In mathematics, an involute is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the locus of a point on a piece of taut string as the string is either unwrapped from or wrapped around the curve. It is a class of curves coming under the roulettefamily of curves. The evolute of an involute is the original curve. The notions of the involute and evolute of a curve were introduced by Christiaan Huygens in his work titled Horologium oscillatorium sive de motu pendulorum ad horologia aptato demonstrationes geometricae.
Adding an arbitrary but fixed number to the integral results in an involute corresponding to a string extended by . Hence, the involute can be varied by constant and/or adding a number to the integral. If one gets
Properties of involutes
In order to derive properties of a regular curve it is advantageous to suppose the arc length to be the parameter of the given curve, which lead to the following simplifications: and , with the curvature and the unitnormal. One gets for the involute: and the statement:
At point the involute is not regular,
and from follows:
The normal of the involute at point is the tangent of the given curve at point.
For a circle with parametric representation, one has Hence, and the path length is. Evaluating the above given equation of the involute, one gets for the parametric equation of the involute of the circle. The figure shows involutes for , , and . The involutes look like Archimedean spirals, but they are actually not. The arc length for and of the involute is
The parametric equation describes a semicubical parabola. From one gets and. Extending the string by extensively simplifies further calculation, and one gets Eliminating yields showing that this involute is a parabola. The other involutes are thus parallel curves of a parabola, and are not parabolas, as they are curves of degree six.
For the catenary, the tangent vector is, and, as its length is. Thus the arc length from the point is Hence the involute starting from is parametrized by and is thus a tractrix. The other involutes are not tractrices, as they are parallel curves of a tractrix.
The parametric representation describes a cycloid. From, one gets and Hence the equations of the corresponding involute are which describe the shifted red cycloid of the diagram. Hence
The involutes of the cycloid are parallel curves of the cycloid
Involute and evolute
The evolute of a given curve consists of the curvature centers of. Between involutes and evolutes the following statement holds:
Application
The involute has some properties that makes it extremely important to the gear industry: If two intermeshed gears have teeth with the profile-shape of involutes, they form an involute gear system. Their relative rates of rotation are constant while the teeth are engaged. The gears also always make contact along a single steady line of force. With teeth of other shapes, the relative speeds and forces rise and fall as successive teeth engage, resulting in vibration, noise, and excessive wear. For this reason, nearly all moderngear teeth bear the involute shape. The involute of a circle is also an important shape in gas compressing, as a scroll compressor can be built based on this shape. Scroll compressors make less sound than conventional compressors and have proven to be quite efficient. The High Flux Isotope Reactor uses involute-shaped fuel elements, since these allow a constant-width channel between them for coolant.
