Computer modelling of architectural forms based on ruled surfaces with imaginary axes

Abstract

One of the methods for constructing ruled surfaces is to isolate them from a linear congruence by immersing a guide line into the congruence body. The article considers a linear congruence defined by four intersecting lines. A theorem on the existence of a beam of planes intersecting a ruled algebraic surface of the order k+2 along algebraic curves of the order k (Theorem 1) is proved. The power frame of such surfaces is formed by rectilinear rods and arcs of second-order curves, which is their technological advantage. An algorithm is proposed for the transition from a linear congruence defined by four lines to an identical congruence defined by collinear fields ∏↔∏’. Such a transition makes it possible to solve the practically important problem of constructing a ruled surface passing through two conical sections. The theorem of an existing congruence defined by collinear fields with second-order curves drawn in them (Theorem 2) is proved. Based on Theorem 2, the construction of a ruled surface passing through predetermined conic sections r, r’ is performed. Examples of constructing biaxial ruled surfaces with real and imaginary axes are considered. Architectural forms based on ruled surfaces with imaginary axes are proposed. The practical use of ruled surfaces with imaginary axes allows expanding the scope of ruled structures in architectural design.