Confusion about hough transformation for ellipse
It's a little mis-leading and confusing.

The usual way to describe an ellipse is using cartesian ($x,y$) coordinates.  Another way to represent an ellipse is using polar coordinates ($r,\theta$) (where $r$ is the distance (radius) from the origin, and $\theta$ is the angle.

What the equation is saying is that the cartesian coordinates of the ellipse are parameterized by $\theta$ in the two equations. That is, as you change $\theta$ from 0 to $2\pi$, the values for $x$ and $y$ will lie on the ellipse, for constants ($a_0,b_0,a_x,b_x,a_y,b_y$).