Tractrix Curve
Drag a point with a fixed-length rope, explore the tangent triangle and compare coordinates, scaling and arc length along a tractrix.
Free theory articleDrag a point with a fixed-length rope, explore the tangent triangle and compare coordinates, scaling and arc length along a tractrix.
Concept & interpretation
A tractrix models a point P dragged by an ideal taut rope of fixed length a. The other end Q moves along a straight guide. The rope points along P’s tangent, so P continually changes direction as it follows Q. The Wolfram MathWorld description gives this geometric construction and its hyperbolic parametrization.
Try the dragging experiment
Open the interactive Tractrix Curve module and compare the same numerical readouts as you change the controls. The canvas fits the selected rope length with equal horizontal and vertical scales, so use the coordinate values to judge changes in absolute size.
- Set Pull progress t to 0 with Rope length a at 2. Observe a vertical rope, zero horizontal lag and zero travelled arc.
- Move Pull progress t to 1, then 2. Compare Point P: height, Puller Q: x and Measured rope. The first decreases, the second increases and the rope stays at 2.
- Toggle Show tangent triangle to compare the two components with the rope. Toggle Show full curve to hide the future reference path without removing the visited cyan path.
- At t = 1, set Rope length a to 1 and then 2. Check that Point P: x, height, Horizontal lag and Travelled arc all double. The automatic framing keeps the shape visible.
- Press Play to advance along the displayed interval. Either slider pauses playback. At t = 3 playback stops; use Reset experiment to restore t = 1, a = 2 and both overlays. Reset view only restores the canvas position after panning.