Tractrix Curve
Drag a point with a fixed-length rope, explore the tangent triangle and compare coordinates, scaling and arc length along a tractrix.
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A straight pull, a curved response
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.
Coordinates and the constant-length constraint
Use a horizontal guide y = 0, start P at (0, a), and write Q = (at, 0). Here t ≥ 0 measures the puller’s position in rope-length units. The dragged point has coordinates x = a(t − tanh t), y = a / cosh t. Its horizontal lag behind Q is a tanh t.
The squared rope length is a²(tanh² t + sech² t) = a². The dashed triangle in the module shows those horizontal and vertical components; its hypotenuse is the amber rope. This identity explains why a changing angle does not require a changing rope length.
Why the rope is tangent
Differentiating P gives P′(t) = (a tanh² t, −a sech t tanh t). For t > 0 this is tanh t times Q − P, so the point’s instantaneous direction is parallel to the rope. The slope is −1 / sinh t. At t = 0 the limiting tangent is vertical, even though this particular parametrization has P′(0) = 0.
The speed with respect to the position parameter is |P′(t)| = a tanh t. Integrating from 0 to t gives the travelled arc s = a ln(cosh t). This is the cyan path length, not the distance travelled by Q, which is at. The distinction matters when comparing the two readouts.
Worked example: a two-unit rope
Choose Rope length a = 2 and Pull progress t = 1. Compute the coordinates first, then check the rope from its components. All displayed values are rounded to six decimal places; the geometric identities refer to the unrounded quantities.
- tanh(1) ≈ 0.761594156 and sech(1) ≈ 0.648054274.
- Q = (2, 0), while P ≈ (0.476812, 1.296109).
- The horizontal lag is 2 − 0.476812 ≈ 1.523188. Thus √(1.523188² + 1.296109²) ≈ 2, matching the measured rope.
- The travelled arc is 2 ln(cosh 1) ≈ 0.867562. It is shorter than the puller’s two-unit displacement.
At t = 2 with the same rope, P ≈ (2.071945, 0.531604), Q = (4, 0), and the travelled arc is about 2.650005. P gets closer to the guide while the measured rope remains 2. Doubling a at any fixed t doubles every coordinate and distance, including the arc.
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.
What this model leaves out
This is an analytic geometric dragging model, not a simulation of mass, friction, rope elasticity or slack. The playback parameter t describes position rather than elapsed physical time; its animation timing should not be interpreted as the motion of a real object under specified forces.
The controls show only 0 ≤ t ≤ 3 and 0.5 ≤ a ≤ 4. For every finite t the height a / cosh t is positive; reaching the right edge of playback does not mean reaching the guide. The guide is an asymptote of the ideal infinite curve. The rendered paths use short line segments, while the readouts use analytic formulas and floating-point arithmetic. Six-place rounding can make component calculations differ slightly from the displayed rope value.
References and coordinate conventions
These educational references describe the fixed tangent-segment construction and its parametrization. Some drawings exchange the horizontal and vertical coordinates. This module puts the straight guide on y = 0, so read each reference’s axes before comparing its picture with the canvas.
- Wolfram MathWorld: Tractrix — construction, hyperbolic coordinates and arc length.
- University of St Andrews, MacTutor: Tractrix — the dragging problem, asymptote and a parametrization with the axes exchanged.
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Tractrix Curve module.
Try Tractrix Curve on Riano →