The hyperbola

Prof Eduard Belinsky

To construct the hyperbola is not as easy as the ellipse despite the simplicity of the mathematical definition. The hyperbola is a curve with the property that the absolute value of the difference of the distances from any two fixed points to any point on the curve is constant. This can be achieved mechanically using string, as in the diagram below. The two fixed points are called its foci, F₁ and F₂. When the pointer moves from P₀ to P such that the weight moves vertically up a distance k, then PF₁-PF₂ = (P₀F₁+k)-(P₀F₂+k)=P₀F₁-P₀F₂=constant.

The hyperbola is another kind of conic section . If the intersecting plane is inclined to the axis of the cone such that it meets both branches of the cone then the curve of intersection looks like a hyperbola. In order to prove that it really is a hyperbola, we consider the two inscribed spheres that touch both the cone (at circular curves of contact) and the intersecting plane. This time the spheres occupy different branches of the cone but lie on the same side of the plane at points F₁ and F₂. Let B be a point of the curve and let BO be the line on the surface of the cone which meets the two circles of contact at P₁ and P₂. Then BF₁=BP₁ and BF₂=BP₂ (tangents from B to sphere are equal) and thus BF₂-BF₁ = BP₂ - BP₁ = P₁P₂ = constant. So this curve is really a hyperbola.

One interesting observation is that the inscribed spheres always touch the plane of intersection at the foci of the hyperbola.

Let us fix two foci and consider the `family' of all ellipses having both foci in common and the `family' of all hyperbolas having the same pair of foci. There are exactly two curves of the system of confocal ellipses and hyperbolas passing through every point of the plane.

If we have the tangent to the hyperbola and tangent to the ellipse at the point P of their intersection we will see that they are perpendicular to each other. Thus the confocal ellipses and hyperbolas form an `orthogonal net of curves'.

Another simple example of orthogonal families of curves are concentric circles and the straight lines passing through their common centre.

Applications of orthogonal families:

1. The contour lines on a map are orthogonal to the lines of steepest descent. You can check this by rolling a ball (or yourself) down a hill.
2. The equipotentials of a magnet placed on a piece of paper. The orthogonal family to this are the lines of force, which you will see if you throw iron filings around the magnet.

The hyperbola appeared as a section of a cone. A cone is a surface of revolution. From the hyperbola, rotating it about the line connecting the foci or about the perpendicular of that line, we obtain the hyperboloid of two sheets/one sheet, respectively.

It is a surprising fact that there are infinitely many straight lines on the hyperboloid. Indeed this surface can also be generated by rotating a straight line about a skew axis! You can make a model (looking like the cooling towers of the Bulawayo Power Station) out of two circular discs, a small wooden rod, some string and some wire.

And the third conic section is the parabola... in the next issue.