Скачать или смотреть How to draw cycloid with lamp camera bicycle on photo. Math curves in nature light photography

Drawing cycloid can be a really great task to be done at schools during math lessons. Light painting cycloids using light and camera on photography is what we found the most interesting and spectacular way of drawing this mathematical curve.
Compass, pencil, ruler, geometry tools, paint - you don't have to use any of traditional methods.
Led lamp, stick, tape, cylindrical vase, camera, tripod - are what we use in our short tutorial on easy way how to draw cycloid in photo.
How to paint cycloid with light in photograph - here you can find step by step easiest tutorial (camera settings etc.) at the example of photograph "Cyclists on cycloids" by Bruno Stangret pol. Cykliści na cykloidach (competition ambassador: Michał Stangret). This photo was awarded special prize by the President of the Polish Mathematical Society prof. dra hab. Wacław Marzantowicz in International Photographic Competition "Mathematics in Focus" - X edition. The competition is part of the international Mathematics in focus scientific and didactic project. The organizer is the University of Szczecin and the Maria Grzegorzewska University (Academy of Special Education) in Warsaw. Prof. Małgorzata Makiewicz is the project leader. The aim of the project is to build a common ground between mathematics and the art of photography, support mathematical education, mathematical thinking, popularize knowledge and mathematical culture The competition won the first place in the category of the best University Scientific Project (University Laurel Poland) in the University Laurels 2019 competition organized by the Forum of Polish Universities.

Brief history of Cyclists on cycloids (Cykliści na cykloidach) photo, plus some extra information on cycloids you can find below.
The bicycle ride turned out to be a math lesson when we noticed that the safety wheel lights draw an interesting shape while riding. In the photo you can see two red chains of curves consisting of many arches connected with each other.
The smaller chain of arches was painted with the light emitted by a red lamp fixed to the wheel of a smaller bicycle. The bigger chain of arches was drawn by a red lamp put at the wheel of a larger bicycle (as close to the wheel rims as possible)
These arches are mathematical curves called cycloids.
A cycloid is the curve traced by a point on the rim of a wheel as the wheel rolls along a straight line without any slipping.
The cycloidal shape can also be seen in the arcades of the building* captured in the background of the photo.
A characteristic cycloidal shape can also be noticed in the shape of the tow bar connecting both bikes, thanks to which a smaller bike can easily and stably be pulled by the bigger bike (left part of the photo).
Historically, the cycloid shape has been employed to great success in many physical contexts, has many interesting properties, intrigues with new and new features, and still is being widespreadly implemented by engineers and designers in new fields.
That all have its roots in the works of Galileo who also gave the current name of the curve in 1599 (from Greek kykloeidēs, kyklos meaning ring, circle).
There are a lot of interesting anecdotes on the curve. One of them dates back to June 1696 when Johann Bernoulli in Acta Eruditorum posed a problem like that: Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time?
He addressed the question to the most brilliant mathematicians in the world ant this was really a challenge. Leibniz wanted Johann Bernoulli to allow a longer time for the challenge, than the set six months. Eventually five solutions were obtained, from: Gottfried Leibniz, Guillaume de L'Hopital, Isaac Newton, Jacob Bernoulli (Johann's brother) and of course Johann Bernoulli.
The solution turned out to be a brachistochrone curve which is a part of an inverted cycloid, now called the curve of fastest descent (from the Greek brakhisto "the shortest" and chronos "time").

In the background of the photo you can see the back / garden facade of the building where Warsaw School of Economics (Poland) is located, designed by Jan Witkiewicz Koszczyc before the Second World War.