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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.)
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.
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
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
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.
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 ...
They come in various forms, including evolute shells where all whorls are exposed, and strongly involute shells with only the outer whorl visible. Shells may be heavily ribbed, with some featuring nodes and spines, while others are completely smooth.
The Centroceratidae, which range from the Middle Devonian to the Lower Permian, are characterized by gyroconic, evolute tarphyceraconic, and involute nautiliconic shells with compressed whorls, typically with a quadrangular whorl section in which the flanks converge on venter that is much narrower than the dorsum and ventral and umbilical shoulders are sharply angular, or rarely rounded.