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

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Anupam Saxena Associate Professor Indian Institute of Technology KANPUR 208016

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

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

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

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

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

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 =

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

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

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

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

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

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