I am programming as a hobby and trying something with elliptical orbits of satellites.\nI need to understand how to measure the length of an arc on an ellipse, e.g. from 0 to pi/3.\nNow I've come upon Elliptic Integrals and an associated brick wall. \n\nFor the "Complete Elliptic Integral of the second kind" the length of an arc from 0 to pi/2, I've found a series the program can use to approximate the value of this kind of arc.\n\nHowever, for the INCOMPLETE sort, there is nowhere a description of how the value is calculated, no series, no formula. Nothing.\n\nThey tell you in most places either :\n1. Look it up\nor\n2. use a website to calculate it, such as [URL]https://keisan.casio.com/[/URL]\n\nThe casio.com website can indeed calculate the value where k = 0.6 and the limits are 0 to pi/3 bringing the value:\n0.98922159...... so there must be an infinite series one utilizes to calculate this.\n\nIt is most certainly not an elementary function. So, what is the series?