Cassini Oval Calculator
Free online cassini oval 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：Cassini 卵形线在线计算器
Cassini 卵形线是到两定点距离之积为常数的轨迹：(x²+y²)² − 2a²(x²−y²) = a⁴(e⁴−1)。乘积条件让它与椭圆（距离之和）分道扬镳——土星环缝的测量、双天线干涉的等响应线都用它。离心率 e 是唯一的形状旋钮：e < 1 分裂成两个卵，e = 1 恰好是伯努利双纽线（面积 2a²），e > 1 捏成一个花生形单卵。
极坐标径向为二次方程 r² = a²(cos 2θ ± √(cos²2θ + e⁴ − 1))：e < 1 时双卵不含原点，每条射线穿过环两次，面积 = 外支减内支的环差积分 A = 2a²∫√(cos²2θ−(1−e⁴))dθ；e > 1 单卵含原点，只取正根 A = (1/2)∫r²dθ。gen198 用双根极坐标与直角切片（外支减内支水平切片）两条路径交叉验证一致。
参考值（a = 1）
e = 2（单卵）：A = 12.367658048844；e = 1（双纽线）：A = 2
e = 0.8（双卵）：A = 0.682857658624；e = 0.5：A = 0.098960391097
How to use the Cassini Oval 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 Cassini Oval 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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