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embedded ICC: The shape of a half pipe

Rim asks (we have translated your email SMS the Castilian language, I hope you do not mind, Rim ):

Good for those who are on the other side. I have a question you do not know if it is useless but the less curious. What is the concrete form of the curve is of the skate park? Yes it makes a "U". I ask because today my math teacher has hinted that it is a concrete and studied, ie it is not made at random, and I did not know people asked where to go because I thought of you. That is my question and if possible also why. And the easiest way with 16 years old do not give to much, hehe. Well, a kiss and a greeting and thank you for those hours I spend discovering things with you.
PS: Sorry for spelling mistakes, but they are 23:14 and I have no desire to start writing properly, how bad are the phones:-p

As your question seems curious. But I'm afraid there is not much mystery behind it. Half pipe English means "half pipe." I guess because they started the first skaters to do tricks with a scooter used what they found there. And what is the half pipe? For half-circle, course. Here the shape of your half pipe. Raised edges are shaped like a quarter circle. Later they added a line in the center part to allow time to prepare for the next stunt.

If you search on Google such as making your own half pipe, there are some pages that give you more or less detailed levels. In one of these planes [. Png, 2.12 MB] clearly shows that the shape is an arc:



Let's look a bit more detail:


We can see clearly that the wooden supports are actually a cirunferencia radios. As supports are perpendicular to the ramp structure (to avoid time and maximize the resistance), we conclude that the ramp is circular.

snowboarding also inherited this way to his half pipes. Currently there is a company ( Pipe Dragon ) that almost all the half pipes of snow in important events. Did the half pipes of the Olympic Games Nagano, Salt Lake City and Turin. To make a half pipe, first dig a rectangular path on the slope and then go over the edges with the Pipe Dragon:



arm is seen that they use to shape the edges of the track is shaped quarter-circle:



And as they say in the your machine specifications: they have arms 12 and 17 feet radius. If you just give the radio, is that it is a circle.

Rim But this question suggests to me a few ideas, all ICC. Why is a half circle? I have it of course, began as a half circle because that was what was (and remains of large pipes), and this has continued with "tradition" and because it is easier to build. But there are other possibilities.

To examine the first, we must go back to the year 1696. That same year Johann Bernoulli published a "challenge" in Eruditorum Act. It is to find a curve on a ball later the shortest possible time to slide from a point A to point B, acting only gravity. It was called the problem of brachistochrone , Greek for "the shortest time." Galileo had hypothesized that this curve is an arc, but I was wrong. Five mathematicians solved the problem: Newton, Jacob Bernoulli (Johann's brother), Leibniz and L'Hôpital, in addition to Johann Bernoulli himself. The solution is that the tour is a piece of cycloid. What is a cycloid? I will not put the formula, but an illustrative image: the cycloid is the path following a point on the rim of a wheel when it rotates without slipping:






If we turn to the cycloid, be a kind of bowl. Dpositada a marble in the bowl, off center, take exactly the same to reach the lowest point, regardless of how high we let (¡!). This is the problem of tautochrone (same time), decided by Huygens.

image Mathematikum museum, the first museum in the world of mathematics in the city of Giessen, Germany. The blue ramp consists of a straight line and a brachistochrone. The ball falls through the brachistochrone falls before that of the straight line, although it travels farther. In the two-lane ramp that is red. It is a tautochrone. If two balls are placed at different heights reach the bottom at the same time. The ball travels higher up but reaches farther faster because the slope is greater.

Updated : Odo (thanks!) Provide us with a great animation on three marbles tautochrone swinging on a curve, with different amplitudes. Shows how the periods are the same for different amplitudes:



Returning to the cycloid, if the item is not on the edge wheel, but inside, we would have contracted cycloid reduced (at school told me hypocycloid called, but I'm not sure). Well Updated hypocycloid not called, but small cycloid. See this comment Dani (thanks!):






What does this new special curve: As the shape of the soundboard of violins are made of sections of cycloid hypocycloids reduced. That's nothing.

One last special curve on which Masgon Patxi and I have talked over once the catenary. When a cable hung from two points, it is curved by the weight: Galileo supueso (again, incorrectly) that the form adopted cable TV. In fact, a hyperbolic cosine. This curve has the advantage that if it invests an arc that has no transverse tension. Gaudi, Gaudi again, used it extensively.

Esquemilla a bridge with suspension cables in the form of catenary

As CPI data, in the jargon of the trains is called the catenary to the whole supply: the wires (these do have a catenary), the towers that hold the cables and all fishing. But at the RENFE (Please Push Our Railways Damaged) can be forgiven any synecdoche, what the hell.

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