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lecnotes6 - 13.472J/1.128J/2.158J/16.940J COMPUTATIONAL...

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Unformatted text preview: 13.472J/1.128J/2.158J/16.940J COMPUTATIONAL GEOMETRY Lecture 6 N. M. Patrikalakis Massachusetts Institute of Technology Cambridge, MA 02139-4307, USA Copyright c 2003 Massachusetts Institute of Technology Contents 6 B-splines (Uniform and Non-uniform) 2 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 6.2.1 Knot vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 6.2.2 Properties and definition of basis function N i,k ( u ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 6.2.3 Example: 2 nd order B-spline basis function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 6.2.4 Example: 3 rd order B-spline basis function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 6.2.5 Example: 4 th order basis function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 6.2.6 Special case n = k- 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 6.3 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 6.4 Approaches to design with B-spline curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 6.5 Interpolation of data points with cubic B-splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 6.6 Evaluation and subdivision of B-splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6.6.1 De Boor algorithm for B-spline curve evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6.6.2 Knot insertion: Boehms algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 6.7 Tensor product piecewise polynomial surface patches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 6.7.1 Example: B ezier patch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 6.7.2 Example: B-Spline patch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6.7.3 Example: Composite B ezier patches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 6.8 Generalization of B-splines to NURBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6.8.1 Curves and Surface Patches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6.8.2 Example: Representation of a quarter circle as a rational polynomial . . . . . . . . . . . . . . . . . . . . . . 20 6.8.3 Trimmed patches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Trimmed patches ....
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