involute

{{See also|Arc length}}

Let $\backslash vec\; c(t),\backslash ;\; t\backslash in\; [t\_1,t\_2]$ be a regular curve in the plane with its curvature nowhere 0 and $a\backslash in\; (t\_1,t\_2)$, then the curve with the parametric representation

$\backslash vec\; C\_a(t)=\backslash vec\; c(t)\; -\backslash frac\{\backslash vec\; c\text{'}(t)\}$

is an involute of the given curve.

The vector corresponding to the end point of the string ($\backslash vec\; C\_a(t)$) can be easily calculated using vector addition, and one gets

$\backslash vec\; C\_a(t)=\backslash vec\; c(t)\; -\backslash frac\{\backslash vec\; c\text{'}(t)\}$

}}

|}

Adding an arbitrary but fixed number $l\_0$ to the integral $\backslash Bigl(\backslash int\_a^t|\backslash vec\; c\text{'}(w)|\backslash ;\; dw\backslash Bigr)$ results in an involute corresponding to a string extended by $l\_0$ (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 $a$ and/or adding a number to the integral (see Involutes of a semicubic parabola).

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

:$\backslash begin\{align\}$

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{align}

File:Involute(in red) of parabola(dark blue).png

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, which lead to the following simplifications: $\backslash ;|\backslash vec\; c\text{'}(s)|=1\backslash ;$ and $\backslash ;\backslash vec\; c\text{'}\text{'}(s)=\backslash kappa(s)\backslash vec\; n(s)\backslash ;$, with $\backslash kappa$ the curvature and $\backslash vec\; n$ the unit normal. One gets for the involute:

:$\backslash vec\; C\_a(s)=\backslash vec\; c(s)\; -\backslash vec\; c\text{'}(s)(s-a)\backslash$ and

:$\backslash vec\; C\_a\text{'}(s)=-\backslash vec\; c\text{'}\text{'}(s)(s-a)=-\backslash kappa(s)\backslash vec\; n(s)(s-a)\backslash ;$

and the statement:

• At point $\backslash vec\; C\_a(a)$ the involute is not regular (because $|\; \backslash vec\; C\_a\text{'}(a)|=0$ ),

and from $\backslash ;\; \backslash vec\; C\_a\text{'}(s)\backslash cdot\backslash vec\; c\text{'}(s)=0\; \backslash ;$ follows:

• The normal of the involute at point $\backslash vec\; C\_a(s)$ is the tangent of the given curve at point $\backslash vec\; c(s)$.
• The involutes are parallel curves, because of $\backslash vec\; C\_a(s)=\backslash vec\; C\_0(s)+a\backslash vec\; c\text{'}(s)$ and the fact, that $\backslash vec\; c\text{'}(s)$ is the unit normal at $\backslash vec\; C\_0(s)$.

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.

This section is based on.{{Cite book |last=Arnolʹd |first=V. I. |url=https://www.worldcat.org/oclc/21873606 |title=Huygens and Barrow, Newton and Hooke : pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals |date=1990 |publisher=Birkhaüser Verlag |isbn=0-8176-2383-3 |location=Basel |oclc=21873606}}

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 $f:\; \backslash R^2\; \backslash to\; \backslash R^3$ defined by $$(s,\; t)\; \backslash mapsto\; (x(s)\; +\; t\backslash cos(\backslash theta),\; y(s)\; +\; t\backslash sin(\backslash theta),\; t)$$where $(x(s),\; y(s))$ is the arclength parametrization of the curve, and $\backslash theta$ is the slope-angle of the curve at the point $(x(s),\; y(s))$. This maps the 2D plane into a surface in 3D space. For example, this maps the circle into the hyperboloid of one sheet.

By this map, the involutes are obtained in a three-step process: map $\backslash R$ to $\backslash R^2$, then to the surface in $\backslash R^3$, then project it down to $\backslash R^2$ by removing the z-axis: $$s\; \backslash mapsto\; (s,\; l-\; s)\; \backslash mapsto\; f(s,\; l-\; s)\; \backslash mapsto\; (f(s,\; l-\; s)\_x,\; f(s,\; l-\; s)\_y)$$where $l$ is any real constant.

Since the mapping $s\; \backslash mapsto\; f(s,\; l-s)$ has nonzero derivative at all $s\backslash in\; \backslash R$, cusps of the involute can only occur where the derivative of $s\; \backslash mapsto\; f(s,\; l-s)$ is vertical (parallel to the z-axis), which can only occur where the surface in $\backslash R^3$ 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.

