APPROXIMATION SPLINE Does not necessarily pass through all control points General representation is P(u) = Σ Fi(n,u) Pi where- Fi is the blending function. n is no. of control points. Interpolation curve Pn Approximation curve P1 Po

PROPERTIES OF APPROXIMATION SPLINE Curve always starts at Po and end at Pn…..I.e curve always passes through first and last control points. If kth derivative at any end point is needed, then only the first / last (k+1) points should be considered. Curve is symmetric about ‘u’ and ‘(1-u)’. Curve should be constrained in a finite space.

where Bi,n(u) is the blending function. BEZIER CURVES Definition- P(u) = Σ Bi,n(u) Pi where Bi,n(u) is the blending function. i=0 n

p(u) = (1-u)2 P0 + 2u(1-u)P1+ u2 P2 n = 2 We have P0, P1, P2 B0,2 = 2C0 u0 (1-u)2 = (1-u)2 B1,2 = 2C1 u1 (1-u)1 = 2u(1-u) B2,2 = 2C2 u2 (1-u)0 = u2 p(u) = (1-u)2 P0 + 2u(1-u)P1+ u2 P2

VARIATION OF BLENDING FUNCTIONS

-To ensure the continuity, P0, P1, P4 and P5 should be collinear.