With this apparatus placed on the stage of an overhead projector, you can draw ellipses of various eccentricity and project them onto a screen. You choose how far apart the foci are by where you set the nails. The string keeps the pen point in such a path that the sum of its distances from the two foci is constant.
Moving objects that are subject to a central force and have sufficient energy travel in paths that are ellipses. A good example is the motion of the planets of the solar system around the sun, whose orbits are elliptical.
An ellipse is the locus of all points in a plane, the sum of whose distances from two fixed points is constant. Its equation is: x2/a2 + y2/b2 = 1, where a and b are constants. The two fixed points are called foci, and they always lie on the major (long) axis of the ellipse. The larger of a and b gives the length of the semimajor axis, and the smaller gives the length of the semiminor axis. Taking a as the semimajor axis, the sum of the distances from any point on the ellipse to the two foci is 2a. If we take the intercept of the ellipse at the semiminor axis, we then have a right triangle with the semiminor axis, b, the hypoteneuse from the intercept to the focus, and the base from the focus to the center of the ellipse. The length of the hypoteneuse is merely a (the same as the distance to the other focus), so the square of the distance, c, from the center to the focus is c2 = a2 - b2, or c = √(a2 - b2).
The eccentricity of the ellipse, that is, how flat it is, or how much it deviates from being a circle, is defined as e = c/a. If c = 0, the two foci coincide, and the ellipse is a circle. At the other extreme, c = a, and the ellipse flattens to a straight line segment between the foci.
On this apparatus, the sum of the distances from the foci to the point on the ellipse is fixed by the length of the string (which equals 2a). By selecting different holes in which to anchor the ends of the string, however, you can vary the distance between the foci, and thus change the eccentricity of the ellipse. The farther apart you place the foci, the more eccentric, or flatter, is the ellipse. The closer together you place the foci, the less eccentric, or more nearly circular, is the ellipse.
References:
1. Thomas, George B., Jr. Calculus and Analytic Geometry (Reading, Massachusetts: Addison-Wesley Publishing Company, Inc., 1972)