Computer Aided Engineering Design

SPLINES 163

[Hint: For ti–1≠ti, use normalization; for ti–1 = ti, find C from the condition M2,i(ti) = N 2 ,i(ti) = 0]. For

t∈ [ti–1,ti), would one need to evaluate M1,i(t) or N1,i(t) at t = ti. Would the right open parenthesis in

[tt–1,ti) serve the purpose of C.

5. The recursion relation for a normalized B-spline basis function is given as
   N1,i(t) = δi such that δi = 1 for t∈ [ti–1,ti)
   = 0, elsewhere

Nt

tt

tt

Nt

tt

tt

ki ik Nt

i ik ki

i

, i ik ki

• –1 – –1, –1 –+1 –1,

() =

• 
  
  
  
  • () +
    
    
    • 
      
    
    
    
    
  
  
  
  


• 
  

  ()

  
  

wherek is the order (degree+1) of the spline and i the last knot over which Nk,i (t) is defined. Show in a

general case that the sum of all non-zero B-spline basis functions over a knot span [tj,tj+1) is 1.

6. Show that the derivative of a B-spline basis function of order kis given as
   d
   dtNt Nt

k

ttN

k

() = () = kj kj j jk kj ttj jk Nkj

• 1


• –
  
  
  • 1
    ,,′ –1 – –1, –1 – –+1 –1,
  
  
  
  

and thus the derivative of a B-spline curve b(t) = Σ

i

n

=0 ()Ntpp i,+ bi is

d

dt

tNi

n

() = =0 ppii

–1

bqΣ –1, +

where qbbi

pi i

i i

p

= tt

( – 1)

+ – +1 ( – )+1

7. Write a generic code to compute the normalized B-spline basis function Nk,i(t). Device ways to make the
   computations robust. [Hint: Note that Nk,i(t) = 0 for t∉ [ti–k,ti). Further, for t∈ [ti–k,ti), computing Nm,j(t)
   requiresNm–1,j–1(t) and Nm–1,j(t),m = 1,... , k,j = i–k+1,... , i, which form a triangular pattern shown
   in Table 5.2. Judge if all basis functions are needed for computations, or some are known to be zero a
   priori].


8. Exploreknot insertion and blossoming as alternative methods to compute B-spline basis functions.


9. How would one get a Bernstein polynomial (Bézier) curve from a B-spline basis function (a B-spline)
   curve? Explain and illustrate.


10. Given the control polyline for a B-spline curve, is it possible to (graphically) obtain the Bézier control
    points for the same curve? If so, under what conditions? (We may want to work out an example).


11. Write a procedure to implement Eq. (5.34) for a general 2-D case.


12. In Exercise 11, provide for (a) data points to be interactively relocated and flexibility and (b) curve
    clamping at the ends using knot multiplicity.


13. Letr 0 ,r 1 ,r 2 ,r 3 ,r 4 be five control points specified by the user. Show that the equations of 3 B-spline curve
    segments with uniform knot vectors are given by

r 1 (u) =^13 (3 –u)^2 r 0 +^12 [(u– 1) (3 – u) + (4 –u) (u– 2)] r 1 +^12 (u– 2)^2 r 2

r 2 (u) =^1

2

(4 –u)^2 r 1 +^12 [(u– 2) (4 – u) + (5 –u) (u– 3)] r 2 +^12 (u– 3)^2 r 3

r 3 (u) =^12 (5 –u)^2 r 2 +^12 [(u– 3) (5 – u) + (6 –u) (u– 4)] r 3 +^12 (u– 4)^2 r 4

(a) Each of the segments can be normalized with 0 ≤u < 1 by substituting (u + 2) in place of u. Show that

the matrix form of the curve segments is given by