Two involutes (red) of a parabola
In the differential geometry of curves, an involute (also known as evolvent) is a curve obtained from another given curve by one of two methods.
- By attaching an imaginary taut string to the given curve and tracing its free end as it is wound onto that given curve; or in reverse, unwound. For example, an involute approximates the path followed by a tetherball as the connecting tether is wound around the center pole. If the center pole has a circular cross-section, then the curve is an involute of a circle.
- By a line segment that is tangent to the curve on one end, while the other end traces out the involute. The length of the line segment is changed by an amount equal to the arc length traversed by the tangent point as it moves along the curve.
The evolute of an involute is the original curve, less portions of zero or undefined curvature.
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).
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.
The integral describes the actual length of the free part of the string in the interval
and the vector prior to that is the tangent unitvector. Adding an arbitrary but fixed number to the integral results in an involute corresponding to a string, which is extended by . Hence:
the involute can be variied by parameter 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. Because of the simplifications in this case: and , whith the curvature and the unit normal, one gets for the involute:
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 and
- the involutes are parallel curves, because of and the fact, that is the unit normal at .
Involutes of a circle
For a circle with parametric representation one gets
. Hence and the integral is . The equations of the involutes are:
The diagram shows involutes for (green), (red), (purple) and (light blue). The involutes are similar to Archimedean spirals but they are actually not.
The arc length of the involute with is
Involutes of a semicubic parabola
Involutes of a semicubic parabola (blue). Only the red curve is a parabola.
The parametric representaion describes a semicubic parabola. From one gets and . Extending the string by causes an essential simplification of the calculation and one gets
Eliminating parameter yields the equation of a parabola:
- The involutes of the semicubic parabola are parallel curves of the parabola
(Parallel curves of a parabola are not parabolas anymore !)
Remark: The evolute of the parabola is the semicubic parabola (see section involute and evolute.)
The red involute of a catenary (blue) is a tractrix.
Involutes of a catenary
For the catenary one gets and because of the length of the tangent vector is and the integral
Hence the parametric representation of the corresponding involute is
which describes a tractrix.
- The involutes of the catenary are parallel curves of the tractrix
Involutes of a cycloid
Involutes of a cycloid (blue): Only the red curve is another cycloid
The parametric representation describes a cycloid. From one gets and . (Trigonometric formulae were used.)
Hence the equations of the corresponding involute are
which describe the shifted red cycloid of the diagram.
- The involutes of the cycloid are parallel curves of the cycloid
Involute and evolute
The evolute of a given curve consists of the curvature centers of . Between involutes and evolutes the following statement holds:
- A curve is the evolute of any of its involutes.
Tractrix (red) as an involute of a catenary
The evolute of a tractrix is a catenary
The involute has some properties that makes it extremely important to the gear industry: If two intermeshed gears have teeth with the profile-shape of involutes (rather than, for example, a traditional triangular shape), they form an involute gear system. Their relative rates of rotation are constant while the teeth are engaged, and also, the gears always make contact along a single steady line of force. 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 gear teeth bear the involute shape.
Mechanism of a scroll compressor
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 involute is an example of a roulette wherein the rolling curve is a straight line containing the generating point.
- ^ McCleary, John (1995). Geometry from a Differentiable Viewpoint. Cambridge University Press. p. 73.
- ^ K. Burg, H. Haf, F. Wille, A. Meister: Vektoranalysis: Höhere Mathematik für Ingenieure, Naturwissenschaftler und ..., Springer-Verlag, 2012,ISBN 3834883468, S. 30.
- ^ R. Courant:Vorlesungen über Differential- und Integralrechnung, 1. Band,Springer-Verlag, 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)