For the first type, one can start by the involute of a circle, with equation$$\backslash begin\{align\}$$

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

Y(t) &= r(\sin t - (t - a)\cos t)

\end{align}then set $a\; =\; 0$, and expand for small $t$, to obtain$$\backslash begin\{align\}$$

X(t) &= r + r t^2/2 + O(t^4)\\

Y(t) &= rt^3/3 + O(t^5)

\end{align}thus giving the order 3/2 curve $Y^2\; -\; \backslash frac\{8\}\{9r\}\; (X-r)^\{3\}\; +\; O(Y^\{8/3\})\; =\; 0$, a semicubical parabola.

File:Involutes of a cubic curve.svg

For the second type, consider the curve $y\; =\; x^3$. The arc from $x=\; 0$ to $x\; =\; s$ is of length $\backslash int\_0^s\; \backslash sqrt\{1\; +\; (3t^2)^2\}dt\; =\; s\; +\; \backslash frac\{9\}\{10\}\; s^5\; -\; \backslash frac\; 98\; s^9\; +\; O(s^\{13\})$, and the tangent at $x\; =\; s$ has angle $\backslash theta\; =\; \backslash arctan(3s^2)$. Thus, the involute starting from $x=\; 0$ at distance $L$ has parametric formula$$\backslash begin\{cases\}$$

x(s) = s + (L-s-\frac{9}{10}s^5 + \cdots)\cos\theta \\

y(s) = s^3 + (L-s-\frac{9}{10}s^5 + \cdots)\sin\theta

\end{cases}Expand it up to order $s^5$, we obtain$$\backslash begin\{cases\}$$

x(s) = L - \frac 92 L s^4 + (\frac 92 L - \frac{9}{10}) s^5 + O(s^6)\\

y(s) = 3Ls^2 - 2 s^3 + O(s^6)

\end{cases}which is a cusp of order 5/2. Explicitly, one may solve for the polynomial expansion satisfied by $x,\; y$:$$\backslash left(x\; -\; L\; +\; \backslash frac\{y^2\}\{2L\}\; \backslash right)^2\; -\; \backslash left(\backslash frac\; 92\; L\; +\; \backslash frac\{51\}\{10\}\; \backslash right)^2\; \backslash left(\backslash frac\{y\}\{3L\}\; \backslash right)^5\; +\; O(s^\{11\})\; =\; 0$$or $$x\; =\; L\; -\; \backslash frac\{y^2\}\{2L\}\; \backslash pm\; \backslash left(\backslash frac\; 92\; L\; +\; \backslash frac\{51\}\{10\}\; \backslash right)\; \backslash left(\backslash frac\{y\}\{3L\}\; \backslash right)^\{2.5\}\; +\; O(y^\{2.75\}),\backslash quad\; \backslash quad\; y\; \backslash geq\; 0$$which clearly shows the cusp shape.

Setting $L=0$, we obtain the involute passing the origin. It is special as it contains no cusp. By serial expansion, it has parametric equation$$\backslash begin\{cases\}$$

x(s) = \frac{18}{5} s^5 - \frac{126}{5} s^9 + O(s^{13}) \\

y(s) = -2s^3 + \frac{54}{5} s^7 - \frac{318}{5} s^{11} + O(s^{15})

\end{cases}or $x\; =\; -\backslash frac\{18\}\{5\; \backslash cdot\; 2^\{1/3\}\}y^\{5/3\}\; +\; O(y^3)$

File:Evolvente-kreis.svg

For a circle with parametric representation $(r\backslash cos(t),\; r\backslash sin(t))$, one has

$\backslash vec\; c\text{'}(t)\; =\; (-r\backslash sin\; t,\; r\backslash cos\; t)$.

Hence $|\backslash vec\; c\text{'}(t)|\; =\; r$, and the path length is $r(t\; -\; a)$.

Evaluating the above given equation of the involute, one gets

:$\backslash begin\{align\}$

X(t) &= r(\cos (t+a) + t\sin (t+a))\\

Y(t) &= r(\sin (t+a) - t\cos (t+a))

\end{align}

for the parametric equation of the involute of the circle.

The $a$ term is optional; it serves to set the start location of the curve on the circle. The figure shows involutes for $a\; =\; -0.5$ (green), $a\; =\; 0$ (red), $a\; =\; 0.5$ (purple) and $a\; =\; 1$ (light blue). The involutes look like Archimedean spirals, but they are actually not.

