Involute
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.
It is generalized by the roulette family of curves. That is, the involutes of a curve are the roulettes of the curve generated by a straight line.
The notions of the involute and evolute of a curve were introduced by Christiaan Huygens in his work titled Horologium oscillatorium sive de motu pendulorum ad horologia aptato demonstrationes geometricae (1673), where he showed that the involute of a cycloid is still a cycloid, thus providing a method for constructing the cycloidal pendulum, which has the useful property that its period is independent of the amplitude of oscillation.^{[2]}
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Involute of a parameterized curve
Let be a regular curve in the plane with its curvature nowhere 0 and , then the curve with the parametric representation
is an involute of the given curve.
Proof The string acts as a tangent to the curve . Its length is changed by an amount equal to the arc length traversed as it winds or unwinds. Arc length of the curve traversed in the interval is given by
where is the starting point from where the arc length is measured. Since the tangent vector depicts the taut string here, we get the string vector as
The vector corresponding to the end point of the string () can be easily calculated using vector addition, and one gets

Adding an arbitrary but fixed number to the integral results in an involute corresponding to a string extended by (like a ball of wool yarn having some length of thread already hanging before it is unwound). Hence, the involute can be varied by constant and/or adding a number to the integral (see Involutes of a semicubic parabola).
If one gets
Properties of involutes
In order to derive properties of a regular curve it is advantageous to suppose the arc length to be the parameter of the given curve, which lead to the following simplifications: and , with the curvature and the unit normal. One gets for the involute:
 and
and the statement:
 At point the involute is not regular (because ),
and from follows:
 The normal of the involute at point is the tangent of the given curve at point .
 The involutes are parallel curves, because of and the fact, that is the unit normal at .
The family of involutes and the family of tangents to the original curve makes up an orthogonal coordinate system. Consequently, one may construct involutes graphically. First, draw the family of tangent lines. Then, an involute can be constructed by always staying orthogonal to the tangent line passing the point.
Cusps
This section is based on.^{[3]}
There are generically two types of cusps in involutes. The first type is at the point where the involute touches the curve itself. This is a cusp of order 3/2. The second type is at the point where the curve has an inflection point. This is a cusp of order 5/2.
This can be visually seen by constructing a map defined by
By this map, the involutes are obtained in a threestep process: map to , then to the surface in , then project it down to by removing the zaxis:
Since the mapping has nonzero derivative at all , cusps of the involute can only occur where the derivative of is vertical (parallel to the zaxis), which can only occur where the surface in has a vertical tangent plane.
Generically, the surface has vertical tangent planes at only two cases: where the surface touches the curve, and where the curve has an inflection point.
cusp of order 3/2
For the first type, one can start by the involute of a circle, with equation
cusp of order 5/2
For the second type, consider the curve . The arc from to is of length , and the tangent at has angle . Thus, the involute starting from at distance has parametric formula
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Examples
Involutes of a circle
For a circle with parametric representation , one has . Hence , and the path length is .
Evaluating the above given equation of the involute, one gets
for the parametric equation of the involute of the circle.
The term is optional; it serves to set the start location of the curve on the circle. The figure shows involutes for (green), (red), (purple) and (light blue). The involutes look like Archimedean spirals, but they are actually not.
The arc length for and of the involute is
Involutes of a semicubic parabola
The parametric equation describes a semicubical parabola. From one gets and . Extending the string by extensively simplifies further calculation, and one gets
Eliminating t yields showing that this involute is a parabola.
The other involutes are thus parallel curves of a parabola, and are not parabolas, as they are curves of degree six (See Parallel curve § Further examples).
Involutes of a catenary
For the catenary , the tangent vector is , and, as its length is . Thus the arc length from the point (0, 1) is
Hence the involute starting from (0, 1) is parametrized by
and is thus a tractrix.
The other involutes are not tractrices, as they are parallel curves of a tractrix.
Involutes of a cycloid
The parametric representation describes a cycloid. From , one gets (after having used some trigonometric formulas)
and
Hence the equations of the corresponding involute are
which describe the shifted red cycloid of the diagram. Hence
 The involutes of the cycloid are parallel curves of the cycloid
(Parallel curves of a cycloid are not cycloids.)
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Involute and evolute
The evolute of a given curve consists of the curvature centers of . Between involutes and evolutes the following statement holds: ^{[4]}^{[5]}
 A curve is the evolute of any of its involutes.
Application
The most common profiles of modern gear teeth are involutes of a circle. In an involute gear system the teeth of two meshing gears contact at a single instantaneous point that follows along a single straight line of action. The forces exerted the contacting teeth exert on each other also follow this line, and are normal to the teeth. The involute gear system maintaining these conditions follows the fundamental law of gearing: the ratio of angular velocities between the two gears must remain constant throughout.
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 planar gear systems are either involute or the related cycloidal gear system.^{[6]}
The 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.
The High Flux Isotope Reactor uses involuteshaped fuel elements, since these allow a constantwidth channel between them for coolant.
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Source: "Involute", Wikipedia, Wikimedia Foundation, (2023, March 23rd), https://en.wikipedia.org/wiki/Involute.
References
 ^ Rutter, J.W. (2000). Geometry of Curves. CRC Press. pp. 204. ISBN 9781584881667.
 ^ McCleary, John (2013). Geometry from a Differentiable Viewpoint. Cambridge University Press. pp. 89. ISBN 9780521116077.
 ^ Arnolʹd, V. I. (1990). Huygens and Barrow, Newton and Hooke : pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals. Basel: Birkhaüser Verlag. ISBN 0817623833. OCLC 21873606.
 ^ K. Burg, H. Haf, F. Wille, A. Meister: Vektoranalysis: Höhere Mathematik für Ingenieure, Naturwissenschaftler und ..., SpringerVerlag, 2012,ISBN 3834883468, S. 30.
 ^ R. Courant:Vorlesungen über Differential und Integralrechnung, 1. Band, SpringerVerlag, 1955, S. 267.
 ^ V. G. A. Goss (2013) "Application of analytical geometry to the shape of gear teeth", Resonance 18(9): 817 to 31 Springerlink (subscription required).
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