Want a spline with the following conditions
f(0)=0 f(1)=1
f'(0)=0 f'(1)=0
f''(0)=0 f''(1)=rho
results in
y=ax5+bx4+cx^3
a=6+rho/2
b=-15-rho
c=10+rho/2
rho defines the curvature at x=1.
Usable range of rho is +infinity to -20
Beyond x<-20, the spline dips below zero.
Beyond x>0, the spline goes above 1.
theoretically interesting value of rho are -7.5 and -12 where y/x^2
quadratic undergoes interesting changes.
at -20 (the sharpest) we get
y=-4x5+5x4
The an S-shaped curve with no degrees of freedom
f(0)=0 f(1)=1
f'(0)=0 f'(1)=0
is
-2t3+3t2