Lemniscate Constant Calculator
Free online lemniscate constant 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：双纽线常数在线计算器
双纽线常数 ϖ（读作 varpi，形似 π 的手写体）是 Bernoulli 双纽线 r² = cos 2θ 一瓣弧长的 2 倍，等于 π/AGM(1, √2)。Gauss 在 1799 年 3 月 30 日的日记里写道：AGM(1, √2) 与 π 的关系「必将开辟分析学全新的领域」——这确实是椭圆函数论的起点，比 Abel 与 Jacobi 早了将近三十年。
页面同时给出三个口径：ϖ = π/AGM(1, √2) = 2.6220575542921198、Gauss 常数 G = 1/AGM(1, √2) = 0.8346268416740731、以及 Γ(1/4)²/(4√(2π)) 的 β 积分交叉验证（两条路径一致到 1e-12）。部分弧长积分 ∫ du/√(1−u⁴) 在 t = 1 处端点奇异，直接 Simpson 会除零，因此逐点积分只在 t 小于 1 时给出。
参考值
ϖ = 2.6220575542921198，G = 0.8346268416740731
Γ(1/4) = 3.6256099082219083119，Γ(3/4) = 1.2254167024651776451
ϖ/2 = Γ(1/4)√π/(4Γ(3/4))：AGM 与 Γ 函数在双纽线处会师
How to use the Lemniscate Constant 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 Lemniscate Constant 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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