About a practical method of determining the length of an ellipse

Abstract

The article presents a practical highly accurate way to determine the length of an ellipse, which does not require the use of tables for elliptical integrals of the second kind in the Legendre form. On the basis of the postulate accepted by the authors that the ellipse is a projection of a circle, a new universal formula for calculating the length of the ellipse is derived. Since the ellipse is a projection of a circle inclined at an arbitrary angle to the plane, the equation of the length of the ellipse is analytically and functionally related to the equation of the length of the circle. The line of projection of a circle is a “double” and continuous Jordan curve, and the ratio of the lengths of the axes of the ellipse (projection of the circle) decreases from 1 to 0. If the small axis is zero, the length of the “double” line of projection is twice the diameter of the circle. Increasing the angle of inclination, the “double” line of projection of the circle bifurcates, forming an ellipse, the ratio of the axes of which increases from zero, at α = 90°, to one, at an angle of α = 180°. Thus, the main parameter of the proposed calculation is the ratio of the lengths of the semiaxes of the ellipse. This allowed us to derive a nonlinear equation for calculating the length of the ellipse with high accuracy over the entire range of changes in the ratio of its axes.