Question

(C) (a) Compare the splines created by B-spline and Bezier spline techniques for the of control. points is given by Po (4,4,4). P = (6,8,6) and P2 (10,3,4). Compute Bazier curve withwintermediate points. same control points. A set

Answer 1

With 3 points we can create quadratic B-spline and quadratic bezier curve.

Following are equations for both:

1. Bezier spline ( t varies from 0 to 1)

B(t) ( ) Po2(1 t)tP + t P2 B(t)(1 t4, 4, 4] 2(1 t)t6, 8, 6]10,3,4] B(t) 4 (1- t)12*t(1 -t) 10t2,4 (1-t)16* t(1-t)3t2,4* (1 t212 t(1 )4t B(t)= 2t + 4t 4,-9t28t4,-424t4]

2. B spline ( t varies from 0 to 1)

Case 1:

control points = 3 ; degree = 2

This will result in same equation as of bezier curve as bezier curve is special case of b-spline.

Hence:

B(t) = 2t2+4t 4,-9t +8t 4,-4t2 4t 4]

Case 2:

control points = 3 ; degree = 1

This will be linear interpolation of all intermediate points.

Two Intermediate points = Midpoint of P₀ and P₁ , Midpoint of P₁ and P₂ .

i.e. I1 = (5,6,5) and I2 = (8,5.5,5)

Linear Equation (t = 2 to 3) :

B(t) = [ 3t -1 , -0.5t + 7, 5 ]

Comparison:

Hence if degree of B-spline is 2 then both methods (B-spline and Bezier curve spline) give same curve.

Now,

Bezier curve through intermediate points is just straight line joining both.

Equation (t = 2 to 3) :

B(t) = [ 3t -1 , -0.5t + 7, 5 ]

Comparison:

Hence if degree of B-spline is 1 then both methods (B-spline and Bezier curve spline) give different curves. And b-spline curve matches with bezier curve through intermediate points.