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.
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
If function is a natural parametrization of the curve (i.e. for all s), then : parametrizes the involute.
Involute of a circle
involute of a circle forms a shape which resembles an Archimedean spiral.
§ In polar coordinates the
involute of a circle has the parametric equation:
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.
of a catenary
involute of a catenary through
its vertex is
a tractrix. In cartesian coordinates the
of a cycloid
One involute of a cycloid is a congruent cycloid. In cartesian coordinates the
Where t is
the angle and r is the radius
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.
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.