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© Copyright 2003, Jim Loy
The hyperbola (above left) is a Conic Section. That means that we can produce one by slicing a cone with a plane. A cone, in this case is two cones, nose to nose. Slicing it produces two identical pieces (nappes) of the hyperbola. Sometimes we just draw one half of the hyperbola, but the whole hyperbola consists of the two pieces. Above right is the traditional way to define a hyperbola, as a conic section. It is the set of points whose distance from a point (the focus) and a line (the directrix) is in a constant proportion (for the hyperbola in the diagram, the proportion is 1.0625). A proportion of 1 gives a parabola, and less than 1 gives an ellipse. The next diagram is a hyperbola with a greater proportion.
One way to define a hyperbola is the set of points (locus) such that the difference of the distances between these points and two fixed points (the two foci) is a constant. Such a locus is shown above left. The curve approaches a pair of straight lines (asymptotes) which intersect at the center of the hyperbola, as shown in that diagram. Different hyperbolas have different shapes, depending on the angle between the asymptotes, just as different ellipses have different shapes. Above right is a special hyperbola, a right hyperbola, the asymptotes being at right angles. It has a simple equation: y = a/x, which is a second degree equation, even though it has no squares in it.
And below left is a pencil, the path of a moving line, in which the line (the dark line in the diagram) wobbles around a circle. This can be done in a number of ways. In this case (shown below right), I drew a circle (its center is at one focus of the hyperbola) with a moving point on it. I then drew the segment between the moving point and the other focus. I then drew the perpendicular bisector of that segment. As the point moves on the circle, this line fills in the space between the two halves of the hyperbola. This can be done with paper folding, too (fold the point onto points all around the circle).
These diagrams were drawn with Paint Shop Pro, the second and third with Cinderella, the last two with Geometer's Sketchpad.