Trochoid — Definition, Formula & Examples

A trochoid is the curve traced by a point attached to (but not necessarily on the rim of) a circle as that circle rolls along a straight line. Depending on whether the point is on the rim, inside, or outside the circle, the trochoid takes different shapes.

Given a circle of radius $r$ rolling without slipping along a straight line, a trochoid is the locus of a point at distance $d$ from the center of that circle. When $d = r$ , the curve is a cycloid; when $d < r$ , it is a curtate (contracted) trochoid; when $d > r$ , it is a prolate (extended) trochoid.

Key Formula

x = r\theta - d\sin\theta, \quad y = r - d\cos\theta

Where:

$r$ = Radius of the rolling circle
$d$ = Distance from the center of the circle to the tracing point
$\theta$ = Angle (in radians) through which the circle has rotated

How It Works

Imagine a wheel of radius

r

rolling along the

x

-axis. Attach a pen at distance

d

from the wheel's center. As the wheel rolls, the pen traces a trochoid. When

d = r

, the pen sits on the rim and draws a cycloid with sharp cusps touching the line. When

d < r

, the pen is inside the wheel, producing smooth undulating waves that never touch the line. When

d > r

, the pen extends beyond the rim, creating loops that cross themselves.

Worked Example

Problem: A circle of radius 3 rolls along the x-axis. A point is fixed at distance 2 from the center. Find the coordinates of the tracing point when the circle has rotated through θ = π/2.

Identify parameters: Here r = 3 and d = 2. Since d < r, this is a curtate trochoid.

Compute x: Substitute into the x-equation.

x = 3\cdot\frac{\pi}{2} - 2\sin\!\left(\frac{\pi}{2}\right) = \frac{3\pi}{2} - 2

Compute y: Substitute into the y-equation.

y = 3 - 2\cos\!\left(\frac{\pi}{2}\right) = 3 - 0 = 3

Answer: The point is at

\left(\dfrac{3\pi}{2} - 2,\; 3\right) \approx (2.71,\; 3)

Why It Matters

Trochoids model real-world rolling motion, from the path of a valve on a bicycle tire to the tooth profiles in gear design. In calculus, they provide rich exercises in parametric differentiation, arc length, and area computation.

Common Mistakes

Mistake: Assuming every trochoid has cusps or loops.

Correction: Cusps appear only when d = r (cycloid). A curtate trochoid (d < r) has smooth waves, while a prolate trochoid (d > r) has self-intersecting loops. Always compare d to r before sketching.