We consider the Keplerian distance d in the case of two elliptic orbits, i.e., the distance between one point on the first ellipse and one point on the second one, assuming they have a common focus. The absolute minimum \(d_{\textrm{min}}\) of this function, called MOID or orbit distance in the literature, is relevant to detect possible impacts between two objects following approximately these elliptic trajectories. We revisit and compare two different approaches to compute the critical points of \(d^2\), where we squared the distance d to include crossing points among the critical ones. One approach uses trigonometric polynomials, and the other uses ordinary polynomials. A new way to test the reliability of the computation of \(d_{\textrm{min}}\) is introduced, based on optimal estimates that can be found in the literature. The planar case is also discussed: in this case, we present an estimate for the maximal number of critical points of \(d^2\), together with a conjecture supported by numerical tests.

我们考虑两个椭圆轨道情况下的开普勒距离d ，即第一个椭圆上的一点和第二个椭圆上的一点之间的距离，假设它们有共同的焦点。该函数的绝对最小值\(d_{\textrm{min}}\)，在文献中称为 MOID 或轨道距离，与检测大约遵循这些椭圆轨迹的两个物体之间可能的影响有关。我们重新审视并比较两种不同的方法来计算\(d^2\)的关键点，其中我们对距离d进行平方以包括关键点之间的交叉点。一种方法使用三角多项式，另一种方法使用普通多项式。检验计算可靠性的新方法\(d_{\textrm{min}}\)是根据文献中的最佳估计引入的。还讨论了平面情况：在这种情况下，我们提出了对\(d^2\)临界点的最大数量的估计，以及数值测试支持的猜想。