> That comes, of course, from the property of the foci > of an ellipse, that the total distance from a point > on the ellipse to the two foci is constant. Of > course, the fact that your string has fixed length > keeps the sum of those distance (the string itself) > constant. > > A parabola only has one focus but there is also a > o a line (the "directrix") such that the total > distance from a point on the parabola to the focus > and directrix is constant. The distance from a point > to a line is measured PERPENDICULAR to the line so > you would have to design some mechanism to insure > that your string stayed perpendicular to the line. > Perhaps use a toy railroad track to represent the > e directrix and have the string tied to a rail truck > that is free to move along the rails. > > A hyperbola has two foci, like an ellipse, but it > it is the DIFFERENCE between the two distances, from > point on the hyperbola to the two foci, that is > constant. I just can't think of a mechanism that > will keep the difference of the two distances > constant.
Methods of achieving both of these objectives have been given in some of the older geometry texts.
The hyperbola is achieved by fixing the pencil to a small localised loop on the string which passes back to pins at the foci. Then feed equal amounts of string to each line and the difference remains constant, giving the hyperbola.
The parabola uses a similar technique, with the pencil fixed to a local loop on the string. One string feeds along the axis and around a pin at the focus. The other end feeds thru the right angle corner of a set square which slides along the directrix. The pencil is kept against the set square, while equal amounts of string are fed to each line, giving the parabola.