Ruled algebraic surfaces with a main frame from three superellipses

Abstract

An opportunity of conversion of algebraic surfaces with a main frame from three superellipses of general type into ruled surfaces of several views is shown. It is necessary to take one, two, or all of three superellipses in the form of a rhombus, i.e. it is necessary to assume exponents in explicit algebraic equations of suitable superellipses equal to one. It was illustrated that having taken one and the same main frame from three plane curves lying in the main coordinate planes, one can construct three algebraic surfaces of different orders. So, it is possible to introduce into practice great number of ruled surfaces with the preliminary given main frame from three superellipses. Some of them must be in the form of straight lines. As a result, fifteen shapes, i.e. five threes of ruled algebraic surfaces with a main frame from three superellipses were obtained with the help of three explicit equations or with the help of three systems of parametric equations. These surfaces contain a polyhedron on given rhombus plane, some types of cylindroids and conoids, and ruled surfaces not described in scientific literature before. All surfaces were visualized for concrete examples. Earlier, Professor A.V. Korotich introduced into practice a new group of surfaces which he called “Ruled quasipolyhedrons from conoids.” Some of the ruled algebraic surfaces presented in this paper can be put in this group of ruled quasipolyhedrons.