Computer Aided Engineering Design
SPLINES 143 It is required to show that (a) ψ(t) is a quadratic spline, (b) ψ(t) = 0 for t < t 0 and t > t 3 , (c) ψ(t) is ...
144 COMPUTER AIDED ENGINEERING DESIGN =f[t 0 ]g[t 0 ,t 1 ,... , tr] + f[t 0 ,t 1 ]g[t 1 ,... , tk] +... + f[t 0 ,t 1 ,... , tk–1 ...
SPLINES 145 t t ii t t i i i i i i i MtdtMt dtMtt t –1 –1 ∫∫1, () = 1, () = 1,()( – –1) = 1 or Mti tt i i 1, –1 () =^1 for t∈ [ ...
146 COMPUTER AIDED ENGINEERING DESIGN ⇒ Nt tt tt tt Nt tt tt tt Nt tt ki i ik ik i ik ki i ik i i ik ki i ik , –1, – ...
SPLINES 147 thus, N2,i(t)≥ 0 for t in [ti–k,ti) (5.30) Likewise, N2,i–1(t)≥ 0 for t in [ti–k,ti) (5.31) Next, assume that Eqs. ( ...
148 COMPUTER AIDED ENGINEERING DESIGN (E) Partition of unity*: The sum of all non-zero order p B-spline functions over the span ...
SPLINES 149 = –3 –1 –3 –3 –2 –3 –2 –1 –1 – 2 –1 –2 –2 –1 – 2 tt tt tt tt tt tt tt tt tt ...
150 COMPUTER AIDED ENGINEERING DESIGN of knots. The first normalized spline on the knot set [t 0 ,tm) is Np,p(t) while the last ...
SPLINES 151 (Figure 5.14b) making the latter of multiplicity k=2, the total number of non-zero B-splines over ti–4 gets reduced ...
152 COMPUTER AIDED ENGINEERING DESIGN Example 5.7 discusses two kinds of B-spline curves, viz. unclampedandclamped. In the forme ...
SPLINES 153 (C) Strong convex hull property: The B-spline curve, b(t) is contained in the convex hull defined by the polyline [b ...
154 COMPUTER AIDED ENGINEERING DESIGN (D) b(t) is Cp-k–^1 continuous at a knot of multiplicity k Ift = ti, a knot of multiplicit ...
SPLINES 155 5.9 Design Features with B-Spline Curves Generating unclamped and clamped B-spline curves, illustrated in Example 5. ...
156 COMPUTER AIDED ENGINEERING DESIGN clamped B-splines have already been shown in Figure 5.16. Figure 5.20 shows a schematic of ...
SPLINES 157 barycentric property of B-spline basis functions that over the span [t 5 ,t 6 ), there are four non- zero such funct ...
158 COMPUTER AIDED ENGINEERING DESIGN property in Section 5.7G1. Shape functions N4,7(t) and N4,10(t) are only C^0 (position) co ...
SPLINES 159 uabai k i kk k n kk = + ( – ) | – | | – | =1 –1 =1 –1 1 2 1 2 Σ Σ bb bb , i = 1,... , n (5.39) withu 0 as a. Note th ...
160 COMPUTER AIDED ENGINEERING DESIGN tn+1 = tn+2 =... = tn+p=b (5.42) Thus, the first internal knot tpis the average of p–1 par ...
SPLINES 161 With 7 data points and 4 as the order of the piecewise curves, the number of knots required are 11. We can use u 0 , ...
162 COMPUTER AIDED ENGINEERING DESIGN weights in Eq. 5.46 to 1 yields the B-spline curve), discussion in this chapter pertaining ...
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