Involute Function Calculator

Calculate the involute function (inv α) used in gear design and mechanical engineering.

Enter an angle to find its involute function value. This is critical for designing gear teeth profiles.

Practical Examples

See how the involute function is calculated for common engineering values.

Standard Gear Pressure Angle

degrees

Calculate the involute for a standard 20° pressure angle, widely used in gearing.

α: 20

Unit: degrees

Alternative Pressure Angle

degrees

Calculate the involute for a 14.5° pressure angle, another common standard.

α: 14.5

Unit: degrees

Angle in Radians

radians

Calculate the involute directly from an angle of 0.5 radians.

α: 0.5

Unit: radians

High-Contact Ratio Angle

radians

Calculate the involute for a higher angle of 30° (π/6 radians), sometimes used in specific gear designs.

α: 30

Unit: degrees

Other Titles
Understanding the Involute Function: A Comprehensive Guide
An in-depth look at the involute function, its derivation, and its critical role in modern mechanical engineering, particularly in gear design.

What is the Involute Function?

  • Core Definition
  • The 'Unwinding String' Analogy
  • Key Mathematical Properties
The involute function, denoted as inv(α) or involute(α), is a mathematical function crucial in the field of mechanical engineering, especially for designing the profiles of gear teeth. It is defined by a simple yet powerful trigonometric formula:
inv(α) = tan(α) - α
In this formula, the angle α must be in radians. The function calculates the difference between the tangent of an angle and the angle itself, which geometrically represents the 'roll' angle of an involute curve.
The 'Unwinding String' Analogy
The easiest way to visualize an involute curve is to imagine a string tightly wound around a cylinder (the 'base circle'). If you unwind the string while keeping it taut, the path traced by the end of the string is an involute curve. The involute function helps define the shape of this curve, which is the ideal shape for gear teeth because it ensures a constant velocity ratio and smooth power transmission between meshing gears.
Key Mathematical Properties
The involute function is fundamental because it relates the pressure angle of a gear to the tooth thickness. As the pressure angle changes, the involute function value changes, directly impacting the gear's performance, strength, and contact ratio.

Real-World Examples

  • Gear tooth design to ensure constant velocity ratio
  • Path of a point on a string unwinding from a cylinder
  • Used in scroll compressors for fluid displacement
  • Cam mechanism design for specific follower motion

Step-by-Step Guide to Using the Involute Function Calculator

  • Entering the Angle
  • Selecting the Correct Unit
  • Interpreting the Results
This calculator simplifies the process of finding the involute function value. Follow these steps for an accurate calculation.
Step 1: Entering the Angle (α)
In the 'Angle (α)' input field, enter the numerical value of your angle. This is often the pressure angle of the gear system you are analyzing, but it can be any angle.
Step 2: Selecting the Correct Unit
It is critical to specify whether the angle you entered is in degrees or radians. Use the dropdown menu to select the correct unit. The calculator will automatically handle the conversion to radians, as the formula inv(α) = tan(α) - α requires α to be in radians. Failure to select the correct unit is a common source of error.
Step 3: Interpreting the Results
After clicking 'Calculate', the tool will display two values: the final 'Involute Value (inv α)' and the intermediate 'Angle in Radians' used for the calculation. The involute value is the primary result, used in further gear design equations.

Real-World Applications of the Involute Function

  • Primary Role in Gear Design
  • Applications in Cams and Splines
  • Use in CNC Machining and Inspection
The involute function is not just an abstract mathematical concept; it is a cornerstone of modern mechanical design.
Primary Role in Gear Design
The profile of most modern gear teeth is an involute curve. This shape allows for smooth, efficient power transmission even with minor variations in the distance between the gear centers. The involute function is used in formulas that determine gear tooth thickness, contact ratio, and interference points.
Common Pressure Angles:
20°: The most common pressure angle in modern gearing, offering a good balance of strength and efficiency. 14.5°: An older standard, now less common but still found in legacy systems. 25°: Used in high-load applications where greater tooth strength and rigidity are required.
Use in CNC Machining and Inspection
When manufacturing or inspecting gears, CNC machines and coordinate-measuring machines (CMMs) use the involute function to generate the precise tool paths required to cut the tooth profile or to verify that a manufactured gear meets its design specifications.

Common Misconceptions and Correct Methods

  • The Critical Degree vs. Radian Conversion
  • The Role of the Pressure Angle
  • It's Not Just a Simple Tangent
The Critical Degree vs. Radian Conversion
The most frequent mistake when calculating the involute function manually is forgetting to convert the angle to radians before applying the formula. For example, inv(20°) is not tan(20) - 20. The correct calculation is tan(20 π/180) - (20 π/180). Our calculator handles this automatically, but it's essential to understand the principle.
The Role of the Pressure Angle
While any angle can be an input, in the context of gears, the angle α is the pressure angle. This is the angle between the line of action (the line along which force is transmitted) and a line tangent to the pitch circles of the meshing gears. It is a fundamental parameter of the gear system.
It's Not Just a Simple Tangent
One might assume the angle subtraction is a minor part of the formula, but it is geometrically significant. The subtraction of α accounts for the 'unrolling' of the angle itself from the curve, creating the characteristic shape of the involute.

Mathematical Derivation and Examples

  • Derivation from Parametric Equations
  • Worked Example 1: Standard Gear
  • Worked Example 2: Custom Application
The involute of a circle with radius 'r' can be described by the parametric equations:
x(t) = r(cos(t) + t * sin(t))
y(t) = r(sin(t) - t * cos(t))
Here, 't' is the angle parameter in radians. The involute function emerges from the geometry of these curves, specifically in relation to the gear's base circle and pressure angle.
Worked Example 1: Standard 20° Pressure Angle
Let's calculate the involute for α = 20°.
Step 1: Convert degrees to radians. α_rad = 20 * (π / 180) ≈ 0.34906585 radians. Step 2: Calculate the tangent. tan(0.34906585) ≈ 0.36397023. Step 3: Apply the formula. inv(20°) = 0.36397023 - 0.34906585 ≈ 0.01490438.
Worked Example 2: Angle in Radians
Let's calculate the involute for α = 0.4 radians.
Step 1: Angle is already in radians. α_rad = 0.4. Step 2: Calculate the tangent. tan(0.4) ≈ 0.42279322. Step 3: Apply the formula. inv(0.4) = 0.42279322 - 0.4 = 0.02279322.