Cycloid and trochoid

Set wheel radius r and pen offset d, or choose a pen position: the complete path updates directly. The angle and synchronized coordinates are in Settings. Play advances at 1.2 rad/s, Pause stops at the current angle, and Repeat starts at zero after one revolution. Playback illustrates geometry, not mechanical dynamics. PNG exports the current view.

The wheel rolls to the right without slipping along y=0. Its center is C=(rθ,r), contact K=(rθ,0), and pen P=(rθ−d sinθ,r−d cosθ). Center-to-contact length is r; the pen arm is d. A rotating rim marker remains visible for d=0. The instantaneous contact is not one fixed material point. d=r produces cusps, 0<d<r a curtate trochoid, d>r loops, and d=0 a line. After one turn the pen is translated by 2πr; this is not a closed plane curve.

Equal x/y scales and initial bounds include the full path and wheel. Drag pans, Ctrl/Meta-wheel or two-finger pinch zooms; ordinary wheel scrolls the page. Fit resets the view. A correctly scaled small wheel can look tiny when d is large. The ground line is a reference, not a physical wall. Pink is the pen, orange the contact, green the rotating rim, and the small contrasting dot the center. Coordinates distinguish overlapping markers.

Finite decimal point/comma inputs: 0.05≤r≤5, 0≤d≤8, angle 0…2π radians. Units are freely chosen lengths. The path has 720 segments; probes use the actual angle. Invalid angles hide moving geometry and coordinates until corrected; invalid dimensions clear the old curve. Only raw dimensions, selected preset and manually entered/scrubbed angle persist, including invalid strings. Animation and camera are not saved; loading recomputes from inputs, and hiding pauses playback. Sources: MathWorld: Trochoid · OpenStax Calculus 2, 7.1.