Computer Aided Engineering Design

1. Computer Aided Engineering Design AnupamSaxena Associate Professor Indian Institute of Technology KANPUR 208016

3. Normalized B-splines More popular Nk,i(t) = (titi k)Mk,i(t)

4. Normalized B-splines N1,i(t) = i such that i = 1 for t [ti1, ti) = 0, elsewhere

5. Lecture #26 B-Spline basis functionsProperties of the Normalized B-Splines

6. Properties of Normalized B-splines Nk,i(t) is a degree k1 polynomial in t Non-negativity: For all i, k and t, Nk, i(t) is non-negative In a given knot span tik < tik +1 < …< ti N1,i(t) = 1, for t [ti 1, ti); = 0, elsewhere N1,i(t)  0 in [ti k, ti) • N1, i1(t) = 1, for t [ti 2, ti 1); = 0, elsewhere N1, i2(t) = 1, for t [ti 3, ti 2); = 0, elsewhere N1, i1(t)  0 and N1, i2(t) 0 in [ti k, ti)

7. Properties of Normalized B-splines t [ti 2, ti) and = 0, elsewhere for t [ti 2, ti 1) for t [ti 1, ti) N2,i(t)  0 for t in [ti2, ti) Perform induction to prove for Nk,i(t)

8. Example N4,4(t) N1,1(t) N1,2(t) N1,3(t) N1,4(t) N2,3(t) N3,3(t) N2,2(t) N3,3(t) N3,4(t) 0 1 2 3 4

9. ti-4 ti-3 ti-2 ti-1 ti ti+1 ti+2 ti+3 ti+4 t Properties of Normalized B-splines Nk,i(t) is a non-zero polynomial in (tik,ti) On any span [ti, ti+1), at most p order p normalized B-Splines are non-zero provides local control for B-spline curves N4,i+2(t) N4,i+3(t) N4,i+1(t) N4,i+4(t) If [ti, ti+1) is contained in [trp,tr), there should be one order p B-spline with tias the first knot and one with ti+1 as last knot For any r, Np,r(t) ≥ 0 in the knot span [trp,tr) rp = i and r = i+1 provide the range r = i+1, …, i+p p splines

10. ti-4 ti-3 ti-2 ti-1 ti ti+1 ti+2 ti+3 ti+4 Properties of Normalized B-splines • Partition of Unity: The sum of all non-zero order pbasis functions over • the span [ti, ti+1) is 1 N4,i+2(t) N4,i+3(t) N4,i+1(t) N4,i+4(t) t B-spline basis functions add to unity within a subgroup Not all B-spline basis functions add to one as opposed to Bernstein polynomials

11. Properties of Normalized B-splines For number of knots as m+1 and the number of degree p–1basis functions as n+1, m = n + p The first normalized spline on the knot set [t0, tm) is Np,p(t) the last spline on this set is Np,m(t) m p+1 basis splines n+1 = m p+1 Multiple knots If a knot ti appears k times (i.e., tik+1 = tik+2 = ... = ti), where k > 1, tiis termed as a multiple knot or knot of multiplicity k for k = 1, ti 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

12. 1 0.8 N3,i(t) 0.6 0.4 0.2 0 1 2 3 4 Properties of Normalized B-splines • At a knot i of multiplicity k, the basis function Npi(t) is Cp 1 k • continuous at that knot

13. 1 0.8 N3,i(t) 0.6 0.4 0.2 0 1 2 3 4 Properties of Normalized B-splines Symmetricity is maintained when knots are moved to the left

14. ti-7 ti-6 ti-5 ti-4 ti-3 ti-2 ti-1 Properties of Normalized B-splines • At each internal knot of multiplicity k, the number of non-zero order • p basis functions is at most p k N4,i-3 N4,i-2 N4,i-1 non-zero splines over a simple knot ti4 p  k = 4  1 = 3 non-zero splines over a double knot ti4 p  k = 4  2 = 2 p  k = 4  3 = 1 non-zero splines over a triple knot ti4 p  k = 4  4 = 0 non-zero splines over a quadruple knot ti4