Everyone knows what a circle is. In constructing the circle by means of a thread we have to hook the closed thread around a fixed point, the centre, and keep it stretched while drawing the circle.
We obtain a similar curve if we keep the closed thread stretched around two fixed points. This curve is called an ellipse and the two fixed points are called foci. This thread construction characterizes the ellipse as the curve with the property that the sum of the distances from two given points to any point on the curve is constant. If the distance between the two points is diminished until the points coincide, we obtain the circle as a limiting case of the ellipse. For example, if you look at a coin from an angle, you see an ellipse. But as you increase this angle to 90° you see a circle. Similarly, look at the shadow on the ground cast by a circular disk held in the sunlight (we can assume the suns rays are parallel). We are going to look at some of the mathematics behind these two practical experiences: that is, the mathematics of the sections of a cone and a cylinder by a plane.
An ellipse is a closed curve which is convex everywhere (convex means that the line joining two arbitrary points of the ellipse is always inside the ellipse). At each point of the ellipse we can draw a tangent that remains outside the ellipse (except at the point of contact). In practical everyday terms, this means that if you ride a bicycle around an elliptical track, your bike will be leaning inward all the time.Let us consider a point A on an ellipse, to which T1T2 is tangent. F1 and F2 are the foci of the ellipse. The lines F1A and F2A are called focal radii of the point A. The focal radii form equal angles with the tangent, ie angle T1AF1 = angle T2AF2 (prove it!).
This property of the tangent to an ellipse has an application to optics: if a source of light is located at one of the foci of a mirror having the form of an ellipse, the reflected light will converge at the other focus. Imagine that you are in an elliptical room with mirror-wall all around you. If you place a candle at one focus, then you'll find that the other focus becomes a very, very bright place! What would you see if you stood at one focus? And then if you walked away?
The circular cylinder is the simplest curved surface in three dimensional space. It can be obtained from both the circle and the straight line. Firstly, by translating a circle through its axis. Secondly, by rotating a straight line about an axis parallel to it. Thus the cylinder is a surface of revolution. If a cylinder intersects a plane at right angles to its axis, we get a circle. But if its axis is neither perpendicular (nor parallel) to the plane, then we get a curve that looks like an ellipse. We shall prove that this curve really is an ellipse.
To do this, we take a sphere that just fits into the cylinder, and move it within the cylinder until it touches the intersecting plane at the point F1. We then take another such sphere and do the same thing with it on the other side of the plane. We obtain the point F2. Let B be any point on the curve of the intersection of the plane with cylinder. Consider the straight line through B lying on the cylinder (i.e parallel to the axis). It meets the spheres at two points, P1 and P2. BP1 and BF1 are tangents to the same sphere and must therefore be equal. Similarly BF2=BP2. It follows that
But by the rotational symmetry of the figure, the distance P1P2 is independent of the point B on the curve. Therefore for any point B on the curve, the sum of the distances from F1 and F2 is constant, that is, the curve is an ellipse with the foci F1 and F2.
The circular cone is, next to the circular cylinder, the simplest surface of revolution. It is obtained by rotating a straight line about an axis that intersects it . Thus the tangents from a fixed point to a fixed sphere form a circular cone. The rays passing through the circumference of a circle from a point source also produce a cone.
If we intersect the cone with a plane perpendicular to its axis, then we obtain a circle in the section. If the intersecting plane is slightly inclined, the section becomes an ellipse. Can you prove this, using the same ideas we used in the case of a cylinder? And try it also with a torch shining against a wall at night!
Now, if we incline the intersection plane more and more, we obtain two other curves. They are the hyperbola and the parabola, which we will consider in future issues of Zimaths.
The author, who joined the UZ Maths Department in 1996 from Israel, got much of the information for this article from "Geometry and imagination" by D. Hilbert and S. Cohn-Vossen (Chelsey Publishing Co. NY 1956)