2. Conic sections

A light in the dark


If we light up a wall with an electric torch keeping it perpendicular to the wall, the lit portion is more or less circular. Let's now begin tilting the torch upwards: the circle deforms and assumes an oblong shape, like a serving tray or a stadium: it's an ellipse.

If we keep tilting the torch, the ellipse gets longer and longer. While one of its ends remains near us, the other moves further and further: if the wall were infinite, the lit area would become bigger and bigger, until for a certain inclination of the torch, it would become infinite. The figure thus obtained is a parabola.

If we tilt the torch even further, the lit area becomes even bigger, and it assumes the shape of a hyperbole.

The three figures which are obtained one after the other, or rather the curves that delimit them, are collectively called conical sections, since they are obtained sectioning a cone (in our case the cone of light projected by the torch) with a plane (the wall).

Conical sections are often found in the most common situations: a table lamp draws two hyperboles on the wall, the shadow of a ball is an ellipse, a stone thrown by a sling takes a parabolic path. In the past, the theory of conical sections was essential to build sundials. In its apparent motion, the sun draws a circumference: the rays that pass by the tip of a sundial's stylus then form a cone, that cut by the wall creates a conical section, which at our latitudes is a hyperbole, on which the shadow of the stylus's tip moves.

One can draw an ellipse using the great three-dimensional compasses which the Arab geometers had called the perfect compasses. The inclined rod that rotates around the vertical axis describes a cone, which is intersected by the drawing plane. According to the latter's inclination, one can obtain a circumference (if the plane is horizontal) or an ellipse, longer the more inclined the plane. If one could increase indefinitely the plane's inclination, one would obtain first a parabola and then a hyperbole.

In the same way, depending on the machine's inclination, the plane of the water, which is always horizontal, intersects the water according to an ellipse, a parabola or a hyperbole. A second cone, symmetrical to the first, shows the two branches of the hyperbole.

Other elliptical compasses, some of which are exhibited, can be built using the various properties of this curve: one can also find them on sale. A parabola or a hyperbole are more difficult to draw.

go back to the top of the page

Echoes and reflections

ellisse

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.

ingranaggi

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.

tangente

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.

bacinella

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.

go back to the top of the page

Burning mirrors

ellisse As the generating plane becomes more and more inclined, the ellipsis becomes more and more oblong, and the second focus moves away from the first. When it is transformed into a parabola, the second focus disappears (sometimes one says it has gone to infinite) and there is only one left. And while in an elliptical mirror the rays originating from a focus ended up in the other, in a parabolic mirror the rays that start from the only remaining focus are reflected parallel to the axis, and viceversa the rays parallel to the axis concentrate in the focus.

This latter property of the parabola can be used to build a burning mirror, that is a mirror that concentrates solar rays (which we may consider parallel because of the great distance of the Sun) in the focus, where they can light up inflammable materials. We have built an indoors burning mirror, substituting solar rays with those from a halogen lamp. We have put the lamp in the focus of a second parabolic mirror, from which the light rays come out parallel after being reflected; a second reflection on the first mirror concentrate them on the focus, where in a short time they light up a match.

The same principle is at the base of the parabolic microphone: the sound waves, which can be considered parallel if they come from afar, are reflected on the parabola and concentrated on the focus, where a microphone is places. This mechanism can hear very far away and faint noises. The great radiotelescopes and the parabolic dishes of satellite TV are also built using the focal properties of the parabola.

Parabolas are often found as solutions of technical and scientific problems. A stone thrown obliquely describes a parabola, just as the cable of a suspension bridge also assumes the shape of a parabola.

go back to the top of the page

A mirage

miraggio

Using two identical parabolas we can create an interesting phenomenon. What you see is a sort of black box in the shape of a disc, with a flat hole on which a die is placed. You can light this die with a torch, and at first sight it has all the properties of a real object. But when you try to take or touch it, you see that there is no die: it's an optical illusion.

This mirage has a simple explanation. The box is actually made of two identical, superimposed parabolic mirrors, each of which has its focus more or less in the other's vertex. On the bottom of the lower mirror is the die, while the upper one has a hole through which it is possible to see inside. When one looks through this hole, or lights it up with a torch, the light ray makes two reflections before hitting the die, thus forming a real image of it, absolutely identical to the original.

go back to the top of the page


Rotating

ellisse

In fact, burning mirrors and the mirage are made not with parabolas, but with surfaces obtained rotating parabolas around their axis. These surfaces are called rotation paraboloids. A paraboloid can be obtained rotating a liquid fast enough inside a cylindrical container. If instead the liquid is contained between two close planes, we have a parabola.

Similarly, when we rotate an ellipse or a hyperbole, we obtain a rotation ellissoid or hyperboloid.

volta

These surfaces also have reflecting properties similar to the ones of the paraboloid. We have built an elliptical chamber, obtained by rotating a half-ellipse along the axis. A phenomenon similar to the burning mirror happens in it: if we place ourselves in one of the foci and we speak towards the elliptical wall, even in a very low voice, those who are in the other focus receives the voice very clearly, while whoever is in between hears hardly anything.

iperboloide

The rotation hyperboloid has the notable characteristic of being a lined surface, that is of being constituted only of straight lines, as we can see from the hyperboloid obtained with strings.

iperboloide

This produces an unexpected phenomenon: rotating a straight line opportunely inclined, one can make it go through a hyperbole-shaped slit. The rod, rotating, describes a hyperboloid, which, intersected with a plane, leaves us with the two slits through which the rod passes with no difficulty.

cubo
cubo in rotazione

The same surface is obtained by rotating a cube. While the upper and lower edges, that meet the rotation axis, form a cone, the intermediate ones, which do not meet the axis, generate a hyperboloid, which is visible thanks to the persistence of images on the retina.

go back to the top of the page