Involute

Two involutes (red) of a parabola

Not to be confused withinvolution (mathematics).

Inmathematics,aninvolute(also known as anevolvent) is a particular type ofcurvethat is dependent on another shape or curve. An involute of a curve is thelocusof a point on a piece of taut string as the string is either unwrapped from or wrapped around the curve.

Theevoluteof an involute is the original curve.

It is generalized by theroulettefamily of curves. That is, the involutes of a curve are the roulettes of the curve generated by a straight line.

The notions of the involute and evolute of a curve were introduced byChristiaan Huygensin his work titledHorologium oscillatorium sive de motu pendulorum ad horologia aptato demonstrationes geometricae(1673), where he showed that the involute of a cycloid is still a cycloid, thus providing a method for constructing thecycloidal pendulum,which has the useful property that its period is independent of the amplitude of oscillation.

Involute of a parameterized curve

See also:Arc length

Let${\vec {c}}(t),\;t\in [t_{1},t_{2}]$be aregular curvein the plane with itscurvaturenowhere 0 and$a\in (t_{1},t_{2})$,then the curve with the parametric representation

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

is aninvoluteof the given curve.

Proof The string acts as atangentto the curve${\vec {c}}(t)$.Its length is changed by an amount equal to thearc lengthtraversed as it winds or unwinds. Arc length of the curve traversed in the interval$[a,t]$is given by $\int _{a}^{t}|{\vec {c}}'(w)|\;dw$ where$a$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 ${\frac {{\vec {c}}'(t)}{|{\vec {c}}'(t)|}}\;\int _{a}^{t}|{\vec {c}}'(w)|\;dw$ The vector corresponding to the end point of the string (${\vec {C}}_{a}(t)$) can be easily calculated usingvector addition,and one gets ${\vec {C}}_{a}(t)={\vec {c}}(t)-{\frac {{\vec {c}}'(t)}{|{\vec {c}}'(t)|}}\;\int _{a}^{t}|{\vec {c}}'(w)|\;dw$

Adding an arbitrary but fixed number$l_{0}$to the integral${\Bigl (}\int _{a}^{t}|{\vec {c}}'(w)|\;dw{\Bigr )}$results in an involute corresponding to a string extended by$l_{0}$(like a ball of woolyarnhaving 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 (seeInvolutes of a semicubic parabola).

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

Involute: properties. The angles depicted are 90 degrees.

In order to derive properties of a regular curve it is advantageous to suppose thearc length$s$to be the parameter of the given curve, which lead to the following simplifications:$\;|{\vec {c}}'(s)|=1\;$and$\;{\vec {c}}''(s)=\kappa (s){\vec {n}}(s)\;$,with$\kappa$thecurvatureand${\vec {n}}$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 isnot 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${\vec {c}}(s)$.
• The involutes areparallel 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${\vec {C}}_{0}(s)$.

The family of involutes and the family of tangents to the original curve makes up anorthogonal 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.

Cusps

This section is based on.

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$f:\mathbb {R} ^{2}\to \mathbb {R} ^{3}$defined by$$(s,t)\mapsto (x(s)+t\cos(\theta ),y(s)+t\sin(\theta ),t)$$where$(x(s),y(s))$is the arclength parametrization of the curve, and$\theta$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 thehyperboloid of one sheet.

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

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

cusp of order 3/2

For the first type, one can start by the involute of a circle, with equation$${\begin{aligned}X(t)&=r(\cos t+(t-a)\sin t)\\Y(t)&=r(\sin t-(t-a)\cos t)\end{aligned}}$$then set$a=0$,and expand for small$t$,to obtain$${\begin{aligned}X(t)&=r+rt^{2}/2+O(t^{4})\\Y(t)&=rt^{3}/3+O(t^{5})\end{aligned}}$$thus giving the order 3/2 curve$Y^{2}-{\frac {8}{9r}}(X-r)^{3}+O(Y^{8/3})=0$,asemicubical parabola.

cusp of order 5/2

Tangents and involutes of the cubic curve$y=x^{3}$.The cusps of order 3/2 are on the cubic curve, while the cusps of order 5/2 are on the x-axis (the tangent line at the inflection point).

