Involute

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).^[1]

Involute of a parameterized curve

Let be ${\vec {x}}={\vec {c}}(t),\;t\in [t_{1},t_{2}]$ a regular curve in the plane with its curvature nowhere 0 and $a\in (t_{1},t_{2})$ , then the curve with the parametric representation

${\vec {X}}={\vec {C}}_{a}(t)={\vec {c}}(t)-{\frac {{\vec {c}}'(t)}{|{\vec {c}}'(t)|}}\;\int _{a}^{t}|{\vec {c}}'(w)|\;dw$

is an involute of the given curve.
The integral describes the actual length of the free part of the string in the interval $[a,t]$ and the vector prior to that is the tangent unitvector. Adding an arbitrary but fixed number $l_0$ to the integral results in an involute corresponding to a string, which is extended by $l_0$ . Hence: the involute can be variied by parameter $a$ and/or adding a number to the integral (see Involutes of a semicubic parabola).

If ${\vec {x}}={\vec {c}}(t)=(x(t),y(t))^{T}$ one gets

{\begin{aligned}X(t)&=x(t)-{\frac {x'(t)}{\sqrt {x'(t)^{2}+y'(t)^{2}}}}\int _{a}^{t}{\sqrt {x'(w)^{2}+y'(w)^{2}}}\,dw\\Y(t)&=y(t)-{\frac {y'(t)}{\sqrt {x'(t)^{2}+y'(t)^{2}}}}\int _{a}^{t}{\sqrt {x'(w)^{2}+y'(w)^{2}}}\,dw\;.\end{aligned}}

Properties of involutes

Involutes: properties

In order to derive properties of a regular curve it is advantageous to suppose the arc length $s$ to be the parameter of the given curve. Because of the simplifications in this case: $\;|{\vec {c}}'(s)|=1\;$ and $\;{\vec {c}}''(s)=\kappa (s){\vec {n}}(s)\;$ , with $\kappa$ the curvature and $\vec n$ the unit normal, one gets for the involute:

{\vec {C}}_{a}(s)={\vec {c}}(s)-{\vec {c}}'(s)(s-a)\

and

{\vec {C}}_{a}'(s)=-{\vec {c}}''(s)(s-a)=-\kappa (s){\vec {n}}(s)(s-a)\;

and the statement:

At point ${\vec {C}}_{a}(a)$ the involute is not regular (because $|{\vec {C}}_{a}'(a)|=0$ ),

and from $\;{\vec {C}}_{a}'(s)\cdot {\vec {c}}'(s)=0\;$ follows:

The normal of the involute at point ${\vec {C}}_{a}(s)$ is the tangent of the given curve at point ${\vec {c}}(s)$ and
the involutes are parallel curves, because of ${\vec {C}}_{a}(s)={\vec {C}}_{0}(s)+a{\vec {c}}'(s)$ and the fact, that ${\vec {c}}'(s)$ is the unit normal at ${\vec {C}}_{0}(s)$ .

Examples

Involutes of a circle

For a circle with parametric representation $\;(r\cos(t),r\sin(t))\;$ one gets ${\vec {c}}'(t)=(-r\sin t,r\cos t)^{T}$ . Hence $|{\vec {c}}'(t)|=r$ and the integral is $r(t-a)$ . The equations of the involutes are:

$X(t)=r(\cos t+(t-a)\sin t)$

Y(t)=r(\sin t-(t-a)\cos t)\;.

The diagram shows involutes for $a=-0,5$ (green), $a=0$ (red), $a=0.5$ (purple) and $a=1$ (light blue). The involutes are similar to Archimedean spirals but they are actually not.

The arc length of the involute with $0\leq t\leq t_{2}$ is

L={\frac {r}{2}}{t_{2}}^{2}\;.

Involutes of a semicubic parabola

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

The parametric representaion $\;{\vec {c}}(t)=({\tfrac {t^{3}}{3}},{\tfrac {t^{2}}{2}})^{T}\;$ describes a semicubic parabola. From $\;{\vec {c}}'(t)=(t^{2},t)^{T}\;$ one gets $\;|{\vec {c}}'(t)|=t{\sqrt {t^{2}+1}}$ and $\int _{0}^{t}w{\sqrt {w^{2}+1}}\;dw={\frac {1}{3}}{\sqrt {t^{2}+1}}^{3}-1/3$ . Extending the string by $l_{0}=1/3$ causes an essential simplification of the calculation and one gets

X(t)=\;\cdots \;=-{\frac {t}{3}}

Y(t)=\;\cdots \;={\frac {t^{2}}{6}}-{\frac {1}{3}}\ .

Eliminating parameter $t$ yields the equation of a parabola: $\;Y={\frac {3}{2}}X^{2}-{\frac {1}{3}}\;.$
Hence:

The involutes of the semicubic parabola $\;({\tfrac {t^{3}}{3}},{\tfrac {t^{2}}{2}})\;$ are parallel curves of the parabola $\;y={\frac {3}{2}}x^{2}-{\frac {1}{3}}\;.$

(Parallel curves of a parabola are not parabolas anymore !)

