What is a Spirograph?
Drawing a figure:
||The Spirograph is a mathematical toy, which you can use
for drawing nice figures.
In the simplest case it exists of a fixed circle, used
as a template, and a smaller rolling circle with holes.
Hold the template (clipping). Stick a ballpoint pen through
one of the holes of the smaller circle. Roll it inside the bigger fixed
circle. Draw it on a sheet of paper. Cogs at the edges of the circles guarantee
a reliable unrolling and prevent sliding.
It is a pity that the spirograph introduced here is non-transparent.
It was a gift of my savings bank at the "World Saving Day", if I remember
If you look at the figures more accurately, you see
that they aren't closed after five revolutions. This is intended:
||You can draw rosettes with five points easily.
I chose only three of twelve possible holes. They are
representative for the possible shapes.
||If you go on drawing you'll get some nicely shaped rosettes.
This shows the attraction of spirograph figures.
You can find parameter formulas for the figures, which
describe them mathematically.
||The large circle (radius R) is given and fixed. The small
circle (radius r) rolls on its inner side. There is point P fixed on the
It has the distance a from the centre M of the small
Now you follow the way of point P during rolling.
||In the end you have to use both trigonometric formulas.
The trigonometric formulas guarantee periodical movements.
The variables (R - r) and a determine the height, the ratio (r : R) the
periodicity of the drawings.
||You get the parametric equations for the movement of
the point on the left.
The coordinates x and y of point P depend on the angle
The variables R, r and a are constant.
If you use a program to draw the
graph of the equations, you get the so-called hypocycloids.
There are closed curves in difference to the spirograph drawings,
because the ratio (R : r) is a digit. It is sufficient to choose the numbers
0 to 5*2Pi for t.
||The parameter a is different for each picture.
You also can choose cases, which cannot be drawn with
the spirograph, because the point P lies outside the rolling circle.
The ratio (R : r) is not a digit, so that you get the
graphs from above.
||The shape of a rosette also depends on the number of
If you transfer the equations to graphs
with the help of a program, you get so-called epicycloids.
||You get another kind of cycloids, if you let unroll the
smaller circle (r=1) outside the fixed circle(R=5). This is also
realized at the spirograph with two circles.
||The parameter a is changed.
You also can choose cases, where the point P lies outside
the rolling circle.
||If you roll the circle on a straight line, a fixed point
on the circle line describes the cycloid.
The standard equations of the cycloid are x = r[t sin(t)
] and y = r[1 cos(t) ], where r is the radius of the rolling circle and
t goes through the numbers from 0 to 2Pi for one period.
It is remarkable that the length of a cycloid is eight
times as long as the radius of the producing circle. The surface between
x axis and cycloid is three times as big as the surface of the producing
You get more cycloids if the point P writing the cycloid
is within or outside the rolling circle.
In the first case a shortened cycloid develops, in the
second case an extended one.
The general parametric representation is x = rt-a sin(t)
und y =r-acos(t). R is the radius of the rolling circle and a the
distance of the point P of its center.
A rectangle and two half circles sitting on two opposite
sides belong to this part of the spirograph. The drawing is a combined
epicycles / cycloids.
||There are r=3 and a=1 (blue), a=3 (green) and a=5 (red).
the variable t follows 0<t<10.
||A program found parts of the graph on the left with the
help of the following parametric equations
Epicycloids equations on the left, cycloids equations
on the right.
Repeating Figures top
There is a stencil with simple figures inside and a fixed
circle, which also belongs to the spirograph system.
These drawings aren't interesting from a mathematical view,
but very effective.
||You draw a figure, move the wheel further by one cog
and draw the next figure.
You repeat this procedure as long as cogs are there.
In the end there is a ring for instance.
You can read more on my German
on the Internet
Richard Parris (peanut Software)
W.Leupold...: Analysis für Ingenieur- und Fachschulen,
Verlag Harri Deutsch, Frankfurt/M.
address on my main page
page is also available in German.
2000 Jürgen Köller