Pursuit curves

Choose a model, target path, speed and duration. Changes update the paths directly. Blue is the target; red, green and violet are pursuers. Pale paths show the full route, strong paths the current progress. Play/Pause or the time slider inspect synchronized positions. Settings shows target coordinates and individual target distances; Formula lets you enter x(t) and y(t), with radians, pi, sin, cos and other mathematical functions. PNG exports the current drawing.

Pursuers move at the chosen constant speed toward the current target. RK4 uses a 0.02 s output grid and smaller steps near contact. Capture is approximated at a distance of 0.000001 coordinate units; captured pursuers stay with the target while remaining pursuers continue. The run ends when all have captured it. The circle has radius 0.55 and angular speed 1 rad/s; the straight target moves at 0.2, the figure eight has varying speed. These are geometric coordinates, not guaranteed SI lengths, with no acceleration or obstacles.

Tractrix is an exact separate model: a straight-moving endpoint pulls a point with a taut unit tether. Speed controls the pulling endpoint; q=v*t gives the shifted coordinates (q−tanh(q), −sech(q)). The dashed line is the tether. Target path and formula controls are hidden.

Speed 0.05–3 and duration 0.1–30 s, rounded to the 0.02 s output grid. Playback takes eight real seconds regardless of simulated duration and pauses when hidden. Drag pans; Ctrl/Meta-wheel zooms the view, ordinary wheel scrolls the page; Fit resets the view. Only raw inputs are retained, never the computed paths, playback or camera. Formulas are limited to 512 characters, coordinates to ±1000000, and close-approach integration to 512 substeps per output interval. Rapid or unresolved targets produce an error; no universal integration or capture guarantee. Tractrix · Pursuit curves.