Schedule. Curves & Surfaces. Questions? Last Time: Today. Limitations of Polygonal Meshes. Acceleration Data Structures.

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1 Schedule Curves & Surfaces Sunday Ocober 5 h, * 3-5 PM *, Room TBA: Review Session for Quiz 1 Exra Office Hours on Monday (NE43 Graphics Lab) Tuesday Ocober 7 h : Quiz 1: In class 1 hand-wrien 8.5x11 shee of noes allowed Wednesday Ocober 15h: Assignmen 4 (Grid Acceleraion) due Las Time: Quesions? Acceleraion Daa Srucures Today Review Moivaion Limiaions of Polygonal Models Phong Normal Inerpolaion Some Modeling Tools & Definiions Curves Surfaces / Paches Subdivision Surfaces Procedural Texuring Limiaions of Polygonal Meshes planar faces fixed resoluion deformaion is difficul no naural parameerizaion

2 Can We Disguise he Faces? Phong Normal Inerpolaion No Phong Shading from Assignmen 3 Insead of using he normal of he riangle, inerpolae an averaged normal a each verex across he face Mus be renormalized 1K faces 1K smooh 10K faces 10K smooh Beer, bu no always good enough Sill low resoluion (missing fine deails) Sill have polygonal silhouees Inersecion deph is planar Collisions in a simulaion Solid Texuring... 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 "waerigh" C 1 coninuous curve/surface derivaive is coninuous "looks smooh, no faces" C 2 coninuous curve/surface 2 nd derivaive is coninuous Acually imporan for shading

3 Quesions? Today Review Moivaion Curves Wha's a Spline? Linear Inerpolaion Inerpolaion Curves vs. Approximaion Curves Bézier BSpline (NURBS) Surfaces / Paches Subdivision Surfaces Procedural Texuring 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) BSpline (approximaion) Linear Inerpolaion Simples "curve" beween wo poins 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

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? 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 Cubic Bézier Curve Connecing Cubic Bézier Curves How can we guaranee C0 coninuiy (no gaps)? How can we guaranee C1 coninuiy (angen vecors mach)? Asymmeric: Curve goes hrough some conrol poins bu misses ohers

5 Higher-Order Bézier Curves Cubic BSplines > 4 conrol poins Bernsein Polynomials as he basis funcions 4 conrol poins Locally cubic Curve is no consrained o pass hrough any conrol poins Every conrol poin affecs he enire curve No simply a local effec More difficul o conrol for modeling Cubic BSplines Cubic BSplines Ieraive mehod for consrucing BSplines MIT EECS 6.837, Durand Culer Shirley, Fundamenals of and Compuer Graphics Cubic BSplines Bézier is no he same as BSpline can be chained ogeher beer conrol locally (windowing) Relaionship o he conrol poins is differen Bézier BSpline 5

6 Bezier is no he same as Bspline Bu we can conver beween he curves using he basis funcions: NURBS (generalized BSplines) BSpline: uniform cubic BSpline NURBS: Non-Uniform Raional BSpline non-uniform = differen spacing beween he blending funcions, a.k.a. knos raional = raio of polynomials (insead of cubic) Quesions? Today Review Moivaion Spline Curves Spline Surfaces / Paches Tensor Produc Bilinear Paches Bezier Paches Subdivision Surfaces Procedural Texuring Tensor Produc Bilinear Pach 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

7 Bilinear Pach Bicubic Bezier 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? Ediing Bicubic Bezier Paches Modeling wih Bicubic Bezier Paches Original Teapo specified wih Bezier Paches Curve Basis Funcions Surface Basis Funcions 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

8 Quesions? Today Review Moivaion Spline Curves Spline Surfaces / Paches Subdivision Surfaces Procedural Texuring Chaikin's Algorihm Doo-Sabin Subdivision Doo-Sabin Subdivision Loop Subdivision hp:// Shirley, Fundamenals MIT EECS 6.837, Durand of and Compuer Culer Graphics

9 Loop Subdivision Weird Subdivision Surface Models Some edges can be specified as crease edges hp://grail.cs.washingon.edu/projecs/subdivision/ Jusin Legakis Quesions? Today Review Moivaion Spline Curves Spline Surfaces / Paches Procedural Texuring Procedural Texures Procedural Solid Texures Noise Turbulence f (x,y,z) color Ken Perlin Jusin Legakis Image by Turner Whied Jusin Legakis

10 Quesions? Nex Thursday: Animaion I: Keyframing