The cycloid is a curve generated by a point on the circumference of a circle which rolls on a straight line tangent to the circle, as shown at the left, Fig. 93.

The rolling circle is called the describing or generating circle, the point on the circle, the describing or generating point, and the tangent along which the circle rolls, the director. In order that the curve described by the point may be a true cycloid the circle must roll without any slipping.

Fig. 01. Construction of Rectangular Hyperbola.

## Hypocycloid

In case the generating circle rolls upon the inside of an arc or circle, the curve thus generated is a hypocycloid, Fig. 92. If the generating circle has a diameter equal to the radius of the director circle the hypocycloid becomes a straight line.

Fig. 92. Geometrical Constructions for Cycloid and Hypocycloid.

## Epicycloid

If the generating circle rolls upon the outside of the director circle, the curve generated is an epicycloid, Fig. 93.

## Involute Curves

If a thread of fine wire is wound around a cylinder or circle and then unwound, the end will describe an involute curve. The involute may be defined as a curve generated by a point in a tangent rolling on a circle, known as the base circle, Fig. 94.

The details of the ellipse, parabola, hyperbola, cycloid, and involute will be taken up in connection with the plates.

The most important application of the cycloidal and involute curves is in the cutting of all forms of gears. It has been found that the teeth of gears when cut accurately to either of these curves will mesh with the least friction and run with exceptional smoothness. The actual application of the cycloidal and involute curves to the laying-out of gears is given in Machine Drawing, Part II.