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knot intervals and T-splines Thomas W. Sederberg

knot intervals and T-splines Thomas W. Sederberg. Minho Kim. knot intervals. knot intervals. representation equivalent to a knot vector without knot origin ▲ geometrically intuitive (especially for periodic case) ▼ different representation for odd and even degree

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knot intervals and T-splines Thomas W. Sederberg

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  1. knot intervals and T-splinesThomas W. Sederberg Minho Kim

  2. knot intervals

  3. 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

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

  5. 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(2,3,5,7) d1=2 P(3,5,7,8) d2=2 P1 P2 d-1=1 d0=1 P(1,2,3,5) t=5 P0 t=5 P(5,7,8,9) t=7 t=7 P3 d3=1 P(9,12,14,15) t=9 d7=3 P6 t=9 t=8 t=8 d6=2 P4 P5 d4=1 P(8,9,12,14) P(7,8,9,12) d5=3

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

  7. example: knot insertion • example: knot insertion in d1

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

  9. T-spline

  10. 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

  11. 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

  12. 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?

  13. T-spline vs. NURBS

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

  15. 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

  16. references [1] T. W. Sederberg, J. Zheng, D. Sewell and M. Sabin,"Non-uniform Subdivision Surfaces," SIGGRAPH 1998. [2] G. Farin, J. Hoschek and M.-S. Kim, (ed.) "Handbook of CAGD," North-Holland, 2002. [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, 455-468, 2003. [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 2004. [6] T-Splines, LLC: http://www.tsplines.com

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