# Epicycles

Make a model of the sun, the earth and the moon following the instructions at Geometry - Sun, earth and moon.

## Locus

The trajectory of the moon, which is shown by the trace, depends on the trajectory of the earth, which is a circle. Since the trajectory of the earth is well defined, GeoGebra can calculate the trajectory of the moon and draw it. Use the tool Locus , then click on the moon and the sun in that order.

## Vary the radii

One way to make the model more interesting is to vary the radii. Use the tool Slider
to enter two sliders, one for each radius. Name the radius of the
earth `R`

, and the radius of the moon `r`

. The sliders should not take on negative values.

The circles in the model are defined by a midpoint and by a number representing the radius, this number should be changed to
a slider-value. Right click on the circle representing the path of the earth and choose `Object Properties`

. Under the
tab `Basic`

there is a line `Definition`

. Change the radius to the slider `R`

.
Then change the definition of the other circle.

By varying the sliders you can generate various trajectories.

## Vary the number of months in a year

You can vary the number of months in a year in a similar way, i.e. the number of revolutions the moon makes around the earth when
the earth makes one revolution around the sun. When you enter the slider, you can click the `Integer`

button.

.

Then change the definition of the moon.

.

The circle around the earth, on which the moon moves, is called an epicycle. The trajectories generated when changing the sliders are called epitrochoids.

This model is easier to make if polar coordinates are used. See GeoGebra Tutorial - Polar Coordinates.

## Insert images

Finally you can insert images instead of points. Make images of your own, search the Internet, or use the images below (right click on an image and save). For more information about handling images in GeoGebra, see GeoGebra Tutorial - Insert Images.

## More information

The pattern made by an object on an epicycle is the same kind of pattern that a spirograph can make. The topmost applet is made using trigonometry, as described on the page Trigonometry - Make a Spirograph.

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License