The Problem

In April of 2011, I read an article in the Mathematics Magazine published by the Mathematical Association of America titled Ellipse to Hyperbola: “With This String I Thee Wed” by Tom M. Apostol and Mamikon A. Mnatsakanian pp. 83–97.  In this article the authors introduce a method of tracing an ellipse and a hyperbola using string and a straw.  An ellipse is the set of all points in a plane whose distances to two fixed points (the foci) is a constant sum.  A hyperbola is the set of points in a plane whose distances to two fixed points (the foci) is a constant difference. 

I replicated their process using a construction software called The Geometer's Sketchpad^®.  I created a circle and defined my foci as the points (0, 0) and (1, 0).  A point E is a point that traverses the circle.  A line l is constructed passing through point E and one of the foci (0, 0).  A segment is constructed connecting point E and the remaining focus (1, 0).  A perpendicular bisector of that segment is formed.  The intersection of that perpendicular bisector and line l is point F.  As point E traverses the circle, point F traces an ellipse.

The proof that point F forms an ellipse is as follows:  AE is the length of the radius which in this example is 2.  A segment FB can be formed creating triangle BDF.  Angles BDF and EDF are both right angles because of the perpendicular bisector and for the same reason segments BD and ED are the same length.  Since triangles BDF and EDF share segment FD, the two triangles are congruent thus segments EF and BF are congruent.  So the sum of the lengths of AF and BF will always equal to the length of the radius and by definition of an ellipse that is what is formed by point F as E traverses the circle.

If one of the foci is moved outside the circle, a hyperbola is formed.  In the following diagram, I moved the focus (1, 0) to (3, 0) to illustrate that fact.




I am fascinated by the traces that are formed as E traverses a pathway and I changed E's pathway to create the following graphs.



 

I believe to do an analysis of these graphs I need to have an algebraic model for this construction.  Given a pathway, P(x), a traversing point on P(x), two foci (a, b) and (c, d), a line l passing through (a, b) and the traversing point on P(x), and a perpendicular bisector of the segment with endpoints (c, d) and the traversing point, determine the intersection of line l and the perpendicular bisector.

I will share my work thus far . . . later. :-)