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

2. 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

3. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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