About a few Geometric Shapes and Brachistochrone Problem

What is a Catenary Curve?

In physics and geometry, a catenary is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends.

Catenary curves were for a long time misunderstood to be parabolas, and has a history behind this shape worth reading! (Continue reading on Wikipedia (link))

What is a Catenoid?

A catenoid is a type of surface in topology, arising by rotating a catenary curve about an axis. Solution to minimum surface of revolution problem is a catenoid – that means, a cateniod has minimum area of all the surfaces generated by revolving a curve. The catenoid was the first non-trivial minimal surface in 3-dimensional Euclidean space to be discovered apart from the plane. (Continue reading on Wikipedia (link))

three-dimensional diagram of a catenoid

What is a Trochoid?

A trochoid (from the Greek word for wheel, “trochos”) is the curve described by a fixed point on a circle as it rolls along a straight line. (Continue Reading on Wikipedia (link))

What is a Cycloid?

A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve.

The cycloid has been called “The Helen of Geometers” as it caused frequent quarrels among 17th-century mathematicians. (Continue Reading on Wikipedia (link)). The solution to Brachistochrone Problem, with fixed boundaries is a Cycloid.

Please make a note of properties like Area, Equation, Arc Length etc. on your NET Notebook.

What is Brachistochrone Problem and Why is it historically important?

In mathematics and physics, a brachistochrone curve (from Ancient Greek βράχιστος χρόνος (brákhistos khrónos), meaning ‘shortest time’), or curve of fastest descent, is the one lying on plane between a point A and a lower point B, where B is not directly below A, on which a bead slides frictionlessly under the influence of a uniform gravitational field to a given end point in the shortest time.

The problem can be solved with the tools from the calculus of variations.

Johann Bernoulli was the first to propose the problem, which was attempted by many, incuding famous names like Newton, Jakob Bernoulli (Johann’s brother), Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus and Guillaume de l’Hôpital. It was Jakob Bernoulli who first pointed at Cycloids as solutions to Brachistochrone Problem.

In an attempt to outdo his brother, Jakob Bernoulli posted a tougher version of the same problem – which was solved by Leonhard Euler into what he called the calculus of variations (in 1766 – after he lost vision in both his eyes (Wikipedia link)). Joseph-Louis Lagrange did further work that resulted in modern infinitesimal calculus.

(Continue reading about Brachistochrone Problem on Wikipedia)

The history quoted above seems a bit inaccurate. though, and is better narrated in the following video by 3Blue, 1Brown’s channel below:

An animation illustrating Solution of Brachistochrone Problem, note that the Cycliod (red line) reaches Point B faster than other curves.

Another illustration of Brachistochrone Problem on few named curves:

…and Cycloid wins the race, Circle comes second, and sixth degree polynomial comes third. Though for the advantage of traveling the least distance, note that the bead on line reaches the last!

Can you guess why it may be relevant in designing water park rides, where you slide down on a mat/balloon? In math we trust to make a safer ride, which is fun at the same time!

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