involute

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In the differential geometry of curves, an involute (also known as evolvent) of a smooth curve is another curve, obtained 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. It is a roulette wherein the rolling curve is a straight line containing the generating point.

Alternatively, another way to imagine the involute of a curve is to replace the taut string by a line, and tracing a given point at the line as the line is rolled on the curve, all the time keeping the line as atangent to the curve.

The evolute of an involute is the original curve, less portions of zero or undefined curvature. 

 

If function is a natural parametrization of the curve (i.e. for all s), then : parametrizes the involute.

 

Involute of a circle

The involute of a circle forms a shape which resembles an Archimedean spiral.

§  In polar coordinates  the involute of a circle has the parametric equation:

where  is the radius of the circle and  is a parameter

Leonhard Euler proposed to use the involute of the circle for the shape of the teeth of toothwheel gear, a design which is the prevailing one in current use, called involute gear.

 

Involute of a catenary

The involute of a catenary through its vertex is a tractrix. In cartesian coordinates the curve follows

Involute of a cycloid

One involute of a cycloid is a congruent cycloid. In cartesian coordinates the curve follows:

Where t is the angle and r is the radius

 

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 (rather than, for example, a "classic" triangular shape), they form an involute gear system. Their relative rates of rotation are constant while the teeth are engaged, and also, the gears 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.

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.