Given only an arc of a circle, we can easily reconstruct it fully without any use of analytic geometry - indeed using only compass and straightedge. Note that by "given only an arc", we mean that "from a circle that was originally drawn on the plane, everything except an arc got erased".


**Question:** Given an arc of an ellipse, can we reconstruct it without using analytic or algebraic geometry but *synthetically* - using objects not restricted to compass and straightedge (for example, one could use threads)?


**Sub-questions:** Does it matter if the arc is more than half of the full ellipse? What if more than one arc, say 2 arcs, of the ellipse are given (here, we mean the arcs given only as broken off fragments of the wireframe boundary of the ellipse and their relative position is not known)?


Note: analogous questions could be asked about the other conic sections.



Reconstructing an ellipse from an arc, synthetically
Five points in general position lie on a unique conic. With the help of Pascal's theorem one can construct arbitrarily many points on the same conic.


EDIT. Answering the question about construction of the foci: yes, it is possible. If 5 points on a conic are given, then the coefficients of the corresponding quadratic polynomial are constructible: solution of a linear system requires only rational functions of the coefficients. The center and the principal axes are constructible: rational functions plus square root extraction when finding the eigenvectors. Ellipse vertices are constructible: intersection of a line with a conic requires square roots and arithmetic operations only. The foci are constructible: for the focal distance arithmetic operations and square roots only are needed.


The above results in a terribly long algorithm. I am sure that the descriptive geometers knew a shorter construction in the 19th century already.


