Analytic Parametric Multi-Step Solution of All Area and Moments Integrals of General Green’s Theorem for Arbitrary Ellipse Region, Part 3: Focal Sector

In this, Part 3 of the paper, analytic solution of all six integrals for determination of the area and moments of arbitrary focal sector of ellipse is given, for ellipse defined in parametrical form by trigonometric functions, based on its corresponding integrals of its central sector from Part 1 of the paper, similar to Part 2 for arbitrary segment. Those integrals are here again built from the general moment form of Green’s theorem curve integral for the calculation of the area and moments of a general planar shape enclosed by arbitrary curve, and then applied for the ellipse focal sector calculations. In order to enable final posterior calculation for given focal polar angle, a relation between the circular angle parameter of a Cartesian, parametric representation of ellipse and a focal polar angle is given, similar to that relation given for the central angle value from Part 1 of the paper. In this way, it is possible to analytically calculate all the values of the arbitrary focal sector of ellipse, such as: area itself, its static moments, area centroids and moments of inertia of the focal ellipse area, defined parametrically, by trigonometric functions. This solution can be then applied in various fields of science, with the special focus on the second Kepler’s law in the field of orbital mechanics.