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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.
Hence: the evolute is the envelope of the normals of the given curve. 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.)
Because the evolute of a nephroid is another nephroid, the involute of the nephroid is also another nephroid. The original nephroid in the image is the involute of the smaller nephroid. inversion (green) of a nephroid (red) across the blue circle
Definitions of evolute, involute, and their relationship. 5-6, 8 Evolute of cycloid and parabola. 7, 9a Rectification of cycloid, semicubical parabola, and history of the problem. 9b-e Circle areas equal to surfaces of conoids; rectification of the parabola equal to quadrature of hyperbola; approximation by logarithms. 10-11
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.
Schloenbachia varians comprises a highly variable group of medium-sized, involute and compressed to evolute and highly inflated, weak to strongly keeled ammonites, most all of which bear umbilical and lower lateral tubercles. Compressed forms may be smooth and constricted.
Permian genera include Parastenopoceras, a smooth, involute form with a semiellptical whorl section and ventral saddle; Plummeroceras, a form similar to Domatoceras but more evolute and with a deep ventral lobe; Pselioceras, a smooth evolute form with a perforate umbilicus, ovoid whorl section, and suture crossing straight over the venter; and ...
Evolute shells have very little overlap, a large umbilicus, and many exposed whorls. Involute shells have strong overlap, a small umbilicus, and only the largest and most recent whorls are exposed. Shell structure can be broken down further by the width of the shell, with implications for hydrodynamic efficiency.