In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point on the circumference of a circle—called an epicycle—which rolls without slipping around a fixed circle. It is a particular kind of roulette.
If the smaller circle has radius r, and the larger circle has radius R = kr, then the parametric equations for the curve can be given by either:
in a more concise and complex form
- angle θ is in turns:
- smaller circle has radius r
- the larger circle has radius kr
(Assuming the initial point lies on the larger circle.) When k is a positive integer, the area of this epicycloid is
It means that the epicycloid is larger than the original stationary circle.
If k is a positive integer, then the curve is closed, and has k cusps (i.e., sharp corners).
|To close the curve and|
|complete the 1st repeating pattern :|
|θ = 0 to q rotations|
|α = 0 to p rotations|
|total rotations of outer rolling circle = p + q rotations|
Count the animation rotations to see p and q
The distance OP from (x = 0, y = 0) origin to (the point p on the small circle) varies up and down as
- R = radius of large circle and
- 2r = diameter of small circle
The epicycloid is a special kind of epitrochoid.
We assume that the position of is what we want to solve, is the angle from the tangential point to the moving point , and is the angle from the starting point to the tangential point.
Since there is no sliding between the two cycles, then we have that
By the definition of angle (which is the rate arc over radius), then we have that
From these two conditions, we get the identity
By calculating, we get the relation between and , which is
From the figure, we see the position of the point on the small circle clearly.
- List of periodic functions
- Deferent and epicycle
- Epicyclic gearing
- Multibrot set
- Roulette (curve)
- J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 161, 168–170, 175. ISBN 978-0-486-60288-2.
- Epicycloids and Blaschke products by Chunlei Cao, Alastair Fletcher, Zhuan Ye
- Epicycloid Evolute - from Wolfram MathWorld
- Pietrocola, Giorgio (2005). "Tartapelago". Maecla.
- Weisstein, Eric W. "Epicycloid". MathWorld.
- "Epicycloid" by Michael Ford, The Wolfram Demonstrations Project, 2007
- O'Connor, John J.; Robertson, Edmund F., "Epicycloid", MacTutor History of Mathematics archive, University of St Andrews
- Animation of Epicycloids, Pericycloids and Hypocycloids
- Spirograph -- GeoFun
- Historical note on the application of the epicycloid to the form of Gear Teeth