Analytical method solving system of hyperbolic equations

Abstract

A hyperbola is defined by difference of distances to foci, in which its absolute value is a constant. Solutions of a system of hyperbolic equations (SoHE) can represent for intersection points of two hyperbolas given by four individual points in xy-plane. In this study, analytical method solving SoHE is aimed to find intersection points of two hyperbolas in the general in xy-plane. The demonstrated method is based on two algorithms for two cases, in which the two hyperbolas are/are not perpendicular to each other. According to analytical algorithms solving quadratic and quartic equation in general, the results of analytical method solving SoHE are shown like explicit solutions. These results are requisite for further development in finding intersection points of two hyperbolas in 3-D space in general and finally used in estimating target position using TDOA.