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De Boor Suitable T-Splines
Goal: to develop a de Boor evaluation algorithm for T-spline surfaces.
Yang Zhang | T-spline - De Boor Algorithm
r t ss - - PowerPoint PPT Presentation
r t ss - - PowerPoint PPT Presentation
rtt rst r t ss st Ptrtr
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De Boor Suitable T-Splines
De Boor Algorithm Review
Yang Zhang | T-spline - De Boor Algorithm
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De Boor Suitable T-Splines
De Boor Algorithm Review
Yang Zhang | T-spline - De Boor Algorithm
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De Boor Suitable T-Splines
A few assumptions
◮ We talk only about Bidegree (3, 3). ◮ T-junctions only, NO L-junctions and NO I-junctions.
Yang Zhang | T-spline - De Boor Algorithm
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De Boor Suitable T-Splines
Admissible T-meshes
Admissible T-meshes
Yang Zhang | T-spline - De Boor Algorithm
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De Boor Suitable T-Splines
Admissible T-meshes
◮ No facing T-junctions in the active region ◮ No T-junctions in the frame region
Yang Zhang | T-spline - De Boor Algorithm
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De Boor Suitable T-Splines
Admissible T-meshes
Finding the local knots for an anchor:
Yang Zhang | T-spline - De Boor Algorithm
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De Boor Suitable T-Splines
Analysis Suitable T-meshes
Analysis Suitable T-meshes
Yang Zhang | T-spline - De Boor Algorithm
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De Boor Suitable T-Splines
Analysis Suitable T-meshes
T-junction extensions:
Yang Zhang | T-spline - De Boor Algorithm
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De Boor Suitable T-Splines
Analysis Suitable T-meshes
A T-mesh is analysis suitable if the horizontal T-junction extensions do not intersect the vertical T-junction extensions. No touching allowed.
Yang Zhang | T-spline - De Boor Algorithm
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De Boor Suitable T-Splines
T-spline Surfaces
T-spline Surfaces
Yang Zhang | T-spline - De Boor Algorithm
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De Boor Suitable T-Splines
T-spline Surfaces
◮ Associate to each anchor ν = (i3, j3) a blending function:
Bν(t, s) = N3(i1, i2, i3, i4, i5; t) ∗ N3(j1, j2, j3, j4, j5; s)
◮ For analysis suitable T-splines, the blending functions are linearly
independent and form a partition of unity.
◮ The surface is:
T(s, t) =
- ν
Pν ∗ Bν(s, t)
Yang Zhang | T-spline - De Boor Algorithm
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De Boor Suitable T-Splines
Two Questions
Two questions:
- 1. How to find the indices for the control points?
- 2. How to construct the de Boor algorithm for the control points?
Yang Zhang | T-spline - De Boor Algorithm
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De Boor Suitable T-Splines
How to find the indices for the control points?
Yang Zhang | T-spline - De Boor Algorithm
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De Boor Suitable T-Splines
How to find the indices for the control points?
Yang Zhang | T-spline - De Boor Algorithm
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De Boor Suitable T-Splines
How to find the indices for the control points?
Yang Zhang | T-spline - De Boor Algorithm
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De Boor Suitable T-Splines
T-junction boxes
de Boor algorithm applies:
Yang Zhang | T-spline - De Boor Algorithm
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De Boor Suitable T-Splines
T-junction boxes
de Boor algorithm does not apply:
Yang Zhang | T-spline - De Boor Algorithm
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De Boor Suitable T-Splines
Theorem
Theorem:
- 1. If the 16 anchors can be covered by 4 lines, then the de Boor
algorithm can be applied by first blending along these 4 lines.
- 2. For Analysis Suitable T-meshes, if the horizontal T-junction boxes
do not overlap the vertical T-junction boxes, then the T-mesh are de Boor Suitable and the de Boor algorithm can always be applied.
Yang Zhang | T-spline - De Boor Algorithm
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