Subdivision Surfaces - PDF document

Subdivision Surfaces 1 GeometricModeling Geometric Modeling • Sometimesneedmorethanpolygonmeshes • –Smooth surfaces •Traditional geometric modeling used NURBS NiftilB Sli – N on un if orm ra ti ona B - S p –Demo ProblemswithNURBS Problems • AsingleNURBSpatchiseitheratopological • A either a disk, a tube or a torus •Must use many NURBS patches to model • WhendeformingasurfacemadeofNURBS • When a patches, cracks arise at the seams 3 Subdivision Subdivision “ Subdivisiondefinesasmoothcurveorsurfaceas defines a or as the limit of a sequence of successive refinements ” refinements 4 Example:Geri ’ sGame(Pixar) • Subdivisionusedfor Geris • Subdivision used –Geri’s hands and head –Clothing Tie and shoes 7 Example:Geri ’ sGame(Pixar) Geris Woody’s hand (NURBS)Geri’s hand (subdivision) 8 Example:Geri ’ sGame(Pixar) Geris Sharpandsemi sharpfeatures and - 9 Example:Games Example: SupportedinhardwareinDirectX 11 11 SubdivisionCurveTypes Subdivision •Approximating •Corner Cuttin g g 12 Approximating Approximating Splitting step: split each edge in two 14 Approximating Approximating Startover... Start over Approximating Approximating ...averaging... ...averaging... 18 Approximating Approximating Iftheruleisdesignedcarefully... If carefully... ... the control polygons will converge to a smoothlimitcurve! smooth CornerCutting • Subdivisionrule: Corner • Subdivision –Insert –the old vertices – Connect the new vertices 22 B - SplineCurves B Curves • Piecewisepolynomialofdegree n • Piecewise of n controlpoints B splinecurve B - parametervalue basis functions parameter B - SplineCurves B Curves • Distinguishbetweenoddandevenpoints • odd and even • LinearB • B - –Odd coefficients (1/2, 1/2) –Even coefficient (1) 24 CubicB - B CubicB - B CubicB - B CubicB - B CubicB - B CubicB - B CubicB - B Interpolating Interpolating 42 Interpolating Interpolating 44 Interpolating Interpolating demo SubdivisionZoo Subdivision • Classificationofsubdivisionschemes • Classification of (face Til i ar es M Approximating)Catmull Interpolating Butterfly 1 ) Kobbelt 1 ) Interpolating Butterfly 1 ) Kobbelt 1 ) (vertexMidedge(C 2 ) •Many more... Biquartic (C ) 50 Catmull - ClarkSubdivision • Generalizationof bi cubicB • of bi - B - •Primal, approximation subdivision scheme •Applied to liitf G G 2 con nuous m sur f for the set of finite extraordinary points •Vertices with valence – C 2 co n t e v e r y e e l se C cotuouseeyeeese Catmull - ClarkSubdivision LoopSubdivision Loop • Generalizationof boxsplines • of box •Primal, approximating subdivision scheme •Applied to liitf G G 2 con nuous m sur f for the set of finite extraordinary points •Vertices with valence – C 2 co n t e v e r y e e l se C cotuouseeyeeese Doo - SabinSubdivision • Generalizationof bi quadraticB • of bi - B - •Dual, approximating subdivision scheme •Applied to liitf G G 1 con nuous m sur f aces:for the set of finite extraordinary points •Center of irregular polygons after 1 subdivision step – C 1 co n t e v e r y e e l se C cotuouseeyeeese Doo - SabinSubdivision ButterflySubdivision Butterfly • Primalinterpolatingscheme • •Applied to •Generates C o forthesetoffiniteextraordinarypoints – C o for finite •Vertices of valence = 3 �or 7 – C1continuous everywhere else 64 Comparison Comparison Doo - - Butterfly Butterfly Comparison Comparison • Subdividingatetrahedron • Subdividing a –Same insights –Severe shrinking for approximating schemes 70 SoWhoWins? So Ctll Clkbthitltiitid • L oop an d C a t b t s no t requ i re •Loop best for triangular meshes •Catmull-Clark best for quad meshes–Don’t triangulate and then use Catmull-Clark 72 AnalysisofSubdivision Analysis • Invariantneighborhoods Invariant –How many control-points affect a small neighborhoodaroundapoint? neighborhood a ? •Subdivision scheme can be analyzed by looking at a localsubdivision matrix 73 AnalysisofSubdivision Analysis • Invarianceunderaffinetransformations • Invariance under transformations –transform(subdivide(P)) = subdivide(transform(P)) 76 AnalysisofSubdivision Analysis Conclusion: 1has to be eigenvector of S i genva l ue 0= 1 LimitBehavior - Position Limit Behavior Position • Anyvectorislinearcombinationofeigenvectors: • of eigenvectors: rows of •Apply subdivision matrix: 80 LimitBehavior - Limit Behavior • Setoriginat a : • at a 0 : •Divide b y 1 j y • Limittangentgivenby: • by: LimitBehavior - Limit Behavior • • –All eigenvalues of than 1to ensure existence of a tangent, i.e. •Surfaces: Tangentsdeterminedby and 1 2 Example:CubicSplines • Subdivisionmatrix&rules Example: • Subdivision & Eigenales • v al u Example:CubicSplines • Eigenvectors Example: • Eigenvectors •Limit position and tangent 85 PropertiesofSubdivision Properties • Flexiblemodeling • Flexible –Handle surfaces of arbitrary topology Provablysmoothlimitsurfaces – Provably –Intuitive control point interaction Slbilit • S ca l a –Level-of-detail rendering – A •Usability –Compact representation–Simple and efficient code 86 BeyondSubdivisionSurfaces Beyond • T [ Sdb tl2003] • T - [ S e d erge –Allows control points to be –Can use less control points – Can model different to p olo ies with sin g le surface pgg NURBST-Splines T-Splines BeyondSubdivisionSurfaces Beyond • Howdoyousubdivideateapot? • do you a