The cicloide: what is the shortest way?

This article is my contribution to the second edition of the Carnival of mathematics organized by Juan Pablo.

Introduction

The world of the curves is a really interesting world. We can be forms of many types, from more acquaintances comoun I segment (yes, although to much they surprise a segment it is a curve in the mathematical sense of the concept) or a circumference portion, up to someone the hipopede of Eudoxo or the cuadratriz.

I feel this so extensive world of the curves we can find many with characteristics very interesting. The cicloide is, undoubtedly, one of them. It has a few very curious properties that, on having been seen, collide with our own intuition. This curve is going to be the protagonist of this article.

What is the cicloide?

Let's begin this point presenting to our friend the cicloide:

The cicloide is the curve planned by a point of a circumference (called circumference generatriz) when this one turns on a line (called straight guideline) without sliding for her.

Namely the cicloide is the curve that appears in red in the following graph:

Considering a circumference of radio and a point of the same one placed in the origin of coordinates, the equations paramétricas of an arch of the cicloide generated by this point, on having turned the circumference on the axis, are:

, with

The cicloide has been a curve much studied along the history. Already at the end of the XVIth century, Galilean he had studied this curve, obtaining certain approaches on calculations related to her (in particular on the area shut up by an arch of cicloide). Mersenne, possibly after knowing these studies of Galilean, attracted attention on the mathematicians of this epoch (we are already in the XVIIth century) towards this curve. And many were those who came to the call. So much it was the sense of expectancy created by this curve that finished with being known as the Hellene one of the geometers for the disputes quantity between mathematicians who provoked the studies related to her.

The case is that one of the first ones that they obtained turned out on the cicloide was Roberval. Mersenne proposed in 1628 the study of this curve to him and a few years later, on 1634, Roberval demonstrated that the area shut up by an arch of cicloide is exactly three times the area of the circumference that generates it. Further on also it found a method to plan the tangent one to the cicloide an any point of the same one (problem solved also by Fermat and Discardings) and realized calculation related to volumes of revolution associated with the cicloide.

Roberval did not publish these results in his moment, since he wanted to keep them in certain secret to use them as problems to be proposed the candidates for his chair. For it, when Torricelli (mathematician who also was interested by this curve) published his solutions to several of the questions solved by Roberval without mentioning him, he believed that it was a question of plagiarism. But the Torricelli studies had developed from independent form to those of Roberval. In the end the history was just with the two: Roberval was the first one in finding the solutions and Torricelli the first one in publishing them.

But the thing did not stay there. In 1658 Christopher Wren calculated that the length of an arch of cicloide is four times the diameter of the circumference that generates the above mentioned curve. And many other they were the mathematicians who dedicated part of his time to her, between that there are illustrious Pascal, Huygens, Leibniz, Newton, Jakob and Johann Bernoulli …

What properties does it have?

The big interest caused by this curve comes from the curious characteristics that it possesses. Apart from the already mentioned calculations, the cicloide has two really interesting properties and that, like handyman at the beginning of the article, in certain way they commit an outrage against our intuition. Specifically there are his condition of braquistócrona and his condition of tautócrona. We are going to try to tell what they mean these two properties.

Braquistocronía

The term braquistócrona means the minor time. The problem of the braquistócrona can be enunciated of the following form:

Given a point in a plane and another point of the same placed plane vertically further down that (without going so far as to be vertically just under), to find the curve that it joins and that makes minimal the time that takes a mobile point in coming of to to be submitted to the action of the gravity

The situation of the points is something like that:

At first it would not be strange to think that this curve is a straight line (a segment in this case), since in a plane a straight line represents the shortest distance between two points. But we are not speaking about distances, but of times. Will the answer keep on being also the straight line? Let's see this video in which two appear cicloides and a segment and let's answer later:

Since it is possible to see the balls (the mobile point) they come earlier to the destination when they go down for the cicloide. Namely that in the cicloide the trip time is minor than in a segment. In fact the cicloide minimizes this trip time, that is to say, the cicloide is the braquistócrona. Onlooker: truth?

Tautocronía

The braquistocronía is not the only curious property of the cicloide. In fact it has one that is more surprising if it fits. We might enunciate it of the following way:

Let's suppose that we have a cicloide that "hangs" down and that we drop along her two balls from different points. The question is that it does not matter since we them drop points since the balls come simultaneously to the rock bottom.

This property is named tautocronía (that means the same time). We are going to see it in a video:

To finish I leave this linkage to you. It has seemed interesting to me because puncturing in each of the grids that appear we create cicloides and can see graphically two properties commented previously.

Sources:

• History of the mathematician, of Carl B. Boyer.
• Cicloide in the Spanish Wikipedia.