# Epitrochoids and Hypotrochoids

On the page Geometry - Sun, earth and moon, it is shown how to make a model of the path of the moon without using trigonometry. By using trigonometry, you can define the path of the moon as a curve. The curve is an example of an epicycloid which is a curve closely related to curves called epitrochoids.

Epitrokoids and hypotrokoids are curves that children can draw by using a spirograph.

## Make a curve

Make a model of the sun, earth and moon. Let $$O$$ be the position of the sun, $$A$$ of the earth, and $$B$$ of the moon. Let $$R$$ be the radius as the sun revolves around the sun, $$r$$ the radius as the moon revolves around the earth,and $$m$$ the number of months in a year.

Make a slider tMax representing the time. The curve drawn by the moon, is drawn between the time zero and the time $$tMax$$.

The time corresponds to the angle extended by the earth on its path around the sun. The angular velocity of the moon is $$m$$ times as large. From the picture above, we get that $$A$$ has the coordinates

$A=(R \cos\alpha, R \sin\alpha)$

$$B$$ has the coordinates

$B=(x(A)+ r\cos(m \alpha),y(A)+r\sin(m \alpha))$

If we choose to write the coordinates of $$B$$ in terms of the angle $$\alpha$$, they are

$B=(R \cos\alpha+ r\cos(m \alpha),R \sin\alpha+r\sin(m \alpha))$

The command for making a curve in a GeoGebra, using a parameter $$t$$, is

Curve[x(t), y(t), t, start, stop]


The command for drawing the curve of the moon is hence

Curve[R cos(t)+r cos(m t), R sin(t)+r sin(m t), t, 0, tMax]


## Epitrochoids

An epitrochoid is drawn when a small circle rolls on the outside of a large circle. If $$A$$ is the midpoint of the small circle, a pen is put in a hole $$B$$ at a distance $$d$$ from $$A$$.

If $$A$$ moves in a counterclockwise order, $$B$$ moves in a clockwise order.

Let $$R$$ be the radius of the larger circle, and $$r$$ be the radius of the smaller circle, then the coordinates of $$A$$ are

$A=((R+r)\cos \alpha, (R+r)\sin \alpha)$

The coordinates of $$B$$ are

$B=(x(A)-d\cos(m\alpha), y(A)-d\sin(m\alpha))$

$$m$$ is the number of revolutions made by $$B$$ around $$A$$, as $$A$$ makes one revolution around $$O$$. $$m$$ can be found by taking the ratio of the circumference of the circle made by $$A$$ around $$O$$ to the circumference of the circle made by $$B$$ around $$A$$. Since the circumference is proportional to the radius, $$m$$ is also the ratio of the radii, yielding

$m=\frac{R+r}{r}$

## Hypotrochoids

A hypotrochoid is drawn when a small circle rolls on the inside of a large circle. If $$A$$ is the midpoint of the small circle, a pen is put in a hole $$B$$ at a distance $$d$$ from $$A$$.

If $$A$$ moves in a counterclockwise order, $$B$$ moves in a clockwise order.

Let $$R$$ be the radius of the larger circle, and $$r$$ be the radius of the smaller circle, then the coordinates of $$A$$ are

$A=((R-r)\cos \alpha, (R-r)\sin \alpha)$

The coordinates of $$B$$ are

$B=(x(A)+d\cos(m\alpha), y(A)-d\sin(m\alpha))$

Using the same reasoning as for the epitrochoid, we get that

$m=\frac{R-r}{r}$

## Hypocycloids and epicycloids

When $$d = r$$ the curves are traced out by a point on the rolling circle. I these special cases, the curves are called hypocycloids and epicycloids. For interactive examples of patterns made by rolling hypocycloids and epicycloids see Rolling Hypocycloids and Epicycloids.

# animated gifs:

Hyotrochoid on tumblr.

# reference:

the picture of the spirograph made by Kungfuman