Construction of both Geometric Relationships of Ellipses and Parabola-bounded Regions in Geometric Placement Problems

Abstract

Currently, there is a significant growth of interest in the practical problems of mathematically modeling the placement of geometric objects of various physical natures in given areas. When solving such problems, there is a need to build their mathematical models, which are implemented through the construction of analytical conditions for the relations of the objects being placed and placement regions. The problem of constructing conditions for the mutual non-intersection of arbitrarily oriented objects whose boundaries are formed by second-order curves is widely used in practice and, at the same time, much less studied than a similar problem for simpler objects. A fruitful and worked out method of representing such conditions is the construction of Stoyan’s Φ-functions (further referred to as phi-functions) and quasi-phi-functions. In this article, considered as geometric objects are an ellipse and a parabola-bounded region. The boundaries of the objects under study allow both implicit and parametric representations. The proposed approach to modeling the geometric relationships of ellipses and parabola-bounded regions is based on coordinate transformation, reduction of an ellipse equation to a circle equation with the use of a canonical transformation. In particular, constructed are the conditions for the inclusion of an ellipse in a parabola-bounded region, as well as the conditions for their mutual non-intersection. The conditions for the relationships between the geometric objects under study are constructed on the basis of the canonical equations of the ellipse and parabola, taking into account their placement parameters, including rotations. These conditions are presented in the form of a system of inequalities, as well as in the form of a single analytical expression. The presented conditions can be used in constructing adequate mathematical models of optimization problems of placing corresponding geometric objects for an analytical description of feasible regions. These models can be used further in the formulation of mathematical models of packing and cutting problems, expanding the range of objects and / or increasing solution accuracy and decreasing time to solution.

Keywords

• ellipse
• parabola
• non-intersection
• inclusion
• phi-function