02/21/2002
15-462 Graphics I
6
Deriving B-Splines
•Consider
points
–pi-2,
pi-1,
pi,
pi+1
–p(0)
approx pi-1, p(1) approx pi
–pi-3,
pi-2,
pi-1,
pi
–q(0)
approx pi-2, q(1) approx pi-1
•Condition
1: p(0) = q(1)
–Symmetry:
p(0) = q(1) = 1/6(pi-2 + 4 pi-1 + pi)
•Condition
2: p’(0) = q’(1)
–Geometry:
p’(0) = q’(1) = 1/2 ((pi – pi-1) + (pi-1 – pi-2)) = 1/2 (pi – pi-2)