Trying to define a simple "warp" function I'm trying to define a 2D "warp" function $y=f(x,w)$.  A picture is worth a thousand bytes:

I am looking for a simple function $f(x, w)$ that satisfies the following:

*

*defined over $x = 0.0 \dots 1.0$

*continuous

*$f(0,w) = 0$,

*$f(1,w) = 1$,

*$f'(x,w) \ge 0$ (slope is never negative),

*defines a straight line at $w=0$, i.e. $f(x,0) = x$ [but see note],

*symmetrical about $y=-x+1$, i.e. $f(x,w) = 1-f(1-x,w)$,

*slope at $x=0$ is the inverse of the slope at $x=1$: $f'(0,w) = 1/f'(1,w)$.

Note: despite what's shown in the graphic, I don't really care about the range of w: it could be -inf to inf (with y=x when w=0) or it could be 0 to 1 (with y=x when w=0.5).  The important part is that the curve becomes sharper as w approaches its extreme values.
I suspect this is simple, but I haven't been able to crack it.  Some kind of conic section, perhaps?
 A: You want a quadrant of a superellipse, appropriately transformed.  That is to say, consider the family of curves $$x^m + y^m = 1,$$ for $m \in (0, \infty).$  Then your curve is merely the slightly modified case $$(1-x)^m + y^m = 1,$$ for $x \in [0,1]$, or equivalently, $$y = (1-(1-x)^m)^{1/m}.$$  Now all that remains is to reparametrize $m$ to $w$, for which $w = 0$ corresponds to $m = 1$, and $w(m) + w(1/m) = 0$.  Your $w$ is underspecified, however; if I infer from your diagram that you would like $w \in (-1,1)$, then the following conditions give the function $$w(m) = \frac{m-1}{m+1}.$$  Therefore, your desired family of curves has the equation $$y = (1-(1-x)^{(1+w)/(1-w)})^{(1-w)/(1+w)}, \quad x \in [0,1], w \in (-1,1).$$

It would seem that this function does not have the property of being symmetric for $w$ and $-w$, so when I have a chance I will give it some more thought.  But the family of curves is valid; it just needs re-parametrization.
A: Any Bezier Curve (parametric)  with degree at least 2 and "symmetric & monotonic Bezier points on the edges of the unit square" will suffice.  The degree will work as the needed "w" by swapping x, y. 
One choice: 
x = t^n;
y = 1-(1-t)^n;  n >= 1,   0=<  t =< 1. 
