Involute — Definition, Formula & Examples

An involute is the curve traced by the end of a taut string as it is unwound from another curve. Think of pulling a thread off a spool while keeping it tight — the path the free end traces is the involute of that spool's shape.

Given a curve $C$ (called the base curve) with arc-length parameterization $\mathbf{r}(s)$ , the involute of $C$ is the locus of points $\mathbf{r}(s) - s\,\mathbf{T}(s)$ , where $\mathbf{T}(s)$ is the unit tangent vector to $C$ at $s$ and $s$ is measured from a fixed starting point on $C$ .

Key Formula

\mathbf{P}(t) = \mathbf{r}(t) - s(t)\,\mathbf{T}(t)

Where:

$\mathbf{r}(t)$ = Position vector of a point on the base curve
$s(t)$ = Arc length along the base curve from the starting point to parameter t
$\mathbf{T}(t)$ = Unit tangent vector to the base curve at parameter t

How It Works

To construct an involute, pick a starting point on the base curve and imagine attaching a string along the curve. As you peel the string away while keeping it taut, the free endpoint sweeps out the involute. At each position, the unwound segment of string is tangent to the base curve and has length equal to the arc length from the starting point. This means the involute is always perpendicular to the tangent of the base curve at the point of departure.

Worked Example

Problem: Find the parametric equations for the involute of a circle of radius

a

centered at the origin.

Parameterize the circle: The circle has position vector

\mathbf{r}(t) = (a\cos t,\, a\sin t)

. The unit tangent vector is

\mathbf{T}(t) = (-\sin t,\, \cos t)

\mathbf{r}(t) = (a\cos t,\; a\sin t)

Compute the arc length: Starting from

t = 0

, the arc length is

s(t) = at

s(t) = \int_0^t a\, d\tau = at

Apply the involute formula: Substitute into

\mathbf{P}(t) = \mathbf{r}(t) - s(t)\,\mathbf{T}(t)

x(t) = a\cos t + at\sin t, \quad y(t) = a\sin t - at\cos t

Answer: The involute of a circle of radius

a

is given by

x = a(\cos t + t\sin t)

and

y = a(\sin t - t\cos t)

for

t \geq 0

Why It Matters

Involutes of circles are the standard tooth profile used in spur gears, ensuring constant angular velocity between meshing gears. In calculus courses, they provide a concrete application of arc-length parameterization and the Frenet–Serret formulas. Understanding involutes also deepens your grasp of curvature, since the involute of a curve has its evolute as the original curve.

Common Mistakes

Mistake: Confusing the involute with the evolute.

Correction: The evolute is the locus of centers of curvature of a curve; the involute is the curve traced by unwinding a string from it. They are inverse operations: the evolute of an involute returns the original base curve.