Epicycloid

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The red curve is an epicycloid traced as the small circle (radius r = 1) rolls around the outside of the large circle (radius R = 3).

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.

Equations[edit]

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:

or:

in a more concise and complex form[1]

where

  • angle θ is in turns:
  • smaller circle has radius r
  • the larger circle has radius kr

Area[edit]

(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).

If k is a rational number, say k = p / q expressed as irreducible fraction, then the curve has p cusps.

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

If k is an irrational number, then the curve never closes, and forms a dense subset of the space between the larger circle and a circle of radius R + 2r.

The distance OP from (x = 0, y = 0) origin to (the point p on the small circle) varies up and down as

where

  • R = radius of large circle and
  • 2r = diameter of small circle

The epicycloid is a special kind of epitrochoid.

An epicycle with one cusp is a cardioid, two cusps is a nephroid.

An epicycloid and its evolute are similar.[2]

Proof[edit]

sketch for proof

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

and

.

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.

See also[edit]

Animated gif with turtle in MSWLogo (Cardioid)[3]

References[edit]

  • J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 161, 168–170, 175. ISBN 978-0-486-60288-2.

External links[edit]