Remark: The evolute of the parabola $\;y={\frac {3}{2}}x^{2}-{\frac {1}{3}}\;$ is the semicubic parabola $\;({\tfrac {t^{3}}{3}},{\tfrac {t^{2}}{2}})\;$ (see section involute and evolute.)

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

Involutes of a catenary

For the catenary $\;(t,\cosh t)\;$ one gets $\;{\vec {c}}'(t)=(1,\sinh t)^{T}\;$ and because of $\;\cosh ^{2}t-\sinh ^{2}t=1\;$ the length of the tangent vector is $\;|{\vec {c}}'(t)|=\cosh t$ and the integral $\;\int _{0}^{t}...=\sinh t\;.$ Hence the parametric representation of the corresponding involute is

(t-\tanh t,1/\cosh t)\;,

which describes a tractrix.
Result:

The involutes of the catenary $\;(t,\cosh t)\;$ are parallel curves of the tractrix $(t-\tanh t,1/\cosh t)\;.$

Involutes of a cycloid

Involutes of a cycloid (blue): Only the red curve is another cycloid

The parametric representation $\;{\vec {c}}(t)=(t-\sin t,1-\cos t)^{T}\;$ describes a cycloid. From $\;{\vec {c}}'(t)=(1-\cos t,\sin t)^{T}\;$ one gets $\;|{\vec {c}}'(t)|=\cdots =2\sin {\frac {t}{2}}$ and $\int _{\pi }^{t}2\sin {\frac {t}{2}}\;dw=-4\cos {\frac {t}{2}}$ . (Trigonometric formulae were used.)
Hence the equations of the corresponding involute are

X(t)=\;\cdots \;=t+\sin t

Y(t)=\;\cdots \;=3+\cos t\ ,

which describe the shifted red cycloid of the diagram.
Result:

The involutes of the cycloid $\;(t-\sin t,1-\cos t)\;$ are parallel curves of the cycloid

(t+\sin t,3+\cos t)\;.

Involute and evolute

The evolute of a given curve $c_{0}$ consists of the curvature centers of $c_{0}$ . Between involutes and evolutes the following statement holds^[2]^[3]:

A curve is the evolute of any of its involutes.

Involte and Evolute

Tractrix (red) as an involute of a catenary

The evolute of a tractrix is a catenary

Application

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.^[4]

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.

Generalization

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

References

↑ 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)

External links

Involute at MathWorld

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] McCleary, John (1995). Geometry from a Differentiable Viewpoint. Cambridge University Press. p. 73.

[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)

Differential transforms of plane curves
Unary operations	Evolute Involute Dual curve Inverse curve Parallel curve Isoptic
Unary operations defined by a point	Pedal & Contrapedal curves Negative pedal curve Pursuit curve Caustic
Unary operations defined by two points	Strophoid
Binary operations defined by a point	Roulette Cissoid
Operations on a family of curves	Envelope

Christiaan Huygens
Books	Systema Saturnium (1659) De Vi Centrifiga (1659) Horologium Oscillatorium (1673) Traité de la Lumiére (1692) Cosmotheoros (1698)
In science and natural philosophy	Centripetal acceleration Coupled oscillation Conception of the standardization of the temperature scale Early history of classical mechanics Early history of calculus Huygens' law Huygens' lemniscate Huygens' principle (Huygens–Fresnel principle) Huygens' construction Huygens' tritone Huygens' wavelet Huygens' wave theory Huygens–Steiner theorem Hypothesis of intelligent extraterrestrial life Isochrone curve (tautochrone curve) Foundations of differential geometry of curves (mathematical notions of the evolute and involute of the curve) Scientific foundations of horology Mathematical and physical investigations of properties of the pendulum Modern conception of centrifugal and centripetal forces Music theory of microtones (31 equal temperament) Polarization of light (Iceland spar) Rings of Saturn Titan (moon) Theoretical foundations of wave optics (wave theory of light)
In technology	Inventions by Huygens Aerial telescope Centrifugal governor Cycloidal pendulum Huygens' engine ¹ Huygens' eyepiece Magic lantern ² Spiral balance spring Precision timekeeping (pendulum clock and spiral-hairspring watch)
Recognitions	List of things named after Christiaan Huygens 2801 Huygens Cassini–Huygens Huygens probe Mons Huygens Huygens (crater) Huygens-Fokker Foundation Huygens Gap Huygens Ringlet Horologium (constellation)
Other topics	Cosmos: A Personal Voyage - Episode 6: "Travellers' Tales" (1980 documentary TV series by Carl Sagan) Clocks and Culture, 1300–1700 (1967 history book by Carlo Cipolla) Revolution in Time: Clocks and the Making of the Modern World (1983 history book by David Landes) Scientific Revolution Golden Age of Dutch science and technology Science and technology in the Dutch Republic Académie des Sciences
Related people	Huygens family Galileo Galilei René Descartes Salomon Coster Antonie van Leeuwenhoek Gottfried Wilhelm Leibniz Isaac Newton Robert Hooke Denis Papin Augustin-Jean Fresnel Thomas Young
¹ A rudimentary prototype of internal combustion piston engine. ² An early practical type of image projector and a precursor to both the modern slide projector and the movie projector. Wikiquote Wikisource texts