Tractrix Calculator
Free online tractrix calculator. Enter your values and get instant results with the formula, step-by-step working, a worked example and FAQs — everything runs locally in your browser on Calcton, no signup required.
中文版：曳物线在线计算器
曳物线（tractrix）来自一个可以亲手实验的定义：把长度为 a 的绳子一头拴在狗上、一头拖着一块表沿直线走，表划出的轨迹就是它——所以德语也叫「拽狗曲线」。参数方程 x = a(t − tanh t)、y = a·sech t。
它的标志性性质：曲线上任意一点的切线段到 x 轴的长度恒等于 a。由此衍生出伪球面（把曳物线绕渐近轴旋转）：高斯曲率处处为 −1/a²，是罗巴切夫斯基双曲几何的物理模型，曲率半径处处相同却处处向内弯。
参考值
a = 1、t = 2：x = 1.0359724199242，y = 0.26580222883408
弧长 = ln cosh 2 = 1.3250027473579（与数值积分 ∫tanh t dt 交叉一致）
切线长 = 1 = a（定义性质的数值验证）
How to use the Tractrix Calculator
- Fill in the inputs with your own numbers — every field shows what it expects.
- The result updates instantly as you type; no "Calculate" click needed.
- Read the result card for the main figure, the step-by-step formula, and practical notes.
Frequently asked questions
Is the Tractrix Calculator free?
Yes — all math calculators on Calcton are free, with no sign-up required. Your inputs are processed in your browser and never sent to a server.
Is there a Chinese version of this tool?
Yes. Use the Chinese version link above — it is the same calculator with the interface in Simplified Chinese.
References
Formulas and benchmarks used by this calculator follow these published sources.
KevinContent LeadCalcton Editorial Team · Last updated 2026-09-28
Kevin maintains Calcton's formula verification and content: every calculator cites the standard it implements, worked examples are recomputed programmatically, and health benchmarks follow WHO and international guidelines.
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