Involute Function Calculator

Created by Davide Borchia
Reviewed by Anna Szczepanek, PhD and Rijk de Wet
Last updated: Mar 23, 2022

Not happy with simple curves, mathematicians decided to see what happens when two curves slide on one another, and the result made many engineers extremely happy: discover why with our involute function calculator!

Here you will learn:

  • What the involute function is;
  • What it has to do with involute curves; and
  • How it is related to the gear pressure angle, which is a very important concept in engineering. ⚙️

Roulette and involute

When mathematicians talk about roulette, they are not talking of the casino game, but rather of a particular family of curves obtained by rolling a curve on another fixed curve and following the trajectory of a given, fixed point integral with the rolling curve. The other conditions are that:

  • Both curves must be differentiable — this means that it is "smooth" in its entirety, without points like cusps or interruptions where it would be impossible to define its slope.
  • The curves must be tangent at every moment.

🔎 There is a drawing device that has fascinated kids for decades, which bases its attractiveness on roulette curves: the spirograph. By rolling geared shapes (like circles and ellipses) with pencil holes around each other, it is possible to draw all sorts of mesmerizing curves.

A particular class of roulette curves is obtained by following a line wrapped around a curve (taut string) as it unwraps, or vice-versa: this type of roulette is called the involute.

When we talk of a string wrapped around a circle, the resulting curve, resembling a spiral, is called involute of a circle. Here it is, with its construction.

Construction of the involute of a circle
A taut string unwrapping from a circle creates an involute curve.

The red straight line represents the tensioned string as it wraps/unwraps. The tip of the string draws the involute.

The involute is described by this parametric equation:

x=a(cosθ+θsinθ)y=a(sinθθcosθ)\begin{split} x &=a(\cos{\theta}+\theta\cdot\sin{\theta}) \\ y &=a(\sin{\theta}−\theta\cdot\cos{\theta}) \end{split}

where aa is the radius of the circle and θ\theta the angle at which the string separates from the circle (where circle and line are tangent).

Involute gears

Involute functions are interesting for mathematicians but fundamental for engineers: the main application of the involute function is the construction of involute gears.

It is necessary to introduce a few elements of involute gears:

  • The line of action and the line of centers;
  • The pitch point;
  • The pressure angle.

The line of centers connects the centers of the two gears. While a couple of teeth is in contact as the gears roll, the point of contact moves along an imaginary line called line of action.

This is the core point of the peculiarity of involute gears: the contact between teeth happens only on a straight line — obviously if the teeth' profiles match.

Involute gears
The various elements necessary to identify the pressure angle.

The pitch point is the point on the line of centers where the circumferential speeds of the gears match (this means that points on gears' edges travel at the same speed). In involute gears, this is also the point of intersection between the action line and the line of centers.

Let's draw the perpendicular to the line of centers passing through the pitch point.

At this point, the angle between this line and the action line is called pressure angle. Its value remains constant during the operation of the gears; hence it is characteristic for a given design.

Involute gear design and involute function

The involute gear pressure angle is the fundamental parameter when designing the gear itself, since its value affects the tilting of the line of action and the behavior of the gear during its operations.

Here is how to calculate the involute as a function of the pressure angle.

Involute construction
Geometrical construction of the involute function.

We start by building the circle with center OO and radius OSOS. Then we draw a part of the involute of the circle. We call the origin point SS, then choose a point PP on this curve. The taut string touches the circumference in the point TT.

By construction, the segment TPTP and the arc STST have the same length. It is then possible to write: TP=ST\overline{TP} =\stackrel{\frown}{ST}

In this construction, the angle of pressure is the angle TOP^=α\hat{TOP} = \alpha. The other angle, SOP^=φ\hat{SOP} = \varphi, can be found following the next steps:

TP= STrtanα=r(φ+α)tanα=φ+αφ=Inv(α)=tanαα\overline{TP} =\ \stackrel{\frown}{ST} \\ r\cdot\tan{\alpha}= r \left( \varphi + \alpha\right) \\ \tan \alpha = \varphi +\alpha\\ \varphi = \operatorname{Inv}(\alpha) =\tan{\alpha} - \alpha

The last expression is the involute function. tan\tan{} is the tangent function. The resulting angle is extremely important in the design of involute gears, for example in the calculation of the tooth thickness.

Involute function
Plot of the involute function. The independent variable is the pressure angle.

From the geometrical construction, you can see that φ\varphi (the value of the involute function for a given pressure angle α\alpha) is somehow connected to the "thickness" of the tooth of an involute gear. This explains why the involute function is so important.

An increase in the pressure angle equals an increase in the width of the tooth. A wider tooth is a stronger tooth that can withstand bigger loads. On the other hand, smaller teeth obtained by reducing the pressure angle α\alpha give advantages to the smoothness of the operation.

How does our involute function calculator work?

Using our involute function calculator is really easy. You can input the value of the gear pressure angle α\alpha to obtain right away the value of the relative involute function.

🙋 You can't use this calculator in "reverse" due to the presence of the tangent function in the involute!

Let's try with an example.

The most common gear pressure angle currently used is 20°20\degree. In the involute function calculator, change the units of α\alpha to degrees and input 2020. The result is dimensionless and equal to 0.014\thicksim 0.014.

If you are interested in other calculators, check our gear ratio calculator or discover the sister curve of the involute, the catenary curve!


What is an involute?

An involute is a parametric curve that describes the wrapping/unwrapping of a taut string around a generating curve. The most common involute is the involute of the circle, which returns a curve similar to a spiral.

How do I calculate the involute function?

The involute function is a function that takes as argument an angle — the pressure angle — and returns a value that corresponds to the width of the involute curve at that point. The formula for the involute function is φ = tan(α) -α.

What is the relation between involute function and angle of action?

The angle of action of an involute gear is associated with the width of the teeth of the gear itself through the involute function.

Why is the involute important in engineering?

Engineers use the shape of the involute of a circle to design the teeth of gears that touch only at one point during the entire contact time. Involute gears are widespread and compose the vast majority of gears produced worldwide.

Davide Borchia
Pressure angle α
Involute function Inv(α)
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