The Most Important Curve in Mechanical Engineering
Nearly every gear manufactured today uses the involute curve for its tooth profile. This mathematical curve, first proposed for gears by Leonhard Euler in 1765, has properties that make it uniquely suited for gear teeth.
What Is an Involute?
Imagine tying a string around a cylinder (the base circle) and pulling it taut. As you unwrap the string, the tip traces a curve — that curve is the involute. Mathematically:
- x = rb × (cos θ + θ × sin θ)
- y = rb × (sin θ − θ × cos θ)
Where rb is the base circle radius and θ is the roll angle.
Why Involute?
The involute profile has three critical properties:
- Constant velocity ratio: Two involute gears maintain a perfectly constant angular velocity ratio throughout mesh — essential for smooth power transmission.
- Center distance tolerance: Small changes in the center distance between two gears don't affect the velocity ratio. This is invaluable in manufacturing where perfect positioning is impossible.
- Manufacturability: Involute profiles can be cut with simple straight-sided rack cutters (hobbing), making mass production economical.
The Base Circle
The base circle is the circle from which the involute is generated. Its diameter is:
db = d × cos(α)
Where d is the pitch diameter and α is the pressure angle. The involute curve only exists outside the base circle — below it, a different root fillet profile connects the teeth.
Pressure Angle's Role
The pressure angle determines the base circle size relative to the pitch circle:
- 14.5°: Larger base circle, thinner tooth base, less undercutting risk, but lower load capacity
- 20°: Standard — good balance of strength and smoothness
- 25°: Smaller base circle, thicker tooth base, higher load capacity, but more radial force
Profile Shift
When a gear has very few teeth, the cutting tool can remove material from the tooth base — a problem called undercutting. Profile shift moves the cutter outward by a factor x × m, adding material to the tooth root. This allows gears with fewer teeth to be manufactured without undercutting.
Our Generator
GearForge's spur gear generator uses proper involute mathematics to create accurate tooth profiles. You can visualize the base circle, adjust the pressure angle, and apply profile shift — all in real-time.