knot intervals and T-splines Thomas W. Sederberg

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knot intervals and T-splinesThomas W. Sederberg

• Minho Kim

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knot intervals
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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

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example odd degree

• knot vector 1,2,3,4,6,9,10,11
• knot intervals 1,1,1,2,3,1,1

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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)
d12
P(3,5,7,8)
d22
P1
P2
d01
d-11
P(1,2,3,5)
P0
t5
t5
P(5,7,8,9)
t7
t7
P3
d31
P(9,12,14,15)
P6
d73
t9
t9
t8
t8
P4
d62
P5
d41
P(7,8,9,12)
P(8,9,12,14)
d53
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examplenon-uniform multiple knots

• varying knot intervals
• multiple knots empty knot intervals

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example knot insertion

• example knot insertion in d1

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knot insertion

• Wolfgang Böhm
• from Handbook of CAGD, p.156

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T-spline
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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

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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

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T-spline (contd)

• 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?

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T-spline vs. NURBS
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T-spline vs. NURBS (contd)
T-spline
knot insertion (lossless)
T-spline simplification (lossy)
NURBS
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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

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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