CS-184: Computer GraphicsLecture #14: Subdivision Prof. James O’BrienUniversity of California, BerkeleyV2007-F-14-1.01SubdivisionStart with:Given control points for a curve or surface, find new control points for a sub-section of curve/surfaceKey extension to basic idea:Generalize to non-regular surfaces22Consider NURBS Surface3Control mesh dictates feature size. Coarse meshDisplaced CVLarge bump3Consider NURBS Surface4Control mesh dictates feature size. Fine meshDisplaced CVSmall bumpExcessive detail4Tensor Product Surface Refinement 5Refinement must be constant across u or v directions5Bézier Subdivision6u ∈ [0..1]x(u) =∑ibi(u)pix(u) = [1, u, u2, u3]βzPβz=1 0 0 0−3 3 0 03 −6 3 0−1 3 −3 1Vector of control points6Bézier Subdivision7x(u) = [1, u, u2, u3]βzPβz=1 0 0 0−3 3 0 03 −6 3 0−1 3 −3 1u ∈ [0..1/2]u ∈ [1/2..1]7Bézier Subdivision8x(u) = [1, u, u2, u3]βzPβz=1 0 0 0−3 3 0 03 −6 3 0−1 3 −3 1u ∈ [0..1]u ∈ [0..1]Can’t change these....8Bézier Subdivision9x(u) = [1, u, u2, u3]βzPS1=1 0 0 00 1/2 0 00 0 1/4 00 0 0 1/8u ∈ [0..12]u ∈ [0..1]x(u) = [1,u2,u24,u38]βzPx(u) = [1, u, u2, u3]S1βzPx(u) = [1, u, u2, u3]βzβ−1zS1βzPx(u) = [1, u, u2, u3]βzHz1P9Bézier Subdivision10x(u) = [1, u, u2, u3]βzHz1Px(u) = [1, u, u2, u3]βzP1P1= Hz1PHz1=1 0 0 012120 014121401838381810x(u) = [1, u, u2, u3]βzP2Bézier Subdivision11S2=112141801212380 014380 0 018Hz2=1838381801412140 012120 0 0 1P2= Hz2P11Bézier Subdivision12x(u, v) = [1, u, u2, u3]βzPβTz[1, v, v2, v3]T4 x 4 matrix of control pointsP2·= HZ2P12Bézier Subdivision13x(u, v) = [1, u, u2, u3]βzPβTz[1, v, v2, v3]T4 x 4 matrix of control pointsP21= HZ2P HTZ113Regular B-Spline Subdivision14Orthographic top-down view3D Perspective view14Regular B-Spline Subdivision15Orthographic top-down view3D Perspective view15Regular B-Spline Subdivision16x(u, v) = [1, u, u2, u3]βBP βTB[1, v, v2, v3]TP11= HB1P HTB116Regular B-Spline Subdivision17P11= HB1P HTB1In this parametric view these knot points are collocated.The 3D control points are not.17Regular B-Spline Subdivision18P11= HB1P HTB1P12= HB1P HTB218Regular B-Spline Subdivision19P11= HB1P HTB1P12= HB1P HTB2P22= HB2P HTB219Regular B-Spline SubdivisionP11= HB1P HTB1P12= HB1P HTB2P22= HB2P HTB2P21= HB2P HTB1HB1=12120 018341800121200183418HB2=183418001212001834180 0121220Regular B-Spline SubdivisionLength 16 vector of coarse CPsLength 25 vector of fine CPs25 x 16 subdivision matrixPi+1= H Pi21Regular B-Spline SubdivisionPi+1= H PiInspection would reveal a pattern• Face points• Edge points• Vertex points22Regular B-Spline SubdivisionFace pointf =v1+ v2+ v3+ v44v1v2v3v4fv1v2f1f2ee =v1+ v2+ f1+ f24Edge pointf1f2f3f4pvm1m2m3m4Vertex pointv =f1+ f2+ f3+ f4+ 2(m1+ m2+ m3+ m4) + 4p16mmidpoint of edge, not “edge point”pold “vertex point”23Regular B-Spline SubdivisionRecall that control mesh approaches surface24Regular B-Spline SubdivisionLimit of subdivision is the surface25Catmull-Clark SubdivisionGeneralizes regular B-Spine subdivisionIrregular B-Spline SubdivisionAn irregular patchNon-quad faceExtraordinary vertex26Catmull-Clark SubdivisionGeneralizes regular B-Spine subdivisionRules reduce to regular for ordinary vertices/facesIrregular B-Spline Subdivisionf = average of surrounding verticese =f1+ f2+ v1+ v24v =¯fn+2 ¯mn+p(n − 3)n¯f = average of adjacent face points¯m = average of adjacent midpointsn = valence of vertex27Catmull-Clark Subdivision2828Catmull-Clark Subdivision2929Catmull-Clark Subdivision3030Catmull-Clark Subdivision3131Catmull-Clark Subdivision3232Catmull-Clark Subdivision3333Catmull-Clark Subdivision3434Catmull-Clark
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