The arc length for $a=0$ and $0\; \backslash le\; t\; \backslash le\; t\_2$ of the involute is

: $L\; =\; \backslash frac\{r\}\{2\}\; t\_2^2.$

File:Evolvente-np.svg

The parametric equation $\backslash vec\; c(t)\; =\; (\backslash tfrac\{t^3\}\{3\},\; \backslash tfrac\{t^2\}\{2\})$ describes a semicubical parabola. From $\backslash vec\; c\text{'}(t)\; =\; (t^2,\; t)$ one gets $|\backslash vec\; c\text{'}(t)|\; =\; t\backslash sqrt\{t^2\; +\; 1\}$ and $\backslash int\_0^t\; w\backslash sqrt\{w^2\; +\; 1\}\backslash ,dw\; =\; \backslash frac\{1\}\{3\}\backslash sqrt\{t^2\; +\; 1\}^3\; -\; \backslash frac13$. Extending the string by $l\_0=\{1\backslash over3\}$ extensively simplifies further calculation, and one gets

:

Y(t) &= \frac{t^2}{6} - \frac{1}{3}.\end{align}

Eliminating {{mvar|t}} yields $Y\; =\; \backslash frac\{3\}\{2\}X^2\; -\; \backslash frac\{1\}\{3\},$ 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 {{slink|Parallel curve|Further examples}}).

File:Involute.gif

For the catenary $(t,\; \backslash cosh\; t)$, the tangent vector is $\backslash vec\; c\text{'}(t)\; =\; (1,\; \backslash sinh\; t)$, and, as $1\; +\; \backslash sinh^2\; t\; =\backslash cosh^2\; t,$ its length is $|\backslash vec\; c\text{'}(t)|\; =\; \backslash cosh\; t$. Thus the arc length from the point {{math|(0, 1)}} is $\backslash textstyle\backslash int\_0^t\; \backslash cosh\; w\backslash ,dw\; =\; \backslash sinh\; t.$

Hence the involute starting from {{math|(0, 1)}} is parametrized by

: $(t\; -\; \backslash tanh\; t,\; 1/\backslash cosh\; t),$

and is thus a tractrix.

The other involutes are not tractrices, as they are parallel curves of a tractrix.

File:Evolvente-zy.svg

The parametric representation $\backslash vec\; c(t)\; =\; (t\; -\; \backslash sin\; t,\; 1\; -\; \backslash cos\; t)$ describes a cycloid. From $\backslash vec\; c\text{'}(t)\; =\; (1\; -\; \backslash cos\; t,\; \backslash sin\; t)$, one gets (after having used some trigonometric formulas)

:$|\backslash vec\; c\text{'}(t)|\; =\; 2\backslash sin\backslash frac\{t\}\{2\},$

and

:$\backslash int\_\backslash pi^t\; 2\backslash sin\backslash frac\{w\}\{2\}\backslash ,dw\; =\; -4\backslash cos\backslash frac\{t\}\{2\}.$

Hence the equations of the corresponding involute are

: $X(t)\; =\; t\; +\; \backslash sin\; t,$

: $Y(t)\; =\; 3\; +\; \backslash cos\; t,$

which describe the shifted red cycloid of the diagram. Hence

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

: $(t\; +\; \backslash sin\; t,\; 3\; +\; \backslash cos\; t).$

(Parallel curves of a cycloid are not cycloids.)

The evolute of a given curve $c\_0$ consists of the curvature centers of $c\_0$. Between involutes and evolutes the following statement holds:

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.

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

{{Gallery

|title= Involute and evolute

|width=160 | height=170

|align=center

|File:Involute.gif

|alt2=Involute of a catenary

|Tractrix (red) as an involute of a catenary

|File:Evolute2.gif

|alt3=Evolute of a tractrix

|The evolute of a tractrix is a catenary

}}

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 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.V. G. A. Goss (2013) "Application of analytical geometry to the shape of gear teeth", Resonance 18(9): 817 to 31 [https://link.springer.com/article/10.1007%2Fs12045-013-0106-3 Springerlink] (subscription required).

Image:Two moving spirals scroll pump.gif

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.

• Envelope (mathematics)
• Evolute
• Goat grazing problem
• Involute gear
• Roulette (curve)
• Scroll compressor

{{reflist}}

• [http://mathworld.wolfram.com/Involute.html Involute] at MathWorld

{{Differential transforms of plane curves}}

Category:Differential geometry

Category:Roulettes (curve)