This is problem #48 from Heinrich Dorrie's [i]100 Great Problems of Elementary Mathematics.[/i]
I already worked out a more general solution as part of Van Schooten's Locus Problem: [url]http://www.geogebratube.org/student/m35812[/url] [list] [*]In Van Schooten's Problem, the diameter of the rolling circle is half the length of the arms of the trammel. The second axis has a variable angle, and the segment AB which is fixed to the arms of the trammel is a chord of the rolling circle. If the arms are perpendicular, AB is a diameter, which gets us to Cardan's problem. [/list] In the above diagram... If P lies on the circumference of the disc, the locus is a line. Let P lie on the diameter fixed to the axes. Start at an endpoint, and move P along the diameter until it reaches the circle center. The locus changes smoothly from from line to circle, with everything in between an ellipse, aligned to the reference axes. But the given axes are [i]arbitrary.[/i] Every diameter of the disc slides along perpendicular axes; I have selected one. Every position of P on the inside of the disc thus describes an ellipse, aligned to the diameter line drawn through AP.