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What is a Spirograph?
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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. Drawing a figure:
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
correctly.
Drawing Exercises top
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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 look at the figures more accurately, you see that they aren't
closed after five revolutions. This is intended:
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If you go on drawing you'll get some nicely shaped rosettes. This shows
the attraction of spirograph figures. |
Formulas top
You can find parameter formulas for the figures, which describe them
mathematically.
Names:
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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 inner circle.
It has the distance a from the centre M of the small circle.
Now you follow the way of point P during rolling. |
Calculations:
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In the end you have to use both trigonometric formulas. |
Result:
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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 t.
The variables R, r and a are constant.
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The trigonometric formulas guarantee periodical movements. The variables
(R - r) and a determine the height, the ratio (r : R) the periodicity of
the drawings.
Hypocycloids top
If you use a program to draw the graph of the
equations, you get the so-called hypocycloids.
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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. |
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 ratio (R : r) is not a digit, so that you get the graphs from above.
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The shape of a rosette also depends on the number of revolutions. |
Epicycloids top
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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.
Parameter equations:
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If you transfer the equations to graphs with the
help of a program, you get so-called epicycloids.
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The parameter a is changed.
You also can choose cases, where the point P lies outside the rolling
circle. |
Cycloids top
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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 circle.
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.
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There are r=3 and a=1 (blue), a=3 (green) and a=5 (red).
the variable t follows 0<t<10. |
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.
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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.
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Repeating Figures top
There is a stencil with simple figures inside and a fixed circle, which
also belongs to the spirograph system.
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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. |
These drawings aren't interesting from a mathematical view, but very effective.
Spirograph on the Internet
top
English:
Anu Garg
Java Applet
Richard Parris (peanut Software)
Program WINPLOT
Wikipedia
Spirograph
German:
admaDIC
Spiromat (Applet)
Jutta Sturm
Zykloiden
Peter Müller
Spirograph
Wikipedia
Spirograph
(Spielzeug), Zykloide,
Epizykloiden
References top
W.Leupold...: Analysis für Ingenieur- und Fachschulen, Verlag
Harri Deutsch, Frankfurt/M.
Feedback: Email address on
my main page
This
page is also available in German.
URL of
my Homepage:
http://www.mathematische-basteleien.de/
©
2000 Jürgen Köller
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