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Anupam Saxena Associate Professor Indian Institute of Technology KANPUR 208016

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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

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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

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Given n+1 control points b 0, b 1,..., b n, a knot vector T = {t 0, t 1,..., t m } the B-spline curve, b(t) of order p t0t0 t1t1 t2t2 … tptp tmtm tmptmp … t m p+1

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– very similar to a Bézier curve wherein the basis functions are the Bernstein polynomials – In a Bézier curve, the degree of Bernstein basis functions is the same as the number of control points – For B-spline curves, the degree of the basis functions is an independent choice specified by the user – The number of knots (m+1) get determined by the relation, m = n + p - the total number of basis functions (n+1) = number of control points - P is the order of the B-spline basis functions and hence the curve

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B-spline curve definition is valid for all t in [ , ], b(t) = 0 for t t 0 and t > t m range [t 0, t m ] seems reasonable More restrictive range: in which full support of the basis functions is achieved i.e., over any knot span [t j, t j+1 ) in [t 0, t m ], all p B-splines of order p are non-zero first such span is [t p-1, t p ) where N p,p, …, N p,2p-1 are non-zero last span is [t m-p, t m-p+1 ) where basis functions N p,m-p+1, …, N p,m are non-zero range of definition of a B-spline curve [t p-1, t m-p+1 ) wherein for any t all p basis functions are non-zero t0t0 t1t1 t2t2 … tptp tmtm tmptmp … t m p+1 t p-1 First span to include all p basis functions Last span to include all p basis functions

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[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10) 0 8 6 X(t) Y(t) For data points, (0, 0), (0, 1), (2, 3), (2.5, 6), (5, 2), (6, 0) and (7, 3), design a B-spline curve Number of data points Order of basis function : 7 : 4 Number of knots: 7 + 4 = 11 Our choice to specify knots [0, 3, 3, 3, 4, 5, 6, 7, 8, 9, 10) [0, 3, 3, 3, 4, 5, 6, 7, 7, 7, 10) 3 7 3

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Strong Convex Hull Property: The B-spline curve, b(t) is contained in the convex hull defined by the polyline, [b j, b j+1,..., b j+p-1 ] for t in [t j+p-1, t j+p ). This convex hull is the subset of the parent hull [b 0, b 1,..., b n ] A B-spline curve is a piecewise curve with each component an order p segment Equality m = n + p must be satisfied t0t0 t1t1 tjtj t j+p tmtm tmptmp t m p+1 t j+p-1 t j+2p-1 … bjbj b j+p-1 p basis splines are barycentric Hence, the property

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08 -4 6 3 t < 4 4 t < 5 5 t < 6 6 t < 7 Actual Convex hull

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b(t) is C p-k-1 continuous at a knot of multiplicity k At t = t i, a knot of multiplicity k, since N p,i (t) is C p-k-1 continuous, so is b(t) at that knot Local Modification Scheme: Relocating b i only affects the curve b(t) in the interval [t i, t i+p ) Let the control point b i be moved to a new position b i + v N p,i+p (t) is non-zero in [t i, t i+p )for t [t i, t i+p ), b(t) gets locally modified

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0 8 -4 6

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Anupam Saxena Associate Professor Indian Institute of Technology KANPUR 208016

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Shape manipulation can be done using Control points knots Use strong convex hull property Use knot multiplicity Changing knot positions can also cause shape change but Is non-intuitive and thus avoided

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Recall… Strong Convex Hull Property: The B-spline curve, b(t) is contained in the convex hull defined by the polyline, [b j, b j+1,..., b j+p-1 ] for t in [t j+p-1, t j+p ). This convex hull is the subset of the parent hull [b 0, b 1,..., b n ] WHAT if b i, b i+1,..., b i+p-1, all are in a straight line ? the curve segment lying their convex hull for t in [t i+p-1, t i+p ) will be a straight line What if p 1 of these control points are identical, say, b i = b i+1 =... b i+p-2 ? the convex hull degenerates to a line segment b i b i+p-1 and the curve passes through b i What if b i-1, b i = b i+1 =... b i+p-2 and b i+p-1 are collinear ? the line segment b i-1 b i+p-1 is tangent to the curve at b i

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08 -4 6 246 -2 0 2 4

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08 -4 6 246 -2 0 2 4

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08 -4 6 246 -2 0 2 4

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08 -4 6 246 -2 0 2 4

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02468 -6 -4 -2 0 2 4 6

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Changing knot positions can also cause shape change but is non-intuitive and thus NOT RECOMMENDED Change in curve’s shape using knot multiplicity is predictable Recall At a knot i of multiplicity k, the basis function N p i (t) is C p 1 k continuous at that knot At each internal knot of multiplicity k, the number of non-zero order p basis functions is at most p k

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t0t0 t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 t7t7 t8t8 t9t9 t 10 N 4,4 N 4,5 N 4,6 N 4,7 N 4,8 N 4,9 N 4,10 At t = t 5, N 4,9 = 0N 4,6 (t 5 ) + N 4,7 (t 5 ) + N 4,8 (t 5 ) = 1

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t0t0 t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 t7t7 t8t8 t9t9 t 10 N 4,6 N 4,7 N 4,8 N 4,9 For t 4 = t 5, N 4,8 = 0N 4,6 (t 5 ) + N 4,7 (t 5 ) = 1

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t0t0 t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 t7t7 t8t8 t9t9 t 10 N 4,6 N 4,7 N 4,8 N 4,9 For t 3 = t 4 = t 5, N 4,7 = 0N 4,6 (t 5 ) = 1 Control point associated with N 4,6 (t) is b 2 : the spline passes through b 2

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if t i+1 = t i+2 =,…, = t i+p-1 with t i+p-1 having multiplicity p–1 only one basis function N p,i+p will be non-zero over t i+p-1 from the barycentric property, N p,i+p will be 1 the B-spline curve will pass through b i

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02468 -6 -4 -2 0 2 4 6 t i = i, i = 0, …, 15 range of full support [3, 12) For spline to pass through (0, 0), the knot t 4+4-1 = t 7 = 7 should have multiplicity 3. t 5 = t 6 = t 7 = 7 For spline to pass through (8, 0), the knot t 7+4-1 = t 10 = 10 should have multiplicity 3. t 8 = t 9 = t 10 = 10

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