In this Post I derive the differential equations of motion of a homogeneous elliptic lamina of mass m and the major and minor axes of lengths of a and b which rolls without slipping along the horizontal x axis within the vertical xy plane.

If the initial angular velocity is large enough, the ellipse will roll forever and go to ±∞ in the x direction, otherwise it will just rock.

I have attached two files:

 rolling-ellipse.mw
        Worksheet to solve the differential equations and animate the motion

rolling-ellipse.pdf
         Documentation containing the derivation of the differential equations

And here are two animations extracted from the worksheet.

The attached worksheet develops a procedure for extrapolating boundary data from a square to its interior.  Specifically, let's consider the square [−1,1]×[−1,1] and the continuous functions u₁, u₂, u₃, u₄ defined on its edges. The procedure constructs a continuous function u(x,y) in the interior of the square which matches the boundary data.

The function u(x,y) is necessarily discontinuous at a corner if the boundary data of the edges meeting at that corner are inconsistent.  However, if the boundary data are consistent at all corners, then u(x,y) is continuous everywhere including the boundary.

Here is what the extrapolating function looks like:

The worksheet Extrapolant.mw contains the details of the derivation, and an example.

Edit: The worksheet Extrapolant-ver2.mw presents a slightly different version of the previous result. Here the square's corners are treated symmetrically, leading to a more pleasing interpolating function.

PS: A challenge.  Extend the result to 3D, that is, construct a function u(x,y,z) on the cube [−1,1]×[−1,1]×[−1,1] which matches prescribed functions on the cube's six faces.