Zernike Polynomial Calculator
Free online zernike polynomial 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：Zernike 多项式在线计算器
Zernike 径向多项式 R_n^m(ρ) 是单位圆盘上的正交多项式基，与角向 e^(imθ) 一起构成光学像差的标准语言——眼镜验光里的离焦、散光、彗差，天文自适应光学的 Zernike 模式分解，都建立在它上面。级数定义 R = Σ(−1)^k(n−k)!/[k!((n+m)/2−k)!((n−m)/2−k)!]·ρ^(n−2k)，k 从 0 到 (n−m)/2。
两条铁律：边缘归一 R(1) = 1，正交范数 ∫₀¹R_n^m²ρdρ = 1/(2(n+1))。低阶像差对照：m = 0 是离焦与球差（R₄⁰ = 6ρ⁴−6ρ²+1），m = 2 是散光（R₄² = 4ρ⁴−3ρ²），m = 1 是彗差方向项（R₃¹ = 3ρ³−2ρ）。
参考值
R₄²(0.5) = −0.5、R₃¹(0.5) = −0.625、R₆²(0.5) = 0.484375、R₅³(0.8) = −0.4096
范数验证：∫₀¹R₃¹²ρdρ = 1/(2×4) = 0.125（gen198 与 GL16 数值积分一致到 1e−16）
How to use the Zernike Polynomial 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 Zernike Polynomial 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.
Related calculators
More in this category: Math calculators
About math calculators
Free online math calculators for algebra, statistics, number theory and everyday arithmetic — from quadratic equations and matrices to percentages, fractions and prime factorization, each with step-by-step formulas.