Analytical Determination of Arc Length of an Elliptic Curve

Abstract

Elliptic curves are commonly used to outline various geophysical objects, natural and artificial bodies, trajectories of the planets, satellites, elementary particles. Utilizing these curves, many theoretical and applied problems in the field of mechanics, geophysics, and mathematical physics can be solved. However, the dependences for finding the length of individual sections (arcs) of the ellipse that are in demand for engineering practice have not been developed up to the present day. The article deals with analytical dependences for determining the length of the arc and entire ellipse that are obtained on the basis of the representation of incomplete and complete elliptic integrals of the 2^nd kind via elementary functions. These dependences coincide with the known exact formulas for the boundary counters of an ellipse, namely, a circle and a horizontal segment of a straight line (thin plate). Graphs of the dependence of the ellipse arc length on the central angle and the ratio of the semiaxes are also given. Comparison of the obtained results of solving the test problem by using the proposed method with the reference values found from numerical calculation using the “Mathematic” software has showed a good agreement (≤ 0.5%).