In the differential geometry of curves, an involute of a smooth curve is another curve, obtained by attaching a string to the curve and tracing the end of the string as it is wound onto the curve. It is a roulette wherein the rolling curve is a straight line containing the generating point. In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... In the differential geometry of curves, a roulette is the general concept behind cycloids, epicycloids, hypocycloids, and involutes. ...

Analytically: if function is a natural parametrization of the curve (i.e. for all s), then : parametrises the involute. In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus. ...

The evolute of an involute is the original curve less portions of zero or undefined curvature. In the differential geometry of curves, the evolute of a curve is the set of all its centers of curvature. ... Curvature refers to a number of loosely related concepts in different areas of geometry. ...

Examples:

• the involute of a catenary through its vertex is a tractrix

With r(s) = (sinh ^{− 1}(s),cosh(sinh ^{− 1}(s))) we have and

substitute to get

• one involute of a cycloid is a congruent cycloid.

The involute of a circle has a property that makes it important to the gear industry: if the teeth of two mating gears have the shape of an involute, their relative rates of rotation are constant while the teeth are engaged. With teeth of other shapes, the relative speeds 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. See also involute gear. In mathematics, the catenary is the shape of a hanging flexible chain or cable when supported at its ends and acted upon by a uniform gravitational force (its own weight). ... Tractrix (from the Latin verb trahere `pull, drag) is the curve along which a small object (tractens) moves when pulled on a horizontal plane with a piece of thread by a puller (tractendus) which moves rectilinearly, it is therefore a curve of pursuit. ... Cycloid (red) generated by a rolling circle A cycloid is the curve defined by a fixed point on a wheel as it rolls, or, more precisely, the locus of a point on the rim of a circle rolling along a straight line. ... Spur gears found on a piece of farm equipment. ... The involute gear profile is the most commonly used system for gearing today. ...

External links

• Mathworld
• Flash animation of an Involute Gear where these curves are used.