I am looking for a 2D function, $f(x, y)$ which increases monotonically over the range $(0,0)$ to $(1,1)$. In other words, it will be $0$ at $(0,0)$ and $1$ at $(1,1)$. It will also evaluate to $0$ whenever $x = 0$ or $y = 0$. It's monotonic in the sense that the value at $(0.5,0.6)$ be greater than at $(0.5, 0.5)$. The closest guess that I have to the function that I am looking for is:
But I would also like to be able to "bias" or make the function rise faster with $x$ relative to $y$ or vise-versa. Ideally, I would like to specify the surface with 2 variables. One that controls / scales how $f(x,y)$ rises relative to $x$ and another that controls / scales how $f(x,y)$ rises relative to $y$.
Edit: I am looking for a sigmoid-shaped curve. The function does not need to be exactly equal to 1 at z(1,1). However, z(1,1) to be higher than any other z(x,y) across the space x[0:1], y[0:1]. A closer function than the one above is:
z=1/(1+e^(-(x^a*y^b)) where a and b are used to "bias" or change the slope of the curve with respect to x vs. y. However, that curve is close to 1 at z(2,2) instead of z(1,1). So x and y need to be scaled. Those scaling factors will be a function of a and b. They cannot be computed directly since the exponent needs to be 0 to satisfy the equation exactly. However, if the exponent evaluates to a sufficiently small number, such as 0.01, that should be close enough for my purposes.