** Next:** 8.3 Spirals
**Up:** 8 Special Plane Curves
** Previous:** 8.1 Algebraic Curves

Suppose given a fixed curve *C* and a moving curve *M*, which rolls on
*C* without slipping. The curve drawn by a point *P* kept fixed with
respect to *M* is called a **roulette**, of which *P* is the
**pole**.

The most important examples of roulettes arise when *M* is a circle
and *C* is a straight line or a circle, but an interesting additional
example is provided by the **catenary** *y*=*a* cosh(*x*/*a*), which
arises by rolling the parabola *y*=*x*/(4*a*) on the *x*-axis with pole
the focus of the parabola (that is, *P*=(0,*a*) in the initial
position). The
catenary is the shape taken under the action of gravity by a chain or
string of uniform density whose ends are held in the air.

A circle rolling on a straight line gives a **trochoid**, with the
**cycloid** as a special case when the pole *P* lies on the circle
(Figure 1). If the moving circle *M* has radius *a*
and the distance from the pole *P* to the center of *M* is *k*, the
trochoid has the parametric equation

*x*=*a*-*k* sin ,
*y*=*a*-*k* cos .

**Figure 1:** Cycloid (top) and trochoids with *k*=.5*a* and *
k*=1.5*a*, where *k* is the distance *PQ* from the center
of the rolling circle to the pole.

The cycloid, therefore, has parametric equation

*x*=*a*(- sin ),

*y*=*a*(1- cos ).

One can eliminate to get *x* as a
(multivalued) function of *y*, which takes the following form for the
cycloid:

The length of one arch of the cycloid is 8*a*, and the area under the
arch is 3*a*.

A trochoid is also called a **curtate cycloid** when *k*<*a* (that is,
when *P* is inside the circle) and a **prolate cycloid** when
*k*>*a*.

A circle rolling on another circle and exterior to it gives an
**epitrochoid**. If *a* is the radius of the fixed circle, *b* that
of the rolling circle and *k* is the distance from *P* to the center
of the rolling circle, the parametric equation of the epitrochoid is

*x*=(*a*+*b*) cos - *k* cos ((1+*a*/*b*)),

*y*=(*a*+*b*) sin - *k* sin ((1+*a*/*b*)).

These equations assume that at the start everything is aligned along
the positive *x*-axis, as in Figure 2, left.

**Figure 2:** Left: Initial configuration for epicycloid (black) and
configuration at parameter value (red). Middle: epicycloid
with *b*=½*a* (nephroid). Right: epicycloid with *b*=*a* (cardioid).

Usually one considers the case when *a*/*b* is a rational number, say
*a*/*b*=*p*/*q* where *p* and *q* are relatively prime. Then the rolling
circle returns to its original position after rotating *q* times
around the fixed circle, and the epitrochoid is a closed curve---in
fact, an algebraic curve. One also usually takes *k=b*, so that *P*
lies on the rolling circle; the curve in this case is called an
**epicycloid**. The middle diagram in Figure 2 shows the
case *b*=*k*=½*a*, called the **nephroid**; this curve is the cross
section of the caustic of a spherical mirror. The diagram on the
right shows the case *b*=*k*=*a*,
which gives the cardioid (compare Figure 8.1.6
, middle).

**Hypotrochoids** and **hypocycloids** are defined in the same way
as epitrochoids and epicycloids, but the rolling circle is inside the
fixed one. The parametric equation of the hypotrochoid is

*x*=(*a*-*b*) cos + *k* cos ((*a*/*b*-1) ),

*y*=(*a*-*b*) sin - *k* sin ((*a*/*b*-1) ),

where the letters have the same meaning as for the epitrochoid.
Usually one takes *a*/*b* rational and *k*=*b*. There are several
interesting particular cases:

*b*=*k*=*a*gives a point.*b*=*k*=½*a*gives a diameter of the circle*C*.*b*=*k*=*a*gives the**deltoid**(Figure 3, left), which has the algebraic equation(

*x*+*y*)-8*ax*+24*axy*+18*a*(*x*+*y*)-27*a*=0.*b*=*k*=¼*a*gives the**astroid**(Figure 3, right), an algebraic curve of degree six whose equation can be reduced to . The figure illustrates another property of the astroid: its tangent intersects the coordinate axes at points that are always the same distance*a*apart. Otherwise said, the astroid is the envelope of a moving segment of fixed length whose endpoints are constrained to lie on the two coordinate axes.

**Figure 3:** The hypocycloids with *a*=3*b* (deltoid) and
*a*=4*b* (astroid).

*Silvio Levy
Wed Oct 4 16:41:25 PDT 1995*

This document is excerpted from the 30th Edition of the *CRC Standard Mathematical Tables and Formulas* (CRC Press). Unauthorized duplication is forbidden.