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Anupam Saxena Associate Professor Indian Institute of Technology KANPUR 208016

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Partition of Unity: The sum of all non-zero order p basis functions over the span [t i, t i+1 ) is 1 titi t i+1 t i+2 t i+3 t i+4 t i-1 t i-2 t i-3 t i-4 t N 4,i+1 (t) N 4,i+2 (t)N 4,i+3 (t) N 4,i+4 (t) B-spline basis functions within a subgroup add to unity Not all B-spline basis functions add to one as opposed to Bernstein polynomials

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Partition of Unity: The sum of all non-zero order k basis functions over the span [t i, t i+1 ) is 1 GENERAL PROOF

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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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For number of knots as m+1 and the number of degree p–1 basis functions as n+1, m = n + p The first normalized spline on the knot set [t 0, t m ) is N p,p (t) the last spline on this set is N p,m (t) m p+1 basis splinesn+1 = m p+1

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If a knot t i appears k times (i.e., t i k+1 = t i k+2 =... = t i ), where k > 1, t i is termed as a multiple knot or knot of multiplicity k for k = 1, t i is termed as a simple knot Multiple knots can significantly change the properties of basis functions and are useful in the design of B-spline curves

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01234 1 0.8 0.6 0.4 0.2 N 3,i (t) At a knot i of multiplicity k, the basis function N p i (t) is C p 1 k continuous at that knot

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01234 1 0.8 0.6 0.4 0.2 N 3,i (t) The effect is similar when knots are moved to the left

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At each internal knot of multiplicity k, the number of non-zero order p basis functions is at most p k N 4,i-1 t i-4 t i-1 t i-5 t i-2 t i-6 t i-3 t i-7 N 4,i-2 N 4,i-3 non-zero splines over a simple knot t i 4 p k = 4 1 = 3 non-zero splines over a double knot t i 4 p k = 4 2 = 2 non-zero splines over a triple knot t i 4 p k = 4 3 = 1 non-zero splines over a quadruple knot t i 4 p k = 4 4 = 0

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