Carved Monge Surfaces as New Forms in the Architecture

This paper describes a revision of so-called Monge surfaces through using digital graphic tools, based on the synthetic conception exposed by Gaspard Monge in the 18th and 19th centuries, through their geometric relation with polar surface. As a starting point, we propose a graphical system that integrally solves the generation of these kind of surfaces, their rationalization and discretization, with the objective of their specific application in architecture. The geometric system described allows the generation of a type of surface in which the pair of curves -generatrix and directrix- give rise to the network of principal lines of curvature (LCP) of the final surface. In this way, we propose a process that follows a bottom-up generative system based on Monge surfaces, which offers wide possibilities of formal exploration, while imposing geometrical-constructive properties that are very useful in fabrication and assembly processes.

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Surfaces of congruent sections on cylinders

Кривошапко

Иванов

2020

Vestnik MGSU

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Introduction. The definition of surfaces of congruent sections was first formulated in the work written by I.I. Kotov. These and several other types of surfaces, generated by the motion of a curve, belonged to the class of kinematic surfaces. Such kinematic surfaces as those of plane parallel displacement, surfaces of rotation, Monge surfaces, cyclic surfaces with ge-nerating circles having constant radius, rotative and spiroidal surfaces, helical some helix-shaped surfaces can be included into the class of surfaces that have congruent sections. Materials and methods. Using I.I. Kotov’s methodology, the authors first derived parametrical and vector equations for eight surfaces of congruent pendulum type cross sections of circular, elliptic, and parabolic cylinders and several helix-shaped surfaces. Circles, ellipses, and parabolas, located in the plane of the generating curve of a guiding cylinder or in the planes of a bundle that passes through the longitudinal axis of a cylinder, generate plane curves. Ellipses, analyzed in the article, can be easily converted into circles and this procedure can increase the number of shapes analyzed here. Results. Formulas are provided in the generalized form, so the shape of a plane generating curve can be arbitrary. Some surfaces of congruent sections are determined by two varieties of parametric equations. In one case, the central angle of the guiding cylindrical surface was used as an independent parameter, but in the other case, one of rectangular coordinates of the cylinder’s guiding curve served as an independent parameter. Two types of surfaces are analyzed: 1) when local axes of generating curves remain parallel in motion; 2) when these axes rotate. Conclusions. The analysis of the sources and the results, recommendations and proposals for application of surfaces, having congruent sections, is made with a view to their use in architecture and technology. The list of references has 27 positions, and it shows that the surfaces considered in this paper are being analyzed by architects, engineers, and geometricians both in Russia and abroad.

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Carved Monge Surfaces as New Forms in the Architecture

Cited by 3 publications

References 7 publications

Automatic Modeling of Surfaces with Identical Slopes

Automatic Modeling of Surfaces with Identical Slopes

Monge Surfaces. Generation, Discretisation and Application in Architecture

Surfaces of congruent sections on cylinders

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