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De Boor Suitable T-Splines 1 Goal: to develop a de Boor evaluation algorithm for T-spline surfaces. Yang Zhang | T-spline - De Boor Algorithm

De Boor Suitable T-Splines De Boor Algorithm Review 2 Yang Zhang | T-spline - De Boor Algorithm

De Boor Suitable T-Splines De Boor Algorithm Review 3 Yang Zhang | T-spline - De Boor Algorithm

De Boor Suitable T-Splines A few assumptions 4 ◮ We talk only about Bidegree ( 3 , 3 ) . ◮ T-junctions only, NO L-junctions and NO I-junctions. Yang Zhang | T-spline - De Boor Algorithm

De Boor Suitable T-Splines Admissible T-meshes 5 Admissible T-meshes Yang Zhang | T-spline - De Boor Algorithm

De Boor Suitable T-Splines Admissible T-meshes 6 ◮ No facing T-junctions in the active region ◮ No T-junctions in the frame region Yang Zhang | T-spline - De Boor Algorithm

De Boor Suitable T-Splines Admissible T-meshes 7 Finding the local knots for an anchor: Yang Zhang | T-spline - De Boor Algorithm

De Boor Suitable T-Splines Analysis Suitable T-meshes 8 Analysis Suitable T-meshes Yang Zhang | T-spline - De Boor Algorithm

De Boor Suitable T-Splines Analysis Suitable T-meshes 9 T-junction extensions: Yang Zhang | T-spline - De Boor Algorithm

De Boor Suitable T-Splines Analysis Suitable T-meshes 10 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

De Boor Suitable T-Splines T-spline Surfaces 11 T-spline Surfaces Yang Zhang | T-spline - De Boor Algorithm

De Boor Suitable T-Splines T-spline Surfaces 12 ◮ Associate to each anchor ν = ( i 3 , j 3 ) a blending function: B ν ( t , s ) = N 3 ( i 1 , i 2 , i 3 , i 4 , i 5 ; t ) ∗ N 3 ( j 1 , j 2 , j 3 , j 4 , j 5 ; s ) ◮ For analysis suitable T-splines, the blending functions are linearly independent and form a partition of unity. ◮ The surface is: � P ν ∗ B ν ( s , t ) T ( s , t ) = ν Yang Zhang | T-spline - De Boor Algorithm

De Boor Suitable T-Splines Two Questions 13 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

De Boor Suitable T-Splines How to find the indices for the control points? 14 Yang Zhang | T-spline - De Boor Algorithm

De Boor Suitable T-Splines How to find the indices for the control points? 15 Yang Zhang | T-spline - De Boor Algorithm

De Boor Suitable T-Splines How to find the indices for the control points? 16 Yang Zhang | T-spline - De Boor Algorithm

De Boor Suitable T-Splines T-junction boxes 17 de Boor algorithm applies: Yang Zhang | T-spline - De Boor Algorithm

De Boor Suitable T-Splines T-junction boxes 18 de Boor algorithm does not apply: Yang Zhang | T-spline - De Boor Algorithm

De Boor Suitable T-Splines Theorem 19 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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