Areas for the existence of biquadratic transformations

Analysis and research carried out quadratic transformations of applied geometry shows that quadratic transformations of the plane have been studied sufficiently and have found wide application in science and technology. However, little attention has been paid to the study and application of four to four-digit correspondences to the plane. Therefore, the article is devoted to the development of the theory of four-four-digit correspondences between non-aligned planes and the development of the theory of biquadratic plane transformations. As a result of the consistent implementation of the proposed constructive device, each point A plane П 1 is converted to four points A 1 / , A 2 / and A 3 / , A 4 / plane of П 1 / . Given the two parametric set of points of the combined plane П 1 /= П 1 we obtain a biquadratic transformation of the plane, denoted by the letter L . In a similar way, it can be shown that in the opposite direction each point A / of the plane П 1 / is transformed into four points of the plane П 1 . This transformation is denoted by the letter L / . Also this article is devoted to determining of the area existence in the development of the theory of biquadratic transformation plane, improvement of methods of geometric design of curves and curved surfaces in construction design. To achieve this purpose it’s requiring to solve the following theoretical and practical problems: definition and development of graphic model of biquadratic transformations of the plane, determination of the existence area of the transformation. The results can be widely used in design and research institutes by designing and construction of surfaces of technical forms, also new dome shells and other surfaces in architectural design.