Parameterizing cliffs I am looking for a function $f(x; \alpha, X_1, X_2, Y_1, Y_2)$ that has the following property: For $\alpha=0$ it behaves linearly between $(X_1, Y_1)$ and $(X_2, Y_2)$, and as $\alpha$ gets closer to 1, it approximates a sharp cliff, as in the figure below. The function needs not be defined for $\alpha=1$.
Is there a relatively "simple" function (trigonometrics, powers, logarithms and exponentials are fine) that captures this behavior?


The linear equation is given by
$$\frac{x - x_1}{\lvert x_2 - x_1\rvert} + \frac{y - y_2}{\lvert y_1 - y_2\rvert} = 1.$$
We can get successively 'sharper' curves by using
$$\left(\frac{x - x_1}{\lvert x_2 - x_1\rvert}\right)^{k} + \left(\frac{y - y_2}{\lvert y_1 - y_2\rvert}\right)^k = 1$$
for integer $k \geq 1$, using larger $k$.
Here are the graphs for $k = 1, 2, 3, 4, 5, 6$ through the points $(0, 2)$ and $(3, 0)$:

For $k = 1$, we have the linear equation. For $k = 2$, it's one quarter of an ellipse.
Unfortunately, I'm not sure how to translate my parameter of positive integer $k$-values into the parameter $\alpha \in [0, 1)$. I'll leave this here for now, and think about it.
EDIT: It appears that using $k = \dfrac{1+ \alpha}{1 - \alpha}$ for the exponent works quite well. 
Here is an image with $\alpha \in \{0, 0.05, 0.1, 0.2, 0.3, 0.5, 0.7, 0.9\}$:

And if instead you use the function
$$\left\lvert\frac{x - x_1}{x_2 - x_1}\right\rvert^{k} + \left\lvert\frac{y - y_2}{y_1 - y_2}\right\rvert^k = 1,\quad \alpha = \frac{1 + \alpha}{1 - \alpha} \in [0, 1),$$
you get something that behaves rather well, regardless of whether $x_1 < x_2$, etc. Now these are between the points $(x_1, y_1) = (0, 2)$ and $(x_2, y_2) = (3, 4)$, with some bonus points thrown in:

