Epicycloid

When an externally rolling circle of radius on a circle of radius describes a point on the circumference of an epicycloid, a special case of a cycloid.

In this way, mandala -like figures can draw that resemble flowers.

For the mathematical description of a Epicycloid you need - as it is angle changes - trigonometric expressions.

The equation of a Epicycloid is therefore:

It is

• The radius of the outer circle moving,
• Is the radius of the inner circle,
• Is the polar angle of the point at which contact with the two circuits,
• And the coordinates of the point on the epicycloid.

If an integer, we obtain a closed curve according to a rotation. We set. Then we can write the equation simpler:

The radius R (at the top, the number B) of the following figures of the image is large, the radius of the inner circle. The radius of the small circle is r ( above the number a ). Links results of an integer, so as not to overlap the " petals " on the left and the curve is closed. But right overlap the " petals ", that is, the curve is not closed since = 2.5. is also called the order of the Epicycloid:

If one seeks the point opposite to the currently drawn and an integer, so you can specify the equation in an alternative form:

For the length of the epicycloid and the contents of the enclosed space, we get the formulas

The tangent vector is also normal to the vector. The evolute has the equation

This is the equation of the epicycloid, in the alternative form, reduced to a factor. The evolute is thus a scaled, rotated copy of the original curve.

( If the ratio is a rational number, the curve closes after several revolutions. Is it irrational, it never closes. )

A Epicycloid is also formed by the composition of two rotations that take place in the same direction of rotation.

Special epicycloids

For results in a cardioid ( heart-shaped ). For perimeter and area are obtained:

This curve can also be obtained differently, namely as Kreiskonchoide ( Pascal's snail): One draws from a point on the circumference of a tendon and extended it around the circle diameter. If the string is rotating, the endpoint of the extension describes a cardioid.

When the tip of the cardioid is located at the origin, the equation in polar or Cartesian coordinates:

If we obtain a Nephroide (kidney curve). It has the dimensions