Cycloid / Trochoid

Set the wheel radius r and pen offset d, then select Display. The plot contains the complete path for 0 ≤ θ ≤ 2π and starts a newly parameterized curve at θ = 0, paused. Move the angle slider or enter an exact angle in radians. Play advances at 1.2 rad/s, Pause stops at the current angle, and Repeat starts again at zero after the end. The playback is an illustration, not a mechanical dynamics calculation.

The wheel rolls to the right without slipping along y = 0. Its center is C = (rθ, r), the instantaneous contact is K = (rθ, 0), and the pen is P = (rθ − d sin θ, r − d cos θ). The line from C to K has length r; the line from C to P has length d. The additional rim marker uses d = r and shows the wheel’s rotation even when the pen is at the center. The contact location is not one fixed material point on the wheel.

With d = r the pen traces an ordinary cycloid with cusps. For 0 < d < r the curve is a curtate trochoid; for d > r it is prolate and has loops. With d = 0 the pen moves along a straight line at height r. The four pen-position presets keep the valid radius and set d/r to 1, ½, 1½ or 0. Manual parameter edits select Custom.

One revolution is not a closed plane curve. After θ = 2π the wheel has advanced by 2πr and the pen has the same height as at the start, translated to the right. The full path and the entire wheel are included in the fixed plot bounds. The same scale is used horizontally and vertically. If d is much larger than r, the correctly scaled wheel can look very small. The reference y = 0 is not an impenetrable wall, so an extended pen can pass below it.

The pen is filled pink, the center is a small contrasting core, the contact has an orange ring, and the rotating rim marker has a green ring. At certain angles several points coincide; the concentric markers and their coordinates describe the same positions. The green dashed radius joins the center to the contact, while the pen arm reaches the pink point. The moving probe is evaluated at its actual angle rather than rounded to those path samples.

Use finite decimal values, with a decimal point or comma: 0.05 ≤ r ≤ 5 and 0 ≤ d ≤ 8 in a common freely chosen length unit. The angle must lie between 0 and 2π inclusive. The slider has 1000 intervals; the numeric angle remains continuous. Invalid angles hide the moving geometry and its coordinates until corrected. Parameter edits clear the result, stop playback and reset the next curve’s angle to zero. Reset clears the result while keeping the inputs, including the angle.

Playback stops when the page or tool is hidden and does not catch up after a pause. Valid settings can be stored locally. Returning within the app can preserve the existing curve and angle, paused; a page reload restores the inputs and waits for Display. There is no zoom or wheel-scroll override.

Sources: Eric W. Weisstein, MathWorld: Trochoid, for the parametrization and classification; OpenStax Calculus Volume 2, 7.1, for the rolling center, clockwise rotation and cycloid derivation.