1. 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

2. 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.

3. 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

4. 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

5. VARIATION OF BLENDING FUNCTIONS

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