Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# Problem I. 258. (January 2011)

I. 258. According to the ancient Greek geocentric model, Earth is the center of the universe, and the Moon, the Sun and the other planets orbit around it, while stars on the celestial sphere are fixed. They thought that the orbit of a moving celestial body could only be a perfect figure, in this case, a circle. However, in most cases they could observe that the motion of a planet relative to the fixed stars is not uniform, or even retrograde. To resolve this inconsistency, they though that their motion can be described by several circles instead of one: in the simplest case, the planet is assumed to move uniformly in a small circle (the epicycle), whose center in turn moves uniformly along a larger circle (the deferent, with its center being the Earth itself).

In our figure, Deferens'' is deferent, Epiciklus'' is epicycle, Bolygó'' is planet, Ekliptika'' is ecliptic and Látszólagos helye az égen'' is its apparent position in the sky''.

Since even this model was not a perfect one for some planets, they introduced several more epicycles: the center of a given epicycle moves along the previous epicycle, and so on, and the planet is on the outermost epicycle. The radius of the epicycles, moreover, the orbital periods of the epicycles are allowed to be different. With this system of epicycles they could correctly reproduce all observed planetary motions.

You should create an example similar to ours by GeoGebra (a freely downloadable software, http://www.geogebra.org), displaying the motion of 2 planets. Each planet should have 2 epicycles with modifiable radius by a slider. Additional sliders should correspond to the location of the centers, the orbital periods and the progress of time. The actual and visible motion of the two planets should be traced, but auxiliary points and lines should be hidden. You should use colors to highlight the important elements. You should also put appropriate legends so that one can easily understand your work.

Your GeoGebra solution i258.ggb should be submitted.

(10 pont)

Deadline expired on February 10, 2011.

Sorry, the solution is available only in Hungarian. Google translation

Mintamegoldásként Kocsis Mátyás munkáját i258.ggb és Varga Erik i258(2).ggb megoldását adjuk közre. Az első külső formáját tekintve igényesebb, de nem olyan szép, hogy az animáció több változóval történik. A második megoldás jobban tükrözi a "valóságot", mert egy idő paramétertől függ a teljes mozgás.

### Statistics:

 7 students sent a solution. 10 points: Kocsis 789 Mátyás, Varga 256 Erik. 9 points: Seres Márk Dániel. 8 points: 1 student. 7 points: 1 student. 5 points: 2 students.

Problems in Information Technology of KöMaL, January 2011