Analytic Parametric Multi-Step Solution of All Area and Moments Integrals of General Green’s Theorem for Arbitrary Ellipse Region, Part 1: Central Sector

In this paper, all the integrals and their solutions are given for the analytical calculation of all six area and moments values of the arbitrary ellipse region given in trigonometric parametric form, based on the general moment form of Green’s theorem curve integral obtained from the discrete and differential vector product methods. The actual area and moments values of the arbitrary ellipse regions are then calculated by application of Boolean algebra on the ellipse parts and their remaining sector triangles, in the multi-step calculation procedure that consists of the generation of belonging ellipse integrals and their simple solution for unknown parameter of a standard Cartesian, parametric representation of ellipse, whose bounds are calculated a posterior, by substitution of ellipse parameter with known central polar angle values. In order to enable the final posterior substitution, a single relation between that parameter and central polar angle is established, based on equaling their scaled tangent function values, thus confirming geometric meaning of that parameter as an angle of auxiliary circles in the known La Hire’s ellipse construction method. In this way, it is possible to analytically calculate the area and moments of an arbitrary central or focal sector of ellipse, as well as their belonging arbitrary segments, with integrals and solutions for: areas, their static moments, area centroids and moments of inertia of ellipse parts, for ellipse defined parametrically, by trigonometric functions. And here, in Part 1 of the paper, the results of calculation are shown for the general central ellipse sector.