3. Other curves

Analytical geometry

Not even conical sections can satisfy all the needs of science and technology. It is therefore necessary to go beyond conical sections, and to consider even more complex curves.

A very effective way to represent these curves on the plane consists in using their equation.

If you trace two perpendicular lines on the plane, it is possible to individuate every point through its Cartesian coordinates, that more or less represent the distance of the point from the two axes. They are called thus in honour of R. Descartes (Cartesius) who first used them systematically to describe curves.

The two coordinates are usually indicated as x and y. To each value of the couple (x,y) corresponds a point on the plane. When x and y vary in every possible way, the corresponding point describes the whole plane. If, instead, the coordinates are subject to an equation, the point that they represent is forced to move along a curve, of which the equation constitutes the analytical representation.

For example, if one fixes x through the equation x=1, the corresponding curve is a vertical straight line; while the equation y=3 represents a horizontal line. More generally, a first-degree equation (that is, an equation in which the variables x and y appear at the power of one) represents a straight line, while a second-degree equation creates one of the conical sections (including the circumference).

One can study curves with a third-degree equation, or fourth-degree, or of a higher and higher degree. For these, we find in the Encyclopedie di Diderot e D'Alembert an universal machine that allows, adding layer upon layer, to trace curves of higher and higher degrees. The one that has been realised has three layers, and therefore traces third-degree curves. It is a rather heavy machine, whose complexity is mainly due to the need to reduce to the minimum the friction that would otherwise prevent it from functioning.

Spirals

Other curves have an equation that has no degree (or rather, they cannot be expressed through a polynomial). Some of these are distinguished by some special properties, that make them particularly useful and interesting. One of these is Archimedes' spiral.

An ant starts from the beginning of a turntable's platter, and walks outward in a straight line. But if the platter starts turning at the moment when the ant starts walking, and if both the ant and the turntable maintain a constant speed, the ant will trace a spiral curve, called Archimedes' because it was firstly studied by the mathematician from Syracuse.

We can substitute the ant with a marker, which we move from the centre to the edge with the most constant speed possible. We will then see Archimedes' spiral form, with more spirals the slower we move the marker.

An interesting application of the spiral is found in sewing machines, in the part that spins the thread around the spool. The thread coming from the skein is kept taut, and it is rolled around the spool, which turns and moves up and down, in order to allow an uniform distribution of the thread. And this is where the spiral comes in. To make sure that the thread is spun uniformly on all the parts of the spool, the oscillation movement needs to keep a constant speed. If the oscillation movement were faster in the centre and slower towards the ends, when the spool must change directions (which is what would happen unless one takes specific measures), the thread would not be distributed uniformly, but would tend to pile up at the spool's extremities.

Therefore, one needs a mechanism that makes the spool oscillate with a constant speed. This is obtained by regulating the oscillation of the spool through a profile made of two coupled spiral arcs. Here we see this specific mechanism of the sewing machine, and a working enlarged reproduction of it. The vertical rod moves alternatively up and down, always with the same speed.

Caustics

The description of particular curves could go on for quite some time. Those who are interested can see on the computer some simulations of mechanisms to draw various types of curves. We will proceed with the visit and examine both the general structure of curves and the specific properties of some of them.

We have seen how conical sections, and in particular parabolas, have the ability to concentrate light rays in one point. Other curves do not, in general, have this property. This however does not mean that the reflected rays scatter completely, illuminating space more or less uniformly. Very often they concentrate not in a point, but on a curve: the caustic. Like the focus of conical sections, the name of this curve derives from "burning"; in fact, the term caustic means burning. In reality, the name does not correspond to an actual ability to light up a fire, at least as long as the light source has a low power, like a light bulb.

Caustics are often seen in everyday life, such as when a kitchen bowl is lit obliquely: the rays reflecting on the vertical side draw a curve on the bottom, which is called a reflection caustic, because it is obtained through the reflection of light rays.

Caustics can also be obtained through refraction, when the rays coming from a point penetrate a medium with varying density.

