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Two involutes (red) of a parabola

An involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. It is the path taken by the end of an idealized string as it wraps (or unwraps) around a curve.

Involute curves are described using the differential geometry of curves, and are 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.
  • 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).[1]

Contents

Involute of a parameterized curveEdit

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 varied by parameter   and/or adding a number to the integral (see Involutes of a semicubic parabola).

If   one gets

 

Properties of involutesEdit

 
Involutes: properties

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  , 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   and
  • the involutes are parallel curves, because of   and the fact, that   is the unit normal at  .

ExamplesEdit

Involutes of a circleEdit

 
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 parabolaEdit

 
Involutes of a semicubic parabola (blue). Only the red curve is a parabola.

The parametric representation   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:  

Hence:

  • 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).

Involutes of a catenaryEdit

 
The red involute of a catenary (blue) is a tractrix.

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.

Result:

  • The involutes of the catenary   are parallel curves of the tractrix  

Involutes of a cycloidEdit

 
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.

Result:

  • The involutes of the cycloid   are parallel curves of the cycloid
 

Involute and evoluteEdit

The evolute of a given curve   consists of the curvature centers of  . Between involutes and evolutes the following statement holds:[2][3]

  • A curve is the evolute of any of its involutes.

ApplicationEdit

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. The gears also 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.[4]

 
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 High Flux Isotope Reactor uses involute-shaped fuel elements, since these allow a constant-width channel between them for coolant.

GeneralizationEdit

The involute is an example of a roulette wherein the rolling curve is a straight line containing the generating point.

See alsoEdit

ReferencesEdit

  1. ^ McCleary, John (1995). Geometry from a Differentiable Viewpoint. Cambridge University Press. p. 73.[ISBN missing]
  2. ^ K. Burg, H. Haf, F. Wille, A. Meister: Vektoranalysis: Höhere Mathematik für Ingenieure, Naturwissenschaftler und ..., Springer-Verlag, 2012,ISBN 3834883468, S. 30.
  3. ^ R. Courant:Vorlesungen über Differential- und Integralrechnung, 1. Band, Springer-Verlag, 1955, S. 267.
  4. ^ V. G. A. Goss (2013) "Application of analytical geometry to the shape of gear teeth", Resonance 18(9): 817 to 31 Springerlink (subscription required).

External linksEdit