For the second type, consider the curve$y=x^{3}$.The arc from$x=0$to$x=s$is of length$\int _{0}^{s}{\sqrt {1+(3t^{2})^{2}}}dt=s+{\frac {9}{10}}s^{5}-{\frac {9}{8}}s^{9}+O(s^{13})$,and the tangent at$x=s$has angle$\theta =\arctan(3s^{2})$.Thus, the involute starting from$x=0$at distance$L$has parametric formula$${\begin{cases}x(s)=s+(L-s-{\frac {9}{10}}s^{5}+\cdots )\cos \theta \\y(s)=s^{3}+(L-s-{\frac {9}{10}}s^{5}+\cdots )\sin \theta \end{cases}}$$Expand it up to order$s^{5}$,we obtain$${\begin{cases}x(s)=L-{\frac {9}{2}}Ls^{4}+({\frac {9}{2}}L-{\frac {9}{10}})s^{5}+O(s^{6})\\y(s)=3Ls^{2}-2s^{3}+O(s^{6})\end{cases}}$$which is a cusp of order 5/2. Explicitly, one may solve for the polynomial expansion satisfied by$x,y$:$$\left(x-L+{\frac {y^{2}}{2L}}\right)^{2}-\left({\frac {9}{2}}L+{\frac {51}{10}}\right)^{2}\left({\frac {y}{3L}}\right)^{5}+O(s^{11})=0$$or$$x=L-{\frac {y^{2}}{2L}}\pm \left({\frac {9}{2}}L+{\frac {51}{10}}\right)\left({\frac {y}{3L}}\right)^{2.5}+O(y^{2.75}),\quad \quad y\geq 0$$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$${\begin{cases}x(s)={\frac {18}{5}}s^{5}-{\frac {126}{5}}s^{9}+O(s^{13})\\y(s)=-2s^{3}+{\frac {54}{5}}s^{7}-{\frac {318}{5}}s^{11}+O(s^{15})\end{cases}}$$or$x=-{\frac {18}{5\cdot 2^{1/3}}}y^{5/3}+O(y^{3})$

Examples

Involutes of a circle

Involutes of a circle

For a circle with parametric representation$(r\cos(t),r\sin(t))$,one has ${\vec {c}}'(t)=(-r\sin t,r\cos t)$. Hence$|{\vec {c}}'(t)|=r$,and the path length is$r(t-a)$.

Evaluating the above given equation of the involute, one gets

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

for theparametric equationof 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 likeArchimedean spirals,but they are actually not.

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

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

Involutes of a semicubic parabola (blue). Only the red curve is a parabola. Notice how the involutes and tangents make up an orthogonal coordinate system. This is a general fact.

Involutes of a semicubic parabola

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

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

Eliminatingtyields$Y={\frac {3}{2}}X^{2}-{\frac {1}{3}},$showing that this involute is aparabola.

The other involutes are thusparallel curvesof a parabola, and are not parabolas, as they are curves of degree six (SeeParallel curve § Further examples).

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

Involutes of a catenary

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

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

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

and is thus atractrix.

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${\vec {c}}(t)=(t-\sin t,1-\cos t)$describes acycloid.From${\vec {c}}'(t)=(1-\cos t,\sin t)$,one gets (after having used some trigonometric formulas)

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

and

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

Hence the equations of the corresponding involute are

$X(t)=t+\sin t,$

$Y(t)=3+\cos t,$

which describe the shifted red cycloid of the diagram. Hence

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

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

(Parallel curves of a cycloid are not cycloids.)

Involute and evolute

Theevoluteof a given curve$c_{0}$consists of the curvature centers of$c_{0}$.Between involutes and evolutes the following statement holds:

A curve is the evolute of any of its involutes.

Involute and evolute

• 
  Tractrix (red) as an involute of a catenary
• 
  The evolute of a tractrix is a catenary

Application

The most common profiles of moderngearteeth are involutes of a circle. In aninvolute gearsystem the teeth of two meshing gears contact at a single instantaneous point that follows along a single straight line of action. The forces exerted 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 thefundamental 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 relatedcycloidal gearsystem.

Mechanism of a scroll compressor

The involute of a circle is also an important shape ingas compressing,as ascroll compressorcan be built based on this shape. Scroll compressors make less sound than conventional compressors and have proven to be quiteefficient.

TheHigh Flux Isotope Reactoruses involute-shaped fuel elements, since these allow a constant-width channel between them for coolant.

See also

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

External links

• InvoluteatMathWorld