Lissajous Calculator
Free online lissajous 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.
Lissajous 曲线 x = A sin(at+δ)、y = B sin(bt) 是两个垂直简谐振动的合成——示波器上比较两个信号频率时看到的花纹就是它。频率比 a:b 决定花纹的瓣数结构，相位差 δ 决定开口方向；当 δ/2 的正切为有理数时曲线闭合。
周长没有初等公式，需对速度模 |v| = √(A²a²cos²(at+δ)+B²b²cos²bt) 数值积分。特例 1:1 且 δ = π/2 退化为圆（周长 2π ≈ 6.283185307179586）；1:1 且 δ = 0 退化为线段；其余情形周长随频率比与相位连续变化。
参考值（A = B = 1、δ = π/2）
3:2：周长 = 15.289373927751；2:1：周长 = 9.293567524866；1:1：2π = 6.283185307179586
How to use the Lissajous 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 Lissajous 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 2026-09-29
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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