Graphical and Analytical Definition of Parametric Curves in Explicit Form

Abstract

The aim of the work is to establish the relationship between graphical and analytical methods of solving problems on the example of the definition of parametric curves in explicit form. The idea is to represent the system of parametric equations of a plane curve in the space of Otxy coordinate system as a system of two projections of the curve on a complex Monge drawing. Then the desired curve can be obtained explicitly as a discrete set of points on the profile plane of projections in the Otxy coordinate system. If we consider the system of curve projections as a single curve in three-dimensional space Otxy, then this curve can be flat and spatial. For planar curves located in a partial position plane, it is easy to make the transition from parametric setting of the curve to an explicit form due to the linear relationship between one of the variables (x or y) and the parameter t. If the curve in the Otxy coordinate system belongs to the general position plane, we can use the projection plane replacement method to transform it to a projecting position. This allows a linear relationship to be established between one of the variables and a parameter to obtain the equation of the curve in explicit form, followed by an inverse transformation of the planes of projections. Mathematically, replacing the plane of projections is equivalent to replacing the coordinate system and reduces it to a rotation around the origin of the coordinates. Formulas for transforming the coordinate system, taking into account the peculiarities of using Monge's complex drawing and for determining the angle of rotation of the coordinate system are given.