Calculate the involute function inv(α) = tan(α) - α (in radians) or find the pressure angle from a given involute value using iterative methods.

inv(20°) = 0.014904

Note: Involute is used in gear design. The inverse problem is solved by the Newton-Raphson iteration method.

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.

How to Use This Calculator

1. Calculate Involute from Angle: Enter an angle in degrees (0 to 89.9°) in the first field. Click “Calculate Involute” or press Enter. The result will be the involute value (radians) and the field will update automatically.
2. Find Angle from Involute: Enter an involute value in the second field. Click “Find Angle from Involute” or press Enter. The calculator will find the corresponding angle in degrees using the Newton-Raphson method.
3. Reset: Click the “Reset” button to restore default values (20° and 0.014904).
4. Interpretation: The involute function is essential for gear design. For example, inv(20°) ≈ 0.014904 — this is a standard reference value for 20° pressure angle gears.