What is the Involute Function?

The involute function, denoted as inv(α) or inv α, is a mathematical function widely used in mechanical engineering, particularly in the design of involute gear teeth. It is defined as the difference between the tangent of an angle and the angle itself, when the angle is measured in radians:

inv(α) = tan(α) − α

In gear engineering, the angle α is known as the pressure angle. The involute function arises from the geometry of the involute curve — the path traced by a point on a taut string as it unwinds from a circle (the base circle). This curve is used for gear tooth profiles because it ensures constant velocity ratio and smooth transmission of motion.

The inverse problem — finding the pressure angle α given an involute value — is not straightforward because the equation inv(α) = tan(α) − α is transcendental and has no closed-form algebraic solution. It must be solved numerically, typically using iterative methods such as the Newton-Raphson method or the Laskin method.

This calculator allows you to:

• Calculate the involute for a given angle in degrees.
• Find the angle (pressure angle) for a given involute value using the Newton-Raphson method.

The calculator is useful for engineers, designers, and students working with gear geometry, tooth thickness measurements, and related calculations.