Involute

In the differential geometry of curves, an involute (also known as evolvent) is a curve obtained from another given curve 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. It is a roulette wherein the rolling curve is a straight line containing the generating point. 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.

Alternatively, another way to construct the involute of a curve is to replace the taut string 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. Compare Media:Evolute2.gif and Media:Involute.gif

If the function $r : ℝ \to ℝ n$ is a natural parametrization of the curve (i.e., $| r'(s) | = 1$ for all $s$ ), then

s\mapsto r(s)-sr^{\prime }(s)

parametrizes the involute.

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 parametrically defined curve

Equations of an involute curve for a parametrically defined function $(x (t), y (t))$ are

{\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}}

Examples

Involute of a circle

The involute of a circle by unwinding a string

The involute of a circle (black) is not identical to the Archimedean spiral (red) in a trivial manner.

The involute of a circle resembles, but is not, an Archimedean spiral.

Its successive turns are parallel curves with constant separation distance, a property which is often (inaccurately) ascribed to the Archimedean spiral.

In Cartesian coordinates $x$ and $y$ , the involute of a circle has the parametric equation

{\begin{aligned}x&=r\left(\cos \theta +\theta \sin \theta \right)\\y&=r\left(\sin \theta -\theta \cos \theta \right)\end{aligned}}

where

r

is the radius of the circle, and

θ

is the angle in radians (

θ \in ℝ

). The counterclockwise spiral is made with positive values of

θ

, and the clockwise spiral is made with the negative values of

θ

In polar coordinates $r and φ$ , the involute of a circle has the parametric equation

{\begin{aligned}r&=a\sec \alpha ={\frac {a}{\cos \alpha }}\\\varphi &=\tan \alpha -\alpha \end{aligned}}

where

a

is the radius of the circle and

α

is an angle parameter in radians (

α \in ℝ

) equal to

t - φ

(so

tan α = t

With this parameter

t

(with

t = tan α

;

t \in ℝ

) it can be written in the form

{\begin{aligned}r&=a{\sqrt {1+\ t^{2}}}\\\varphi &=t-\arctan(t).\end{aligned}}

From the Cartesian coordinates and the first polar coordinates, can be seen that

r(\theta )=a{\sqrt {1+\theta ^{2}}}

\theta =t.

Curve length

The arc length of the above curve for $0 \leq t \leq t 1$ is

L={\frac {a}{2}}{t_{1}}^{2}

Involute of a catenary

The involute of a catenary, a tractrix.

The involute of a catenary through its vertex is a tractrix. In Cartesian coordinates the curve follows

{\begin{aligned}x&=t-\tanh(t)\\y&=\operatorname {sech} (t)={\frac {1}{\cosh(t)}}\end{aligned}}

where $t$ is a parameter and $sech$ is the hyperbolic secant ( $1 / cosh(t)$ ).

Derivative

With

r(s)={\Big (}\sinh ^{-1}(s),\cosh \left(\sinh ^{-1}(s)\right){\Big )}

we have

r'(s)={\frac {(1,s)}{\sqrt {1+s^{2}}}}

and

r(t)-tr^{\prime }(t)=\left(\sinh ^{-1}(t)-{\frac {t}{\sqrt {1+t^{2}}}},{\frac {1}{\sqrt {1+t^{2}}}}\right).

Substitute

t={\frac {\sqrt {1-y^{2}}}{y}}

to get

\left(\operatorname {sech} ^{-1}(y)-{\sqrt {1-y^{2}}},y\right).

Involute of a cycloid

One involute of a cycloid is a congruent cycloid. In cartesian coordinates the curve follows

{\begin{aligned}x&=r{\bigl (}t-\sin(t){\bigr )}\\y&=r{\bigl (}1-\cos(t){\bigr )}\end{aligned}}

where $t$ is the angle and $r$ is the radius.

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.

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.

References

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

External links

Mathworld
Application of the involute to gear teeth – a short historical account

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 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 (spacecraft) 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

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