1. Anupam Saxena Associate Professor Indian Institute of Technology KANPUR 208016

2. Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformati ons and Projections 1-2 Modeling of Curves Representati on, Differential Geometry Ferguson Segments Bezier Segments 1- 2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite NURBS

4. Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1- 2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1- 5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite NURBS Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformati ons and Projections 1-2 Modeling of Curves Representati on, Differential Geometry Ferguson Segments Bezier Segments 1- 2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite NURBS

5. Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformati ons and Projections 1-2 Modeling of Curves Representati on, Differential Geometry Ferguson Segments Bezier Segments 1- 2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite NURBS

6. Given n+1 data points p 0, p 1,..., p n fit them with a B-spline curve of given order p  n a set of parameters u 0, u 1,..., u n may be generated the number of knots m + 1 may be computed knot vector [t 0, t 1, …, t m ] may then be computed Basis functions known n + 1 conditions

7. Control points b i ’s are (n+1) unknowns Consider p k = b(u k ) = p0p1p2…pnp0p1p2…pn N p,p (u 0 )N p,p+1 (u 0 ) N p,p+2 (u 0 )… N p,n+p (u 0 ) N p,p (u 1 )N p,p+1 (u 1 ) N p,p+2 (u 1 )… N p,n+p (u 1 ) N p,p (u 2 )N p,p+1 (u 2 ) N p,p+2 (u 2 )… N p,n+p (u 2 ) …… ……… N p,p (u n )N p,p+1 (u n ) N p,p+2 (u n )… N p,n+p (u n ) P == b0b1b2…bnb0b1b2…bn = NB k = 0, …, n

9. Interpolate data points, (0, 0), (0, 1), (2, 3), (2.5, 6), (5, 2), (6, 0) and (7,  3), using a B-spline curve with piecewise cubic polynomial segments Compute distances between successive data points: d 1 =  (1 2 + 0 2 ) = 1 d 2 =  (2 2 + 2 2 ) = 2.83 d 3 =  (0.5 2 + 3 2 ) = 3.04 d 4 =  (2.5 2 + 4 2 ) = 4.72 d 5 =  (1 2 + 2 2 ) = 2.24 d 6 =  (1 2 + 3 2 ) = 3.16 Sum of distances L = 17 Set u 0 = 0 u 1 = u 0 + d 1 /L = 0.058u 4 = u 3 + d 4 /L = 0.682 u 2 = u 1 + d 2 /L = 0.225u 5 = u 4 + d 5 /L = 0.814 u 3 = u 2 + d 3 /L = 0.404u 6 = u 5 + d 6 /L = 1.000 [t 0, …, t m ]  [–2, –1, 0, 0.058, 0.225, 0.404, 0.682, 0.814, 1, 2, 3] 0.69–5.90 –0.201.74 2.802.38 1.928.32 4.772.34 6.13–0.10 8.86–8.46 B =

10. -2 10 -10 10 Plot for t  [0, 1) Interval of full support [0.058, 0.814)

11. Short for Non-Uniform Rational B-Splines Recall from Rational Bézier curves that Likewise, NURBS can be computed as weights w i specified by the user to gain additional design freedom non-uniform: knots are not placed at regular intervals w i = 0: location of b i does not affect the curve’s shape For larger values of w i, the curve gets pushed towards b i Offer great flexibility in design Possess local shape control & all other Properties of B-spline curves Widely used in freeform curve design Can also model analytical curves

13. For data points (0, 0), (0, 1), (2, 3), (2.5, 6), (5, 2), (6, 0) and (7,  3), design NURBS with basis functions of order 4. First set all weights to 1. Increase the weight w 3 corresponding to (2.5, 6) to visualize the shape change 0 8 -4 6 w 3 = 0 10 1 2 [0, 1, 2, …, 10) Knot vector

14. 0 8 -4 6 w 3 = 0 10 1 2 [0, 3, 3, 3, 4, 5, 6, 7, 7, 7, 10) Knot vector

15. For an order p curve … Repeat the first knot ‘0’ p times Repeat the last knot ‘1’ p times Consider n + 1 = p B-spline basis functions/ Control points  number of knots (m + 1); m = n + p = p + p = 2p