# Involute

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. Notations: spreadsheet syntaxis.

# 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

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 crossing the base circle
• Hypotenuse is A1*R1
• A1*R1 is always 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 = R1 * cos (A1) + R1 * A1 * sin (A1) = R1 * (cos (A1) + A1 * sin (A1))

• Y3 = Y1 - Y2 = R1 * sin (A1) - R1 * A1 * cos (A1) = R1 * (sin (A1) - A1 * cos (A1))

• Polar coordinates:
• Points on the involute curve have polar coordinates (R2,A2).
• R1 and A1*R1 are always perpendicular.
• So radius R2 can be calculated using the Pythagorean theorem. This results in:
• R2 = sqrt (R1^2 + (A1 * R1)^2) = R1 * sqrt (1 + A1^2).

• 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). Result:

• A2 = atan2 (X3,Y3)

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