Involute

Two involutes (red) of a parabola

In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the locus of a point on a piece of taut string as the string is either unwrapped from or wrapped around the curve.[1]

It is a class of curves coming under the roulette family of curves.

The evolute of an involute is the original curve.

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).[2]

Involute of a parameterized curve

Let ${\displaystyle {\vec {c}}(t),\;t\in [t_{1},t_{2}]}$ be 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

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

is an involute of the given curve.

 ProofThe string acts as a tangent to the curve ${\displaystyle {\vec {c}}(t)}$. Its length is changed by an amount equal to the arc length traversed as it winds or unwinds. Arc length of the curve traversed in the interval ${\displaystyle [a,t]}$ is given by ${\displaystyle \int _{a}^{t}|{\vec {c}}'(w)|\;dw}$ where ${\displaystyle a}$ is the starting point from where the arc length is measured. Since the tangent vector depicts the taut string here, we get the string vector as ${\displaystyle {\frac {{\vec {c}}'(t)}{|{\vec {c}}'(t)|}}\;\int _{a}^{t}|{\vec {c}}'(w)|\;dw}$ The vector corresponding to the end point of the string (${\displaystyle {\vec {C}}_{a}(t)}$) can be easily calculated using vector addition, and one gets ${\displaystyle {\vec {C}}_{a}(t)={\vec {c}}(t)-{\frac {{\vec {c}}'(t)}{|{\vec {c}}'(t)|}}\;\int _{a}^{t}|{\vec {c}}'(w)|\;dw}$

Adding an arbitrary but fixed number ${\displaystyle l_{0}}$ to the integral ${\displaystyle {\Bigl (}\int _{a}^{t}|{\vec {c}}'(w)|\;dw{\Bigr )}}$ results in an involute corresponding to a string extended by ${\displaystyle l_{0}}$ (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 ${\displaystyle a}$ and/or adding a number to the integral (see Involutes of a semicubic parabola).

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

Involute: properties. The angles depicted are 90 degrees.

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, which lead to the following simplifications: ${\displaystyle \;|{\vec {c}}'(s)|=1\;}$ 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 ${\displaystyle {\vec {c}}(s)}$.
• 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 has ${\displaystyle {\vec {c}}'(t)=(-r\sin t,r\cos t)}$. Hence ${\displaystyle |{\vec {c}}'(t)|=r}$, and the path length is ${\displaystyle r(t-a)}$.

Evaluating the above given equation of the involute, one gets

{\displaystyle {\begin{aligned}X(t)&=r(\cos t+(t-a)\sin t)\\Y(t)&=r(\sin t-(t-a)\cos t)\end{aligned}}}

for the parametric equation of the involute of the circle.

The figure 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 look like Archimedean spirals, but they are actually not.

The arc length for ${\displaystyle a=0}$ and ${\displaystyle 0\leq t\leq t_{2}}$ of the involute is

${\displaystyle L={\frac {r}{2}}t_{2}^{2}.}$
Involutes of a semicubic parabola (blue). Only the red curve is a parabola.

Involutes of a semicubic parabola

The parametric equation ${\displaystyle {\vec {c}}(t)=({\tfrac {t^{3}}{3}},{\tfrac {t^{2}}{2}})}$ describes a semicubical parabola. From ${\displaystyle {\vec {c}}'(t)=(t^{2},t)}$ one gets ${\displaystyle |{\vec {c}}'(t)|=t{\sqrt {t^{2}+1}}}$ and ${\displaystyle \int _{0}^{t}w{\sqrt {w^{2}+1}}\,dw={\frac {1}{3}}{\sqrt {t^{2}+1}}^{3}-{\frac {1}{3}}}$. Extending the string by ${\displaystyle l_{0}={1 \over 3}}$ extensively simplifies further calculation, and one gets

{\displaystyle {\begin{aligned}X(t)&=-{\frac {t}{3}}\\Y(t)&={\frac {t^{2}}{6}}-{\frac {1}{3}}.\end{aligned}}}

Eliminating t yields ${\displaystyle Y={\frac {3}{2}}X^{2}-{\frac {1}{3}},}$ 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 Parallel curve § Further examples).

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

Involutes of a catenary

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

Hence the involute starting from (0, 1) is parametrized by

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

and is thus a tractrix.

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

Involutes of a cycloid

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

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

${\displaystyle |{\vec {c}}'(t)|=2\sin {\frac {t}{2}},}$

and

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

Hence the equations of the corresponding involute are

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

which describe the shifted red cycloid of the diagram. Hence

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

(Parallel curves of a cycloid are not cycloids.)

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: [3][4]

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.[5]

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.