Bivariate function that produces "ridge"-like surface

Question

I'm trying to find the right vocabulary to describe a "ridge" like distribution that depends on two variables and how similar they are to each other. I'm imagining it looks something like the following. The utility of such a "ridge" like function is motivated by physical modeling. If $x$ is some expected distance and $y$ is the observed distance, then the probability $z$ surface should peak when $x - y = 0$ .

I admit this sounds really silly to ask, but I'm not finding relevant answers by searching for "ridge" functions. Would someone be able to point me in the right direction?

Answer 1

Would a sufficiently rotated/scaled Two-dimensional Gaussian function work? Below, I've plotted $$f(x,y) = \exp\left(- \left(a(x - x_o)^2 + 2b(x-x_o)(y-y_o) + c(y-y_o)^2 \right)\right)$$ where

\begin{align} a & = \frac{\cos^2\theta}{2\sigma_X^2} + \frac{\sin^2\theta}{2\sigma_Y^2} \\[4pt] b & = -\frac{\sin2\theta}{4\sigma_X^2} + \frac{\sin2\theta}{4\sigma_Y^2} \\[4pt] c & = \frac{\sin^2\theta}{2\sigma_X^2} + \frac{\cos^2\theta}{2\sigma_Y^2}. \end{align} Here $\theta$ is the angle of clockwise rotation.

See CalcPlot3D for the full example, here I've used the variables $a = \sigma_{x} = 2$ and $b = \sigma_{y} = 1$ centered at the origin and rotated an angle of $t = \theta = \frac{\pi}{4}$.