s Cubic s 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 Bézier is no he same as Bézier is no he same as Relaionship o he conrol poins is differen Bézier Bézier Convering beween Bézier & Convering beween Bézier & original conrol poins as Bézier new conrol poins o mach Bézier Using he basis funcions: new Bézier conrol poins o mach original conrol poins as
NURBS (generalized s) Quesions? : 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) Today Moivaion Spline Curves Spline Surfaces / Paches Tensor Produc Bilinear Paches Bezier Paches Subdivision Surfaces Tensor Produc Of wo vecors: Similarly, we can define a surface as he ensor produc of wo curves... Farin, Curves and Surfaces for MIT EECS 6.837, Durand Compuer and Aided Culer 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?
Bicubic Bezier Pach Ediing Bicubic Bezier Paches Curve Basis Funcions Surface Basis Funcions Bicubic Bezier Pach Tessellaion Assignmen 8: Given 16 conrol poins and a essellaion resoluion, creae a riangle mesh Modeling wih Bicubic Bezier Paches Original Teapo specified wih Bezier Paches resoluion: 5x5 verices resoluion: 11x11 verices resoluion: 41x41 verices Modeling Headaches Trimming Curves for Paches Original Teapo model is no "waerigh": inersecing surfaces a spou & handle, no boom, a hole a he spou ip, a gap beween lid & base Shirley, Fundamenals MIT EECS 6.837, Durand of and Compuer Culer Graphics
Quesions? Bezier Paches? or Today Review Moivaion Spline Curves Spline Surfaces / Paches Subdivision Surfaces Triangle Mesh? Henrik Wann Jensen Chaikin's Algorihm Doo-Sabin Subdivision Doo-Sabin Subdivision Loop Subdivision hp://www.ke.ics.saiama-u.ac.jp/xuz/pic/doo-sabin.gif Shirley, Fundamenals MIT EECS 6.837, Durand of and Compuer Culer Graphics
Loop Subdivision Quesions? Some edges can be specified as crease edges hp://grail.cs.washingon.edu/projecs/subdivision/ Jusin Legakis 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 Bézier curve is bounded by he convex hull of is conrol poins. Nex Tuesday: (no class Thursday!) Animaion I: Paricle Sysems