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In mathematics, an involute (also known as an evolvent) 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. [1] The evolute of an involute is the original curve.
Tractrix with object initially at (4, 0). Suppose the object is placed at (a, 0) (or (4, 0) in the example shown at right), and the puller at the origin, so a is the length of the pulling thread (4 in the example at right).
The involute gear profile, sometimes credited to Leonhard Euler, [1] was a fundamental advance in machine design, since unlike with other gear systems, the tooth profile of an involute gear depends only on the number of teeth on the gear, pressure angle, and pitch. That is, a gear's profile does not depend on the gear it mates with.
At sections of the curve with ′ > or ′ < the curve is an involute of its evolute. (In the diagram: The blue parabola is an involute of the red semicubic parabola, which is actually the evolute of the blue parabola.) Proof of the last property:
The envelope E 1 is the limit of intersections of nearby curves C t.; The envelope E 2 is a curve tangent to all of the C t.; The envelope E 3 is the boundary of the region filled by the curves C t.
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Involute polar angle. Expressed as θ, the involute polar angle is the angle between a radius vector to a point, P, on an involute curve and a radial line to the intersection, A, of the curve with the base circle. [1]