*Reference:*
Burlov V.V., Remontova L.V., Kosolapov V.V., Kosolapova E.V..
The methods of solving problems with second-order curves
// Software systems and computational methods. – 2018. – ¹ 1.
– P. 73-80.

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**Abstract:** The presented article deals with the application of affine transformations in solving problems with second-order curves, namely, stretching and contraction with respect to a straight line, that is, transforming a circle into an ellipse and vice versa.Ellipse finds the widest application in various fields due to the grace of form and its properties: in art, design, architecture, physics and technology, astronomy, its properties are described in fiction. The construction of an ellipse can be done very accurately with the help of improvised tools (pegs, threads, kinks of the circle, using a strip of paper), special adaptations and modern computer systems of mathematical modeling and CAD. The methods of constructing an ellipse are based on its properties, which also determines its shape.Using the laws of affine transformation will help to strengthen the skills of applying the properties of an ellipse and solving problems in determining its basic parameters. Method of work. The article presents methods of transformations aimed at determining the large and small axes of an ellipse, constructing tangents to an ellipse, and determining the points of intersection of a straight line with an ellipse.As a result of the work, algorithms for solving problems have been obtained that allow the authors to determine the intersection point of a straight line with surfaces of the second order (paraboloid, hyperboloid of one sheet) and a truncated cone using the method of a related transformation. The authors also determine the axes of the ellipse, the points of tangency and the intersection of the line with the ellipse. Scientific novelty. A method is proposed that makes it possible to simplify the solution of design problems on the intersection of a second-order surface with a straight line and a second-order surface, which will contribute to an increase in the accuracy and adequacy of their construction.The properties and essence of the affine transformation of an ellipse into a circle are shown and vice versa. The algorithms for solving various geometric problems based on the application of related transformations are demonstrated. The materials of the work are of practical importance, since they significantly broaden the concept of how to solve various problems with second-order curves.

**Keywords:** perpendicular straight lines, related points, conjugated diameters, axis of kinship, ellipse, affine transformation, circle, tangent, curves, intersection

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