Around 1858, the Reverend Hamnet Holditch studied the curves generated by a point on a chord of fixed length sliding inside another curve.
This Demonstration is an animation showing the generation of some Holditch curves inside ellipses of different eccentricities. You can see the effect of the chord length and the position of the generating point on the chord.
The parametric equation of the ellipse is used with the semimajor axis equal to 1 and eccentricity : .
Since there is no simple formula for the chord length along the circumference of an ellipse, you have to continuously solve the equation expressing the intersection of the ellipse and a circle with radius equal to the chord length.
For a description of Holditch's theorem, see  and .