Involute Function Calculator

Compute the involute function inv(α) = tan(α) - α used in gear tooth design and engineering.

Enter Pressure Angle

Result

inv(α)
0.014904
radians
Angle (degrees) 20
Angle (radians) 0.349066
tan(α) 0.36397
inv(α) 0.014904

Step-by-Step Solution

inv(α) = tan(α) - α

Involute Function Table

Understanding the Involute Function

The involute function, written as inv(α) = tan(α) - α, is a fundamental function in gear engineering. It describes the involute curve of a circle, which is the basis for the tooth profile of most modern gears. The angle α is typically the pressure angle of the gear system, measured in radians for calculation purposes.

The involute of a circle is the curve traced by the end of a taut string as it unwinds from the circle. This curve has the unique property that the contact between two mating gear teeth always occurs along a straight line (the line of action), ensuring smooth and constant angular velocity transmission.

Key Concepts in Gear Design

Involute Function

The core formula relating the pressure angle to the involute curve geometry.

inv(α) = tan(α) - α

Standard Pressure Angles

Common pressure angles used in gear design: 14.5, 20, and 25 degrees.

Standard: 20 deg (most common)

Base Circle

The circle from which the involute curve is generated. Its radius defines the gear tooth shape.

r_base = r_pitch * cos(α)

Tooth Thickness

The involute function is used to calculate tooth thickness at various diameters.

t = d * (t_p/d_p + inv(α_p) - inv(α))

Applications of the Involute Function

  • Gear tooth profile design: Generating the exact shape of involute gear teeth.
  • Tooth thickness calculation: Finding tooth thickness at any given diameter.
  • Gear inspection: Measuring and verifying gear tooth geometry.
  • Profile shift calculations: Determining the effect of profile shifts on tooth shape.
  • Backlash analysis: Computing gear backlash in meshing gear pairs.

Why Involute Gears Are Preferred

Involute gear teeth are the industry standard because they maintain a constant velocity ratio even when the center distance between gears varies slightly. They are also easier to manufacture than other tooth profiles, as they can be cut with simple straight-sided cutting tools (rack cutters). The contact between involute teeth always occurs along a straight line, which produces smooth, efficient power transmission.

Converting Between Degrees and Radians

Since the involute function requires the angle in radians, it is important to convert properly: radians = degrees * π / 180. The output inv(α) is always in radians. Common pressure angles are 14.5 degrees (0.25307 rad), 20 degrees (0.34907 rad), and 25 degrees (0.43633 rad).

How to Use This Calculator

  1. Select whether your input is in degrees or radians.
  2. Enter the pressure angle value.
  3. Optionally set the number of table points to generate a reference table.
  4. Click "Calculate Involute" to see the result, step-by-step solution, and reference table.