The curve is formed by the locus of a point, attached to a circle, that rolls on 1) the outside of another circle 2). In the curve's equation the first part denotes
the relative position between the two circles, the second part denotes the rotation of the
The value of the constant b determines the starting point in relation to the
The isoptic of the ordinary epicycloid is an
The curve has a closed form when the ratio of the rolling circle and the other circle
(a) is equal to a rational number. When giving this ratio its simplest form, the numerator
is the number of revolutions around the resting circle, before the curve closes. The
denominator is the number of rotations of the rolling circle before this happens.
A epicycloid with parameters a and b is the same as a hypocycloid
with parameters a+1, 1/b.
In this rational case the curve is algebraic,
The epicycloid curves have been studied by a lot of mathematicians around the
17th century: Dürer (1515), Desargues (1640), Huygens
(1679), Leibniz, Newton (1686), de L'Hôpital (1690), Jakob
Bernoulli (1690), la Hire (1694), Johann Bernoulli (1695), Danilel
Bernoulli (1725) and Euler (1745, 1781).
Apollonius of Perga (about 200 BC), had the idea to describe the celestial
movements as combinations of circular movements. It was Hipparchos of Nicaea
(about 150 BC), the greatest astronomer in Greek antiquity, who worked out this theory in
detail. The results did become famous by the books of Ptolemy (about 150 AD). The
earth is thought as standing in (or nearby) a celestial center, around which the other
celestial bodies rotate. The combination of the rotation of the earth and the planet's
rotation around her makes an epicycloid. This geocentric theory should be the accepted
theory for almost 2000 years. The heliocentric theory (as constructed by Copernicus), was
also discussed by the Greek, but refused for emotional reasons.
Some relations with other curves:
- the radial and the pedal (with the center as pedal point) of the
curve is the rhodonea
- the evolute of an
epicycloid is a similar epicycloid, but smaller in size
The curve is a cycloidal
There are some epicycloids that have been given an own name:
- a = 1/5: the ranunculoid
This curve has been named after the buttercup genus Ranunculus by Madachy
- a = 1/2: the nephroid
- a = 1: the cardioid
For 1 = 1/n, n gives the number of cusps.
Now the point being followed is not lying on the rolling circle. When the point lays
outside the circle (b>1), the curve is called a prolate
epicycloid. When the point lays inside the rolling circle (b<1), the curve
is called a curtate epicycloid.
The variable b is the ratio of the distance from the starting point to the
center of the rolling circle, and the radius of that circle.
With a spirograph set many epitrochoids can be drawn.
In Dürer's 'Instruction in measurement with compasses and straight
edge' (1525) occurs an example of an epitrochoid. He called them spider
lines because of the form of the construction lines he used.
Other mathematicians who studied the curves were: la Hire, Desargues
And now some examples of epitrochoids:
When the two circles have equal radius (a=1), the epitrochoid is a limaçon.
1) Epi = on
2) Let a circle with radius r roll on the outside of a circle with radius R. Take as
center of the coordinate system the center of the rolled circle. Now let the starting
point be on a distance b r from the center of the rolling circle. Then the coordinates of
the epicycloid as a function of the rolled angle t are:
3) Trochus (Lat.) = hoop.
Sometimes the meaning of epicycloid and epitrochoid is interchanged: epitrochoid for the
general case, epicycloid only for the situation that the starting point is lying on the