Echoes and reflections


The simplest way to draw an ellipse is with a piece of string, a bit like with the circumference we drew at the start.

A circumference has all the points at the same distance from the centre, so we can draw it with a string, keeping one end fixed and rotating the other one with a marker. When the circumference gets longer and becomes an ellipse, the centre, so to say, divides up into two points: the foci. These have a characteristic property: if you take any point on the ellipse and you unite them with the two foci, the sum of the lengths of the two segments is always the same.

This property can be used to draw an ellipse on the ground: fix two stakes on the foci and attach to them the two ends of a string. If now we bring a pencil around so that the string is always kept taut, the curve we have drawn is an ellipse, called the gardener's ellipse, because this method is often used to draw elliptical flower beds.


The same property can be used to build elliptical gears. If you take to identical ellipses, dispose them so that each of them can rotate around one of their foci, and if the distance between the stakes is equal to the length of the string that describes the ellipse, the two ellipses always remain tangent, and the rotation of one drags along the other. Moreover, if the first one rotates uniformly, the second has a variable velocity, higher the nearer the tangent point is to the fixed focus. If the two ellipses are very oblong, while the first rotates in 24 hours, the second takes up almost all the time to go half around, and goes the other half in a few minutes. This phenomenon is used for date display mechanisms in watches.


Another important property of the ellipse is that the line perpendicular to the ellipse in any of its points divides the angle formed by the string (that is, by the lines that link the point to the foci) in two halves. This property is relevant to light reflection. When a ray of light reflects on a mirror, be it flat or curved, the perpendicular to the mirror makes equal angles with both the incident ray and with the reflected ray, that is, with the incoming and the outgoing rays. But then, a ray of light which starts from a focus behaves like the string in the gardener's ellipse: after having reflected on the ellipse, it will strike the other focus.


The same is true for any type of ray: light, sound, heat. In every case, all the rays that originate from a focus, after a reflection on the ellipse, will concentrate in the other. This is the reason for the name foci ("fires"): if one places a source of heat on one of the foci, the heat concentrates in the other and it can light up a piece of paper or an inflammable material.

A simple kitchen pan (of an approximately elliptical shape) with its bottom covered in water can be used to illustrate the phenomenon. If you touch the water with a finger on one of the foci (marked by a dot on the bottom), you create concentric waves that, after having reflected on the side of the pan, concentrate on the other focus.

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