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MIT 6 837 - Curves & Surfaces

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Curves & SurfacesLast Time:Questions?TodayLimitations of Polygonal MeshesCan We Disguise the Facets?Gouraud ShadingPhong Normal InterpolationSlide 9Better, but not always good enoughSome Non-Polygonal Modeling ToolsContinuity definitions:Slide 13Slide 14Definition: What's a Spline?Interpolation Curves / SplinesLinear InterpolationInterpolation CurvesInterpolation vs. Approximation CurvesSlide 20Cubic Bézier CurveSlide 22Slide 23Connecting Cubic Bézier CurvesSlide 25Higher-Order Bézier CurvesCubic BSplinesSlide 28Slide 29Slide 30Connecting Cubic BSpline CurvesBSpline Curve Control PointsBézier is not the same as BSplineSlide 34Converting between Bézier & BSplineSlide 36NURBS (generalized BSplines)Slide 38Slide 39Tensor ProductBilinear PatchSlide 42Bicubic Bezier PatchEditing Bicubic Bezier PatchesBicubic Bezier Patch TessellationModeling with Bicubic Bezier PatchesModeling HeadachesTrimming Curves for PatchesSlide 49Slide 50Chaikin's AlgorithmDoo-Sabin SubdivisionSlide 53Loop SubdivisionSlide 55Slide 56Neat Bezier Spline TrickNext Tuesday: (no class Thursday!)MIT EECS 6.837, Durand and CutlerCurves & SurfacesMIT EECS 6.837, Durand and CutlerLast Time:•Expected value and variance•Monte-Carlo in graphics•Importance sampling•Stratified sampling•Path Tracing•Irradiance Cache•Photon MappingMIT EECS 6.837, Durand and CutlerQuestions?MIT EECS 6.837, Durand and CutlerToday•Motivation–Limitations of Polygonal Models –Gouraud Shading & Phong Normal Interpolation–Some Modeling Tools & Definitions•Curves•Surfaces / Patches•Subdivision SurfacesMIT EECS 6.837, Durand and CutlerLimitations of Polygonal Meshes•Planar facets (& silhouettes)•Fixed resolution•Deformation is difficult •No natural parameterization (for texture mapping)MIT EECS 6.837, Durand and CutlerCan We Disguise the Facets?MIT EECS 6.837, Durand and CutlerGouraud Shading•Instead of shading with the normal of the triangle, shade the vertices with the average normal and interpolate the color across each faceIllusion of a smooth surface with smoothly varying normalsMIT EECS 6.837, Durand and CutlerPhong Normal Interpolation•Interpolate the average vertex normals across the face and compute per-pixel shading(Not Phong Shading)Must be renormalizedMIT EECS 6.837, Durand and Cutler10Kfacets1Kfacets1Ksmooth10KsmoothMIT EECS 6.837, Durand and CutlerBetter, but not always good enough•Still low, fixed resolution (missing fine details)•Still have polygonal silhouettes•Intersection depth is planar (e.g. ray visualization)•Collisions problems for simulation•Solid Texturing problems•...MIT EECS 6.837, Durand and CutlerSome Non-Polygonal Modeling ToolsExtrusionSpline Surfaces/PatchesSurface of RevolutionQuadrics and other implicit polynomialsMIT EECS 6.837, Durand and CutlerContinuity definitions:•C0 continuous–curve/surface has no breaks/gaps/holes•G1 continuous–tangent at joint has same direction•C1 continuous–curve/surface derivative is continuous–tangent at join has same direction and magnitude•Cn continuous–curve/surface through nth derivative is continuous–important for shadingMIT EECS 6.837, Durand and CutlerQuestions?MIT EECS 6.837, Durand and CutlerToday•Motivation•Curves–What's a Spline?–Linear Interpolation–Interpolation Curves vs. Approximation Curves–Bézier–BSpline (NURBS)•Surfaces / Patches•Subdivision SurfacesMIT EECS 6.837, Durand and CutlerBSpline (approximation)Definition: What's a Spline?•Smooth curve defined by some control points•Moving the control points changes the curveInterpolation Bézier (approximation)MIT EECS 6.837, Durand and CutlerInterpolation Curves / Splineswww.abm.orgMIT EECS 6.837, Durand and CutlerLinear Interpolation•Simplest "curve" between two pointsQ(t) = Spline Basis Functionsa.k.a. Blending FunctionsMIT EECS 6.837, Durand and CutlerInterpolation Curves•Curve is constrained to pass through all control points•Given points P0, P1, ... Pn, find lowest degree polynomial which passes through the pointsx(t) = an-1tn-1 + .... + a2t2 + a1t + a0y(t) = bn-1tn-1 + .... + b2t2 + b1t + b0MIT EECS 6.837, Durand and CutlerInterpolation vs. Approximation CurvesInterpolationcurve must pass through control pointsApproximationcurve is influenced by control pointsMIT EECS 6.837, Durand and CutlerInterpolation vs. Approximation Curves•Interpolation Curve – over constrained → lots of (undesirable?) oscillations•Approximation Curve – more reasonable?MIT EECS 6.837, Durand and CutlerCubic Bézier Curve•4 control points•Curve passes through first & last control point•Curve is tangent at P0 to (P0-P1) and at P4 to (P4-P3)A Bézier curve is bounded by the convex hull of its control points.MIT EECS 6.837, Durand and CutlerCubic Bézier Curve•de Casteljau's algorithm for constructing Bézier curvesttttttMIT EECS 6.837, Durand and CutlerCubic Bézier CurveBernstein PolynomialsMIT EECS 6.837, Durand and CutlerConnecting Cubic Bézier CurvesAsymmetric: Curve goes through some control points but misses others•How can we guarantee C0 continuity?•How can we guarantee G1 continuity? •How can we guarantee C1 continuity?•Can’t guarantee higher C2 or higher continuityMIT EECS 6.837, Durand and CutlerConnecting Cubic Bézier Curves•Where is this curve–C0 continuous?–G1 continuous?–C1 continuous?•What’s the relationship between: –the # of control points, and –the # of cubic Bézier subcurves?MIT EECS 6.837, Durand and CutlerHigher-Order Bézier Curves•> 4 control points•Bernstein Polynomials as the basis functions•Every control point affects the entire curve –Not simply a local effect –More difficult to control for modelingMIT EECS 6.837, Durand and CutlerCubic BSplines•≥ 4 control points•Locally cubic•Curve is not constrained to pass through any control pointsA BSpline curve is also bounded by the convex hull of its control points.MIT EECS 6.837, Durand and CutlerCubic BSplines•Iterative method for constructing BSplinesShirley, Fundamentals of Computer GraphicsMIT EECS 6.837, Durand and CutlerCubic BSplinesMIT EECS 6.837, Durand and CutlerCubic BSplines•Can be chained together•Better control locally (windowing)MIT EECS 6.837, Durand and CutlerConnecting Cubic BSpline Curves•What’s the relationship between –the # of control points, and –the # of cubic BSpline subcurves?MIT EECS 6.837, Durand and CutlerBSpline Curve Control PointsDefault BSplineBSpline with


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MIT 6 837 - Curves & Surfaces

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