I have some response data across a $2D$ space, which seems to have a ring-like structure (left figure).
To model this response as a $2D$ Gaussian that traverses an elliptical path, I use the following parametric equation:
\begin{equation} f(x,y) = \frac{1}{2\pi\sigma_x\sigma_y}e^{-\frac{1}{2} \left( \left( \frac{x-h(\theta)}{\sigma_x} \right)^2 + \left( \frac{y-g(\theta)}{\sigma_y} \right)^2 \right)} \end{equation} where: \begin{equation} \theta = arctan2(y,x) \\ h(\theta) = r_1\cdot cos(\theta),\\ g(\theta) = r_2\cdot sin(\theta) \end{equation}
As long as the ellipse radii, $r_1$ and $r_2$ are equal, I get a circular ring as expected (right figure). However, when the radii are different from one another, I get the result below. I can't seem to wrap my head around it. What am I missing?
The model "modulates" per $(x,y)$ the center of a $2D$ Gaussian that lies in an ellipse's circumference. To understand the result, I drew two line segments: one goes from the origin to some point, $P=(p_x, p_y)$, and the other from the origin to $C=(h(p_x, p_y), g(p_x, p_y))$.
Now, consider the left panel where $\theta(p_x, p_y)$ is approx. $-90^\circ$. Here, the two lines almost overlap, and P's value is calculated based on a Gaussian that is very near. As $\theta$ changes towards zero (right panel), the two lines move away from each other; the Gaussian that determines P's value lies further away on the ellipse.
At the bottom line, the model is wrong. I seek a model that assigns to each $(x,y)$ the nearest Gaussian that lies on the ellipse circumference.