# Involute

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]

## Involute of a parameterized curve

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

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

is an involute of the given curve.
The integral describes the actual length of the free part of the string in the interval ${\displaystyle [a,t]}$ and the vector prior to that is the tangent unitvector. Adding an arbitrary but fixed number ${\displaystyle l_{0}}$ to the integral results in an involute corresponding to a string, which is extended by ${\displaystyle l_{0}}$. Hence: the involute can be varied by parameter ${\displaystyle a}$ and/or adding a number to the integral (see Involutes of a semicubic parabola).

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

{\displaystyle {\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 ${\displaystyle s}$ to be the parameter of the given curve. Because of the simplifications in this case: ${\displaystyle \;|{\vec {c}}'(s)|=1\;}$ and ${\displaystyle \;{\vec {c}}''(s)=\kappa (s){\vec {n}}(s)\;}$, with ${\displaystyle \kappa }$ the curvature and ${\displaystyle {\vec {n}}}$ the unit normal, one gets for the involute:

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

and the statement:

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

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

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

## Examples

### Involutes of a circle

Involutes of a circle

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

• ${\displaystyle X(t)=r(\cos t+(t-a)\sin t),}$
${\displaystyle Y(t)=r(\sin t-(t-a)\cos t).}$

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

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

${\displaystyle L={\frac {r}{2}}t_{2}^{2}.}$

### Involutes of a semicubic parabola

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

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

${\displaystyle X(t)=\cdots =-{\frac {t}{3}},}$
${\displaystyle Y(t)=\cdots ={\frac {t^{2}}{6}}-{\frac {1}{3}}.}$

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

Hence:

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

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

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

### Involutes of a catenary

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

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

${\displaystyle (t-\tanh t,1/\cosh t),}$

which describes a tractrix.

Result:

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

### Involutes of a cycloid

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

The parametric representation ${\displaystyle {\vec {c}}(t)=(t-\sin t,1-\cos t)^{T}}$ describes a cycloid. From ${\displaystyle {\vec {c}}'(t)=(1-\cos t,\sin t)^{T}}$, one gets ${\displaystyle |{\vec {c}}'(t)|=\cdots =2\sin {\frac {t}{2}}}$ and ${\displaystyle \int _{\pi }^{t}2\sin {\frac {t}{2}}\,dw=-4\cos {\frac {t}{2}}}$ (trigonometric formulae were used).

Hence the equations of the corresponding involute are

${\displaystyle X(t)=\cdots =t+\sin t,}$
${\displaystyle Y(t)=\cdots =3+\cos t,}$

which describe the shifted red cycloid of the diagram.

Result:

• The involutes of the cycloid ${\displaystyle (t-\sin t,1-\cos t)}$ are parallel curves of the cycloid
${\displaystyle (t+\sin t,3+\cos t).}$

## Involute and evolute

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

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

## 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. 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.

## Generalization

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