General Analysis of the Shape of Two Similar Second-Order Surfaces’ Intersection Line

Abstract

The presented paper is devoted to classification questions of fourth-order spatial curves, obtained as a result of intersection of non-degenerate second-order surfaces (quadrics) from the point of view of the forms of the original quadrics generating this curve. At the beginning of the paper is performed a brief historical overview of appearance of well-known and widely used curves ranging from ancient times and ending with the current state in the theory of curves and surfaces. Then a general analysis of the influence of the shape parameters and the relative position of original surfaces on the shape of the resulting curve and some of its parameters (number of components, presence of singular points, curve components flatness or spatiality) is carried out. Curves obtained as a result of intersection of equitype surfaces are described in more detail. The concept of interacting surfaces is introduced, various possible cases of the forms of the quadrics generating the curve are analyzed. A classification of fourth-order curves based on the shape parameters and relative position of second-order surfaces is proposed as an option. Illustrations of the resulting curve shapes with different shape parameters and location of generating quadrics are given. All surfaces and curves are considered in real affine space, taking into account the possibility of constructing them using descriptive geometry methods. Possible further research directions related to the analysis of the curves under discussion are briefly considered. In addition, are expressed hypotheses related to these curves use in the process of studying by students of technical universities the courses in analytical geometry, descriptive geometry, differential geometry and computer graphics. The main attention is paid to forms, therefore a wide variability of the surface shape in the framework of its described equation has been shown, provided by various values of numerical parameters.