Conics are relevant in many undergraduate course on classical physics, from free fall to Rutherford scattering. A pedagogical derivation of the intrinsic expressions of the curvature radius of different types of conic sections is presented. Our proof is carried out without resorting to any coordinate systems, but rather on using only elementary kinematic concepts together with basics of vector calculus and the very definition of conics. As a byproduct application of the present analysis, a simple and compact deduction of the Newton ‘inverse square law’ for gravitation from the three Kepler laws is also presented.

中文翻译：

---

圆锥截面的曲率半径：运动学推导

从自由落体到卢瑟福散射，圆锥曲线与经典物理学的许多本科课程都相关。介绍了不同类型圆锥截面曲率半径的内在表达式的教学推导。我们的证明是在不借助任何坐标系的情况下进行的，而是仅使用基本运动学概念以及矢量微积分的基础知识和圆锥曲线的定义。作为本分析的副产品应用，还介绍了从三个开普勒定律对牛顿“平方反比定律”进行的简单而紧凑的推论。