f(x) = a*x**3 + b*x**2 + c*x + d
f'(x) = 3*a*x**2 + 2*b*x + 1*c
f(0) = 0    # start at 0
f'(0) = 0   # want a flat slope
f(1) = 1    # end at 1
f'(1) = 0   # but flat at this end, as well.
solve: 
d = 0       # from f(0)
c = 0       # from f'(0)
b = 3, a=-2 # from f(1) and f'(1)

##</code><code>##

f(x,y) = a*x**2 + b*y**2 + c*x*y + d*x + e*y + g
f(0,0) = 0
f(1,1) = 1
df/dx(0,0) = 0
df/dy(0,0) = 0
df/dx(1,1) = 0
df/dy(1,1) = 0

##</code><code>##

p:0..1
H(p) = a*p**3 + b*p**2 + c*p**1 + d
    I would recommend corners like:
        H(0) = Hmin
        H'(0) = 0
        H(1) = Hmax
        H'(1) = 0
    to get a similar "s" shape to 3*x**2 - 2*x**3, but scaled so the output go from the min to the max you want
S(p) = e*p**3 + f*p**2 + g*p**1 + h
    S(...) = Smin, 0, Smax, 0
V(p) = j*p**3 + k*p**2 + m*p**1 + n
    V(...) = Vmin, 0, Vmax, 0