Involute - or evolvent - curves are used as the shape of teeth of gears. For an introduction to more advanced math, see https://en.wikipedia.org/wiki/Involute. However, this page is more practical.

What is the involute curve of a circle?

Involute: a quick and dirty example

In plain language. A cord is coiled around a cylinder. At the end of the cord is a pencil. The cord is unrolled while the cord is held taut. The pencil draws a curve. This curve is called involute of a circle - you can make involutes based on other shapes too.

Involute curve with x- and y-axis, without base circle.

Practical use

The main reason for using gears with involute shapes is that there is no friction between teeth - against common believe. The teeth roll on each other when rotating.

It is the gear application that makes a polar calculation based on radii more useful than a Cartesian calculation based on (x,y) - we are only interested in the first small part that starts from the base circle. On the other hand, a polar approach has its limits, see below.

Mathematics

Involute with base circle

Angles are in radians, 360 degrees equals (2 * pi) radians.

Overall, see illustration:

• The blue curve is the involute of a circle.
• Points on that curve have coordinates X3 and Y3.
• From those points runs a cord A1*R1, tangent to the base circle.
• The base circle has a radius R1.

Mathematics of involute

In detail:

• The first triangle:
  • Start at point (0,0)
  • Hypotenuse is R1
  • The angle is A1

  • X1 = R1 * cos (A1)

  • Y1 = R1 * sin (A1)

• The second triangle:
  • Start at the end of R1
  • Hypotenuse is A1*R1
  • A1*R1 is perpendicular to R1
  • Therefore, the angle is again A1

  • X2 = R1 * A1 * sin (A1)

  • Y2 = R1 * A1 * cos (A1)

  • About the length of hypotenuse A1*R1:

    • If A1 goes all the way around from 0 to 2 * pi radians (equals 360 degrees), than the length of the cord changes from 0 to 2 * pi * R1. After all, the circumference of a circle equals 2 * pi * R1.

    • The length of the cord is thus equal to the product of angle A1 and radius R1, hence A1 * R1.

• Cartesian coordinates:
  • Points on the involute curve have Cartesian coordinates (X3,Y3).

  • X3 = X1 + X2

  • Y3 = Y1 - Y2

• Polar coordinates:
  • Points on the involute curve have polar coordinates (R2,A2).
  • About R2:
    • R1 and A1*R1 are always perpendicular.
    • Radius R2 can be calculated using the Pythagorean theorem.

    • R2 = sqrt (R1^2 + (A1 * R1)^2) = R1 * sqrt (1 + A1^2).

  • About A2:
    • This can be calculated by using asin, acos or atan.

    • Unfortunately, these functions do not go all the way around with one argument supplied. For example, A2 = acos (X3 / R2) and that is only true for 0 < A2 < pi. See example.

    • Fortunately, many environments offer a solution when X- and Y values are both entered as arguments. For example, LibreOffice Calc has a formula ATAN2(X-value;Y-value) and LISP for CAD offers (ATAN Y-value X-value).

    Wrongly drawn by using acos(X3/R2) for angle A2

• Additional information:
  • For a gear application, A1 is not interesting while R1 and R2 are important.

    • Polar part R2 is defined as a relation of R1 and A1, as stated before: R2 = R1 * sqrt (1 + A1^2).

    • If we solve this equation in order to find A1, the answer becomes:

    • A1 = 2 * sqrt ((R2^2 - R1^1) / 2)