knot intervals and T-splines Thomas W. Sederberg Minho Kim

knot intervals

representation equivalent to a knot vector without knot origin ▲ geometrically intuitive (especially for periodic case) ▼ different representation for odd and even degree ▼ unintuitive phantom vertices and edges due to end condition

example: odd degree knot vector = [1,2,3,4,6,9,10,11] knot intervals = [1,1,1,2,3,1,1]

example: even degree knot vector = [1,2,3,5,7,8,9,12,14,17] knot intervals = [1,1,2,2,1,1,3,2,3] P(1,2,3,5) P(2,3,5,7) P(3,5,7,8) P(5,7,8,9) P(7,8,9,12)P(8,9,12,14) P(9,12,14,15) P0P0 P1P1 P2P2 P3P3 P4P4 P5P5 P6P6 d 0 =1 d 1 =2 d 2 =2 d 3 =1 d 4 =1 d 5 =3 d 6 =2 d 7 =3 d -1 =1 t=5 t=7 t=8 t=9 t=5 t=7 t=8 t=9

example: non-uniform & multiple knots varying knot intervals multiple knots = empty knot intervals

example: knot insertion example: knot insertion in d 1

knot insertion Wolfgang Böhm –from “Handbook of CAGD,” p.156

T-spline

PB-spline Point Based spline linear combination of blending functions at points arbitrarily located at least three blending functions need to overlap in the domain to define a surface

T-spline splines on T-mesh where T-junctions are allowed based on PB-spline imposes knot coordinates based on knot intervals and connectivity less control points due to T-junctions

T-spline (cont’d) questions –Are the blending functions basis functions? (Are they linearly independent?) –Do they form a partition of unity? –Is it guaranteed that at least three blending functions are defined at every point of the domain?

T-spline vs. NURBS

T-spline vs. NURBS (cont’d) T-spline NURBS knot insertion (lossless) T-spline simplification (lossy)

T-NURCC NURCC with T-junctions –NURCC (Non-Uniform Rational Catmull-Clark surfaces): generalization of CC to non-uniform B-spline surfaces local refinement in the neighborhood of an extraordinary point

references [1] T. W. Sederberg, J. Zheng, D. Sewell and M. Sabin,"Non- uniform Subdivision Surfaces," SIGGRAPH [2] G. Farin, J. Hoschek and M.-S. Kim, (ed.) "Handbook of CAGD," North-Holland, [3] T. W. Sederberg, Jianmin Zheng, Almaz Bakenov, and Ahmad Nasri, "T-splines and T-NURCCS," SIGGRAPH 2003 [4] T. W. Sederberg, Jianmin Zheng and Xiaowen Song, "Knot intervals and multi-degree splines," Computer Aided Geometric Design,20, 7, , [5] T. W. Sederberg, D. L. Cardon, G. T., Finnigan, N. S. North, J. Zheng, and T. Lyche, "T-spline Simplification and Local Refinement," SIGGRAPH [6] T-Splines, LLC: