Rolling Ellipse!

When you have Circlular wheel rolling smoothly, i.e, the path traced by the point through which it is rotated (axle point) traces a straight line and the road is a horizontal line in this case. Infact, for any wheel to roll smoothly, the path traced by the axle point should be a straight line.

Suppose you have to roll an Elliptical wheel smoothly with its axle point at one of its focii. Then what would be the equation of its road in order to accomplish this.

$Note:$ $k$ is the distance of axle point from corresponding directrix of the Ellipse.

$e$ is the eccentricity of the Ellipse.

$a = \sqrt{1 - e^2}.$ Moreover, assume that the road starts such that the ellipse at start is at lowest elevation possible ( does not start rolling from an elevated ground).