By attaching an imaginary taut string to the given curve and tracing its free end as it is wound onto that given curve; or in reverse, unwound. For example, an involute approximates the path followed by a tetherball as the connecting tether is wound around the center pole. If the center pole has a circular cross-section, then the curve is an involute of a circle.
By a line segment that is tangent to the curve on one end, while the other end traces out the involute. The length of the line segment is changed by an amount equal to the arc length traversed by the tangent point as it moves along the curve.
The evolute of an involute is the original curve, less portions of zero or undefined curvature.
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 (1673).
Let be a regular curve in the plane with its curvature nowhere 0 and , then the curve with the parametric representation
is an involute of the given curve.
The integral describes the actual length of the free part of the string in the interval
and the vector prior to that is the tangent unitvector. Adding an arbitrary but fixed number to the integral results in an involute corresponding to a string, which is extended by . Hence:
the involute can be variied by parameter and/or adding a number to the integral (see Involutes of a semicubic parabola).
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. Because of the simplifications in this case: and , with the curvature and the unit normal, one gets for the involute:
and the statement:
At point the involute is not regular (because ),
and from follows:
The normal of the involute at point is the tangent of the given curve at point and
the involutes are parallel curves, because of and the fact, that is the unit normal at .
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 (rather than, for example, a traditional triangular shape), 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 modern gear teeth bear the involute shape.
Mechanism of a scroll compressor
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.