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
• ▼ 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(2,3,5,7)
d1=2
P(3,5,7,8)
d2=2
P1
P(1,2,3,5)
d-1=1
t=5
t=7
P(9,12,14,15)
t=9
d0=1
P0 t=5
P(5,7,8,9)
t=8
t=7
d7=3
P6
t=9
d6=2
P(7,8,9,12)
P3
t=8
P4
P5
P(8,9,12,14)
P2
d5=3
d4=1
d3=1
example:
non-uniform & multiple knots
• varying knot intervals
• multiple knots = empty knot intervals
example: knot insertion
• example: knot insertion in d1
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
knot insertion
(lossless)
T-spline simplification
(lossy)
NURBS
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,"Nonuniform 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