Envelops

>Reflected or refracted rays are all tangent to the caustic they form. More in general, if one takes a family of straight lines (as in the caustics the reflected or refracted rays), they can be uniformly distributed on the plane, as it happens when for example they are parallel, but they can also concentrate onto a curve, to which they are all tangent. This curve is called the envelope of the lines. Thus a reflection caustic is the envelope of the rays reflected by a mirror. One can create envelopes of lines by pulling strings between various points on the plane or in space. The rotating tetrahedron at the entrance, the bicycle wheel above your head, the perpendicular lines to the parabola, show curves obtained as envelopes of straight lines. To build an envelope one need not start with straight lines - the same can be done with curves. Every curve of the family will be tangent to the envelope, that is it will have the same tangent line in the contact point. On the computer, it is possible to see various examples of envelopes of circles and also of more complex curves. An interesting example comes from ballistics. Ignoring the resistance of air, the shell shot by an artillery piece will describe a parabola, the shape of which will depend on the power of the cannon and on the shooting angle. Let us suppose now that we have a cannon which always shoots with the same power, but that can vary the angle. Where must one stand to be sure one is outside the cannon's range? The answer is simple: one traces all the shells' trajectories with varying angles, and one stands outside the region these cover. This region is delimited by the curve enveloping all the parabolas, which in this case is also a parabola. Let us note that the caustics, and envelopes in general, are not curves that physically exist. What exist are the rays of light, or more in general the lines of the family. The curve that they envelop appears only because they concentrate on it. Thus the light rays concentrating on the caustic shed more light on the area of the plane corresponding to the curve, and they draw it. The same happens with the lines normal to the parabola, which can be seen in the entrance hall. They trace a curve that can be seen, although it does not exist - only the lines enveloping it exist. In some cases it is important to avoid the formation of caustics, and to have a light which is as uniform as possible. For example, in photography the uniformity of illumination is essential. The Fortuny lamp, which is still in use, thanks to the shape of its reflecting surface, supplies a light density which is constant in every point.

Curvature and its environs

Just as of all lines that pass through a point P of a curve, the tangent is the one that best approximates the curve, of all the circles that pass through P there is one which best adapts to the curve's shape near P. This circle, whose centre is on the line perpendicular to the curve (or to its tangent, which is the same thing), is called the osculatory circle.

We can thus measure the curvature of a curve. The tangent allows us to determine the direction of a curve C. If we imagine a point moving along C, we can think that in every moment the point moves in the same direction than the tangent. Analogously, the curvature of C will be given by that of its osculatory circle, and since a circle is more curved the smaller its radius, we can measure C's curvature through the inverse of the osculatory circle's radius, also called curvature radius.

As point P varies on the curve, the curvature centres (the centres of the osculatory circles) describe a second curve, which is called the evolute of the first. This curve is also the envelope of the lines perpendicular to the given curve.

Reciprocally, the first curve is the involute of the second. The involute of a curve can be materially obtained by attaching a piece of string to the profile of the curve, then slowly detaching it, being careful to keep the detached part always taut. The free extremity of the string will then describe the involute. Thus it is possible to draw the involute of the circle.

Figures with constant thickness

In addition to parallel lines, there are also parallel curves, which always keep the same distance. The sides of a road are parallel, as are the shores of a canal, and the guides on which the coins slide in a slot-machine. If we want the coins not to rattle while they fall, we must make it so that they always touch the sides of the guide. Round coins are good for that, since they can slide on two parallel curves always touching the two sides.

One might think that circles are the only figures with this property, that is the only curves with a constant thickness. In fact, this is not the case: we can construct many other figures that always touch two parallel curves. The simplest is a sort of triangle with the sides formed by circular arches. sliding inside a guide with parallel sides. From this, cutting near the angles to make them sharp, we obtain the point of an (eccentric) drill to make square holes. One can find drills to make hexagonal holes on sale.
One can also make curves of constant thickness with various shapes. The English 50-pence coins are among those.

The cycloid

A curve with very peculiar properties is the cycloid, that is the curve described by a point on a rotating circumference. The cycloid can be seen by attaching a light bulb to the wheel of a bicycle, possibly in the dark, or just rolling a circle on which we have marked a point.

Although simple to describe, the cycloid is a relatively modern curve. Among the first, if not the first, to examine it was Galileo, who observed how elegantly it described the arch of a bridge. During the whole seventeenth century, the cycloid was an object of study by the best mathematicians, who determined its length, its enclosed area (which Galileo had first guessed as being three times the generating circle, as many geometers later demonstrated independently, among them Torricelli), its balance point and other connected quantities.

But surprises lay in wait until the end of the century, when the cycloid became the solution of two important mathematical problems.

One deals with the fall of weights, and it marks the birth of a totally new branch of mathematics: the calculus of variations.

Let us suppose that we want to send a ball from a point A to a lower point B, but not along the vertical, We can imagine building a profile which unites the points A and B, and of making the ball roll along it. Obviously, there are infinite such profiles. We then ask ourselves: is there one that minimises the trajectory time?

At first sight, one might think that the solution is the straight line joining A and B. However, if we stop and think, we will see that the straight line is indeed the shortest line between two points, but not necessarily the fastest: it could be best to make the ball start more vertically, so that it would instantly gain enough speed to compensate for the longer path.

