Ellipse Foci

An ellipse is defined as the locus of points for which the sum of the distances from each point to two fixed points is equal.

On my Ellipse Device page, I ask the question "Do you know how to draw an ellipse using two pins and a string? Do you want to know?'. Jeanne and Derek took me up on my offer. I prepared a short tutorial for them, complete with illustration. It is included, below. After their kind comment that the tutorial was useful, I was inspired to make this Ellipse Foci animation and merge it with the Ellipse Device . Thank you, Jeanne and Derek, for the stimulus to add to my web site.

Let's say you have a rectangular piece of wood that you want to make into an elliptical plaque. Square up the wood to a convenient size, both length and width. Let's call the length the major axis and the width the minor axis. The center can be found by drawing two diagonals. Where they cross is the center, but this isn't enough for the Ellipse Foci method. We need perpendicular bisectors for both the length and width; the major and minor axis. This phase of the construction is represented by the blue rectangle with the major and minor axis labeled A'A and B'B respectively. The foci, F1 and F2, can be found by first solving for d. Find a and b; square a; square b; subtract b2 from a2; and finally take the square root of the result: d = sqrt(a2 - b2). Now that we have d, finding the foci is easy. We have already found the center (C) of the rectangle. Measure from C along the perpendicular bisector of the MINOR AXIS (line A'A) the distance d, both towards A' and A. These points are the foci (F1 and F2) and also the placement for two pins. In this case, nail two brads in the wood at the foci. Take a piece of strong string, loop it around both brads as shown in the drawing. While someone holds the marking device at point B (the perpendicular bisector of the MAJOR AXIS, line B'B), tie the string at the point B. An alternate method is to nail a third brad at B. Again, tie the string into a loop going around the three brads. Remove the third brad and insert a marking device. While holding the string taut, scribe the ellipse by moving the marking device in a circular path. Remove the remaining two brads, take the blank to the band saw, and cut out your elliptical plaque.

Edwin Attaway <edwin222@EdwinsAnimatedImages.com>