B Spline Methods BSpline have two advantages over

B- Spline Methods • B-Spline have two advantages over Bezier spline: 1. Degree of B-Spline polynomial can be set independently of number of control points. 2. B-spline allow local control over the shape of a spline curve as surface. @Abhishek Khare SVIM CG&M Unit-4

Contd. • @Abhishek Khare SVIM CG&M Unit-4

Contd. Each blending function is defined over d, subinterval of total range of u starting of knot value of uk @Abhishek Khare SVIM CG&M Unit-4

Numerical • Construct the B-spline curve of order 4 and with 4 polygon vertices A(1, 1) B(2, 3) C(4, 3) D(6, 2) • Sol: Let us take u = 0, ¼, ½ , 3/4, 1 P(0) P(1/4) P(1/2) P(1) @Abhishek Khare SVIM CG&M Unit-4

Contd. • • • P(0) = [1, 1] P(1/4) = [123/64, 137/64] P(1/2) = [25/8, 21/8] P(3/4) = [289/64, 163/64] P(1) = [6, 2] @Abhishek Khare SVIM CG&M Unit-4

Numerical • @Abhishek Khare SVIM CG&M Unit-4

Contd. P(t) = (1, 1)(1 -t)3 + (2, 3). 3 t(1 -t)2+(4, 3). 3 t 2(1 -t)+(3, 1)t 3 At mid point t =0. 5 P(0. 5) = (1, 1)(0. 5)3+(6, 9)(0. 5)3+(12, 9)(0. 5)3+(3, 1)(0. 5)3 P(0. 5) = 0. 125[(1, 1)+(6, 9)+(12, 9)+(3, 1)] P(0. 5) = 1/8[22, 20]=(22/8, 20/8) @Abhishek Khare SVIM CG&M Unit-4

Numerical If a Bezier curve passes through (2, 1) and (6, 2) and controlled by the points (3, 2) and (5, 0). Find the equation of Bezier curve. Find midpoint on the Bezier curve. Draw rough sketch of the Bezier curve. Sol: n=3 At mid point t =0. 5 P(0. 5) = (4, 9/8) @Abhishek Khare SVIM CG&M Unit-4