This problem of the brachistochronous curve, or curve of the shortest time (from the Greek brachistos, shortest, and chronos, time), was posited by Johann Bernoulli as a challenge to the mathematicians of his time, and was solved among others by Newton and Leibniz - the fastest curve is a cycloid. Bernoulli was so proud of his discovery, that he had on the title page of his Collected Works the drawing of a dog trying - in vain - to catch up with a cycloid, and the motto: Supra invidiam, above envy.

We have compared the cycloid with the straight line: letting two balls fall at the same time along those two curves, we can see how the time for the cycloid is considerably shorter than the one for the straight line.

The cycloidal pendulum

The cycloid is also concerned in another problem, this time of a more technical kind. Galileo again had observed how the oscillations of a pendulum would take more or less the same time, independently from the amplitude of the oscillations. The first pendulum clocks originated from this observation by Galileo.

In reality, the pendulum's oscillations are not exactly isochronous: the time it takes to complete an oscillation does depend on the oscillation's amplitude, and it is longer for wider oscillations. Only for very small oscillations we can consider the time substantially constant, and it is those small oscillations that are used for pendulum clocks.

One can then wonder: how must a pendulum be so that the all its oscillations require exactly the same time, or in one word (also derived from Greek: isos, equal, and chronos, time) are strictly isochronous?

Let us see things from a slightly different point of view. In a normal pendulum, weight swings freely attached to a point, therefore along a circumference. Its oscillations are only approximately isochronous, and require more time the larger the arc of circumference. In this context, the question becomes: along what type of curve must a body oscillate so that the oscillations are perfectly isochronous?

The answer, once again, is: the cycloid. If we put two balls in two points of the cycloid, and we make them fall at the same time, they will hit each other exactly in the lower point of the curve, even if one started very near and the other very far from this point. In other words, the ball takes the same time to travel over the large arc and the small one, a sign that the oscillations are isochronous. In a circumference this does not happen: the ball that starts nearer arrives earlier.

>So, if we want to build a perfectly isochronous pendulum clock, we must have the weight oscillate along a cycloid. But how to make the weight move along this line without making it slide, that is without using a cycloidal profile? The situation is analogous to that of the first room, when we wanted to draw a straight line without using a profile. There we used a rather complex mechanism, Peaucellier's mechanism. Here we will use another trick: instead of allowing the weight to oscillate freely (in this case it would describe a circumference), we will condition its trajectory by making the string lie on two profiles.

In mathematical terms, the profiles will have to be shaped in such a way that their involute is a cycloid. And here we have a surprise: the involute of a cycloid is a second cycloid equal to the first! Consequently, the profiles must be arcs of a cycloid. In this case (and only in this case) the pendulum will oscillate along a cycloid, and will therefore be isochronous.

We have compared two pendulums: one free, moving along a circumference; and another cycloid, obtained with profiles at the top. Two photocellules measure their periods, that are visualised on the computers. As you may see, while the period of the ordinary pendulum decreases with the amplitude of its oscillations, the one of the cycloidal pendulum is strictly constant.

New tendencies

Modern geometry has continued along this road and made it even more obvious, in both directions: more generality, more complexity. Correspondingly, mathematic techniques have become more and more abstract, to the point of making it difficult, if not impossible, to describe them even approximately. Since we don't want to completely renounce giving an idea of modern developments of curve geometry, we chose two examples that represent the two tendencies: geodetics and fractals.

On a flat surface, as for example in a city square, the shortest path between two points is a straight line. But if we move along a curved surface, like the surface of the Earth, it is not possible to go in a straight line, and in its place there is a curve, called geodetic, whose length is the minimum among all curves that touch both its ends. Just like as we pulled a string taut to get a straight line at the beginning of the exhibition, we could find the geodetic of a convex surface by pulling a string between two points. In the case of the Earth, which is more or less a sphere, the geodetics are the the maximum circles, those that divide the sphere in two equal parts. To go from one point to another it is best to move along the maximum circle that goes through the two points. This is the reason for polar routes in intercontinental air travel: on the maps they seem much longer, because they are drawn on a plane, but it is enough to pull an elastic band on a globe to see how actually the shortest route between Pisa and Los Angeles goes near the North Pole. The second example regards the very concept of curve. Those we have seen so far correspond to the intuitive idea of a curve that we all have; they are a single-dimensioned object, which we can think can be obtained bending a piece of string. If they have some special points, such as the edge of a polygon, these are finite and isolated. Around the end of the nineteenth century, pathological curves with surprising properties begin to pop up. One of which is Peano's curve, which fills a square and poses the problem of the meaning of dimensions: another is Koch's curve, which has no tangent in any point. Finally, in recent years, some new object have emerged, the fractals, which have a fractionary dimension and the surprising property that every part of them, no matter how small, is similar to the whole. These forms, whose generation is relatively simple, produce images of a remarkable beauty: images with which we conclude our visit in the world of curves.