ANALYSIS OF THE PROBLEM OF CONSTRUCTING AN ELIPSE FROM TWO TANGENTS

Abstract

This article discusses the problem of constructing an ellipse by two given tangents and the coordinates of the center of the ellipse, which arises in the development of CAD design systems. When analyzing the sufficiency of the completeness of the initial data, a symmetrical display of the given lines relative to the center of the ellipse is performed, which leads to a parallelogram with an ellipse inscribed in it. The subsequent affine transformation maps the parallelogram into a square with an ellipse inscribed in it. The analysis of the obtained results allows us to conclude that the initial data are incomplete and it is necessary to supplement the initial data with the coordinate of the point of contact of the ellipse with one of the straight lines. When constructing an ellipse from the supplemented initial data, using direct and inverse affine transformations, the points through which the ellipse must pass and the equations of tangents at these points are determined. This makes it possible to compose a system of linear equations in which the unknowns are the coefficients of the general equation of the second-order curve. The resulting solution is analyzed according to the classification of second-order curves for belonging to a real ellipse. The proposed method is compared with the well-known projective-graphical method for solving the same problem.