Epitrochoids and Hypotrochoids
Epitrochoids and Hypotrochoids are members of the
Trochoid family. They are formed by rolling one circle around
another stationary circle, and the specific curve is formed by the
path generated by the end of an arm which is fixed to the rolling
circle. A limited number of these curves can be generated by the
commercially sold Spirograph game, but mathematically, the
possibilities are much richer.
We will first look at an example of an Epitrochoid
which is formed by rolling one circle around the outside of
another circle. Further down this page we will look at
Hypotrochoids which are formed by rolling one circle around the
inside of another. All graphs on this page were generated via
The graph below shows an Epitrochoid that is formed
when a small circle with a radius of 2 is rolled around the
outside of a circle with a radius of 6. An arm with a length of 4
has one end attached to the center of the rolling circle. The path
of the other end of the arm is shown in red and forms the
Epitrochoid. The circumference of the stationary circle is shown
in green while the path taken by the center of the rolling circle
is shown in dark blue.
The following sequence of pictures show various
stages in the generation of the Epitrochoid. In each picture, the
circumference of the stationary circle is shown in green. The
current position of the rolling circle is shown in light blue. The
trace of the path taken by the center of the rolling circle is
shown in dark blue. The current position of the arm attached to
the rolling circle is shown in orange. Finally the path generated
by the outer end of the arm generates the Epitrochoid, and is
shown in red.
Initially the rolling circle starts at the top of the
stationary circle. The orange arm is straight up and overlies part
of the Y axis. There is no red or dark blue line yet as nothing
Here, the rolling circle has started its clockwise
trip around the stationary circle. The dark blue line shows the
path of the center of the rolling circle while the outer end of
the arm is beginning to generate the Epitrochoid. At this stage
the rolling circle has completed 2 % of its trip.
The following sequence of pictures show subsequent
stages in the formation of the Epitrochoid. The legend in the
lower right portion of each picture displays the relative
The radius of the stationary green inner circle is 6.
The radius of the light blue, rolling circle is 2. Thus the
rolling circle will rotate 6 / 2 = 3 times relative to the
stationary circle while it is rolling around the stationary
circle. Since the clockwise direction that the rolling circle is
moving produces a clockwise relative rotation of the arm, the
total number of revolutions of the arm relative to the coordinate
system will be 3 + 1 = 4. The arm will be straight up at trip
proportion locations of 0.25, 0.50, 0.75, and 1.00.
When the earth rotates around the sun a similar
rotation phenomenon exists. The earth rotates ~365.25 times per
year relative to the sun to produce ~365.25 days per year.
However, as measured by the rest of the stars in the universe, our
absolute number of rotations per year is one more: 365.25 + 1.00 =
Finally after the rolling circle has completed 3
revolutions relative to the stationary circle (and completed 4
revolutions relative to the coordinate system), the rolling circle
has returned to the starting position, and the Epitrochoid is
While the above example displays one of the infinite
number of possible Epitrochoids, slight changes in the dimensions
that are used can produce other interesting patterns.
In the picture above, the radius of the rolling
circle has been increased from 2.00 to 2.02. Thus the
distance required to complete 3 rolling rotations around the
stationary circle increases slightly from 6.00 to 6.06. The
diagram shows the result of 30 rolls relative to the stationary
circle. The slightly increased total distance produces a clockwise
rotational precession in the resulting pattern.
Another change in the radius of the rolling circle
produces the above pattern. If the radius had been set to exactly
1.50, then a simple Epitrochoid with 4 loops would have resulted.
However, by setting the radius to 1.51, another precession pattern
For additional information about Epitrochoids and the
mathematics required to generate them, please see http://mathworld.wolfram.com/Epitrochoid.html
If the circle rolls around the
inside of a stationary circle, then the result is a
In the example shown here, the stationary circle has
a radius of 10. A circle with a radius of 4 rolls around the
inside of the stationary circle. Again, the dark blue circle
traces the path of the center of the rolling circle. The arm
length is the same as the radius of the rolling circle, and thus
the end of the arm is essentially the same as a point on the
rolling circle. The red line is a trace of this point, and it
generates a Hypotrochoid en route as it returns to its starting
location. The diagrams below illustrate various stages of this
The picture above shows the starting conditions. The
rolling circle is shown in light blue and the arm is again shown
in orange as it overlies part of the vertical axis.
This next picture shows the interim result after the
rolling circle has completed 20 % of its first circuit around the
inside of the stationary circle. Note that while the rolling
circle is traveling clockwise, the arm on the rolling circle is
rotating counterclockwise. For each trip that the rolling circle
makes around the inside of the stationary circle, the arm will
make 10 / 4 = 2.5 counterclockwise revolutions relative to the
stationary circle. (Count the number of arm contacts with the
green circle.) However, since the relative rotation of the arm is
opposite to the rolling direction of the circle, the absolute
number of arm rotations as measured by the coordinate systems is
one less than this: 2.5 - 1.0 = 1.5.
Note that when the light blue rotating circle has
returned to the starting point, the red trace shows that it has
gone thru 2.5 rotations relative to the green stationary circle,
but has only completed 1.5 rotations relative to the coordinate
system. At this point the Hypotrochoid (red line) is only one-half
complete, and thus the rolling circle will have to make another
circuit to complete the pattern.
As is the case with Epitrochoids, itís interesting to
experiment with various circle sizes to see what happens. One
interesting phenomenon occurs when you set the radius of the
rolling circle (and arm length) to 6 instead of 4. The red line
traces an identical pattern, but this time it traces it backwards.
Also the rolling circle has to make 3 circuits instead of two.
For additional information about Hypotrochoids and
the mathematics required to generate them, please see http://mathworld.wolfram.com/Hypotrochoid.html
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