In order to determine the stochastic visibility (https://link.springer.com/book/10.1007/978-1-4612-2690-1) in Poisson fields where the center of blockages is randomly distributed according to a Poisson point process. We are interested in the following scenario in which a rectangular blockage can obstruct the Fresnel zone between a transmitter and a receiver.
Consider an ellipse with equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$,
and a rectangle of width W and length L tilted at an angle $\theta$ with respect to the horizontal axis (Ox).
This rectangle is shifted on the outer surface of the ellipse for fixed theta as illustrated in the figure.
The movement of the rectangle can be described alternatively as follows: One corner of the rectangle traces part of the ellipse until one side becomes tangent, at this point the rectangle slides along the tangent line, and the process restarts. The path of the rectangle's center seems to consist of four translated arcs of the ellipse connected by four line segments.
Any tips to obtain the parametric equations of the curve drawn by the center of the rectangle are appreciated?
Yes, the path is formed by four arcs of ellipse (when the rectangle doesn't slide) and four segments joining them (when the rectangle slides). If the rectangle doesn't slide, its center is translated by the same vector with respect to the contact point (red arrows below) and it then describes an ellipse (red) which is a translated copy of the original ellipse (green). When the rectangle slides its center describes a segment (green in figure below), bur after sliding the vector (blue) between contact point and center has changed, so the center describes a different translated ellipse. And so on.