Epicycloid

The epicycloid is the path traced by a point on the edge of a circle as it rolls without slipping along the edge of another circle. The two circles are not required to be the same size:

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With the above setup, let t be the angle that circle 1 (having radius R1) has rotated around circle 2 (radius R2). Then the parametric equations for the path a point on the edge of circle 1 traces is:

x=(R1+R2)*cos(t*R1/R2)-R1*cos(t)
y=(R1+R2)*sin(t*R1/R2)-R1*sin(t)

Note that, if R1=R2, then the shape is a cardioid. (In the above setup, I used R1=25chars, R2=24chars.)

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Parametric Cartesian equation: x = (a + b)cos(t) - bcos((a/b + 1)t), y = (a + b)sin(t) - bsin((a/b + 1)t)

There are four curves which are closely related. These are the epicycloid, the epitrochoid, the hypocycloid and the hypotrochoid and they are traced by a point P on a circle of radius b which rolls round a fixed circle of radius a.

For the epicycloid, shown above in clarity-reducing ASCII, the circle of radius b rolls on the outside of the circle of radius a. The point P is on the circumference of the circle of radius b. For the example drawn here a = 8 and b = 5.

Special cases

• a = b, a cardioid is obtained
• a = 2b, a nephroid is obtained
• a = (n - 1)b where n is an integer, then the length of the epicycloid is 8nb and its area is b²(n² + n)

Also, the evolute of an epicycloid is a similar epicycloid.

Ep`i*cy"cloid (?), n. [Epicycle + -oid: cf. F. 'epicycloide.] Geom.

A curve traced by a point in the circumference of a circle which rolls on the convex side of a fixed circle.

⇒ Any point rigidly connected with the rolling circle, but not in its circumference, traces a curve called an epitrochoid. The curve traced by a point in the circumference of the rolling circle when it rolls on the concave side of a fixed circle is called a hypocycloid; the curve traced by a point rigidly connected with the rolling circle in this case, but not its circumference, is called a hypotrochoid. All the curves mentioned above belong to the class class called roulettes or trochoids. See Trochoid.

 

© Webster 1913.