Quadratic birational transformations in computer simulation of building structures

Abstract

Quadratic transformations are a perspective way of form-making in architectural and engineering-construction design. A quadratic (nonlinear) transformation is based on the establishment of one-to-one correspondence between the algebraic n- and 2nd-order curves. In particular, lines correspond to the second-order curves, which are widely used in the design of gridwork building structures and utility lines. Quadratic transformations are used to model the profile of rotor blades of environmentally friendly wind farms. We developed algorithms for building corresponding points for the practical application of nonlinear transformations in the computer simulation of building structures. We have shown that the inversion relative to the circle is a central quadratic transformation with imaginary fundamental point, coinciding with the cyclic points of the plane. We considered an example of using the inversion for building the rotor blade profile of a wind power installation and a cam profile. Quadratic transformation algorithms are practically realized using computer graphics and specialized software tools.