THE EPICYCLOID
BY
DENNIS ASTLEY & EMILY ASTLEY
The epicycloid is the special plane curve defined as the path traced by a point P on a circle that rolls around a fixed circle without slipping.
The point P lies on a circle of radius b, which rolls about the convex side of a circle of radius a.
The Greek mathematician Hipparchus (c. 140 B.C.) was the first to recognize the epicycloid in his astronomical theory of epicycles, in which he developed a model for the motion of the moon. Later, Ptolemy (c. 130 A.D.), a Greek astronomer and geographer, used combinations of epicycles to predict the positions of the sun, moon and planets. This idea was not superceded until Copernicus (1514) theorized that the sun, rather than the earth, was the center of the universe.
The construction of the epicycloid was first described in 1525 by Albrecht D¸rer, a German artist. D¸rer published this and many other curves in the first German mathematics text. GÈrard Desargues (1640), a French engineer, was the first to put the epicycloid to use in a system used for raising water near Paris. The next known practical use envisioned for epicycloids was in the working of mechanical gears, although there is some debate about who first thought of this. Danish astronomer Olaus Roemer is said to have investigated the use of cycloidal curves in the manufacture of gear teeth in 1674. However, it is worth noting that French mathematician Philippe de La Hire, whose father was a student of Desargues, is credited with inventing the epicycloidal profile for gear teeth in 1694, twenty years later.
Deriving Parametric Equations for the Epicycloid
Point P moves around a circle with radius b. The parametric equations for the circle are:
x = b*cos(θ)
y = b*sin(θ)
As the circle with radius b (circle B) rotates counterclockwise around the circle with radius a (circle A), point P not only moves through the angle θ, but also translates through the angle t. Therefore:
x = b*cos(θ+t)
y = b*sin(θ+t)
In order to parameterize the curve in terms of t only, we need to find a substitution for θ. Since circle B moves around circle A without slipping, we know that arc length s must be equal to arc length s':
s = a*t
s' = b*θ
s = s'
a*t = b*θ
θ = (a/b)*t
We can now replace θ to obtain:
x = b*cos([(a+b)/b]*t)
y = b*sin([(a+b)/b]*t)
C, the center of circle B, travels in a circle around circle A with radius a+b:
x = (a+b)*cos(t)
y = (a+b)*sin(t)
If O is the origin, then the vector OC lies along the x-axis in the positive direction and the vector CP lies along the x-axis and points in the negative x-direction. Therefore:
OP = OC + CP
OC = <(a+b)*cos(t),(a+b)*sin(t)>
and
CP = <-b*cos([(a+b)/b]*t),-b*sin([(a+b)/b]*t)>
so
OP = <(a+b)*cos(t)-b*cos([(a+b)/b]*t),(a+b)*sin(t)-b*sin([(a+b)/b]*t)>
As P moves away from its starting position, the vector OP' is the position vector of the point P'. Therefore, we have shown that the parametric equations of the epicycloid are:
x = (a+b)*cos(t)-b*cos([(a+b)/b]*t)
y = (a+b)*sin(t)-b*sin([(a+b)/b]*t)
To obtain an epicycloid with n number of cusps, let b = a/n, so that n rotations of B return the point P to its starting position. Some of these special epicycloids have names of their own, such as the one-cusp cardioid, the two-cusp nephroid and the five-cusp ranunculoid.
n=1 (Cardioid) n=2 (Nephroid) n=3 n=4 n=5 (Ranunculoid)
Here are some examples of the interesting shapes that can be obtained by changing the value of n and t :
n=33/32 n=9/4 n=9*sqrt(ln(pi))/8*sin(e^2)
References
"GÈrard Desargues." From The Galileo Project. http://galileo.rice.edu/Catalog/NewFiles/desargue.html
"Albrecht D¸rer." From MacTutor History of Mathematics Archive. http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Durer.html
"Epicycloid." From MacTutor History of Mathematics Archive. http://www-history.mcs.st-andrews.ac.uk/history/Curves/Epicycloid.html
"Hipparchus of Rhodes." From MacTutor History of Mathematics Archive. http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Hipparchus.html
"Philippe de La Hire." From The Galileo Project. http://galileo.rice.edu/Catalog/NewFiles/lahire_phi.html
Maor, Eli. (1998). Trigonometric Delights. Princeton, NJ: Princeton UP. http://www.pupress.princeton.edu/books/maor/
"Claudius Ptolemy." From MacTutor History of Mathematics Archive. http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Ptolemy.html
Smith, D. E. (1958). History of Mathematics, Volume II. New York: Dover.
Questions, comments or suggestions? Email us!