Classification by Type of the Arches in Gaudí’s Palau Güell

Abstract

Architects and historians commonly claim that when designing the Palau Güell, Gaudí experimented with some architectural elements which would influence his later works, with the arches being prevalent in the building. But, to date, no mathematically thorough study has been conducted in order to determine the type of each of such arches. Furthermore, literature contains subjective and intuitive claims which create ambiguity and contradictions. Owing to the methods used by Gaudí, the arches in Palau Güell are best fit by conical curves (ellipse, parabola, hyperbola) or hyperbolic-cosine curves (catenary, Rankine). Using photogrammetrical and mathematical techniques, in this paper we will classify the arches of this building according to their types, we will eliminate the inconsistencies which currently exist in literature, and we will obtain the corresponding analytical equations, which may be helpful in future studies.

The use of Weber’s focal-directorial plane curves as approximation of top view contour curves at architectural buildings objects

“…After substituting the boundary conditions X A = 36.325 m, Y B = 31.425 m into the system of equations (8), the following values for the Weber"s curve The selected set of points P on the ground-base contour of the object through the Approximation II (see Fig. 6 -Approximation II), gives the following coefficient of determination II R 2 = 0.9992.…”

“…Gonzalez et al determined a curve, which best fits an architectural arch and its analytical equation, and then they apply this method to the 23 arches in Palau Guell, [8]. Authors have concluded that Gaudi experimented with the four types of conic curves (circle, ellipse, parabola, hyperbola) and the two types of hyperbolic-cosine curves (Rankine, catenary).…”

A Focal Curve Approximation of a Borromini Oval Contour

“…The second class of investigations underlines the following question: what geometric shape, that is, curve, best fits the contour of particular architectural object either as the whole or one of its components. This problem has been treated by multiple authors including González et al (2018), Samper et al (2017), Petrović et al (2016Petrović et al ( , 2018 and Huylebrouck (2007). In the research of Samper et al (2017), the comparison of the types of conics and hyperbolic-cosine curves in the approximation of Gaudi's geometric forms has been carried out.…”

“…The importance of such a curve approximation approach can provide a base for further computer modelling for the sake of the preservation of cultural heritage. For example, González et al (2018) determined a curve which best fits an architectural arch and its analytical equation, and then applied this method to the 23 arches in Palau Guell. The authors concluded that Gaudi had experimented with the four types of conic curves (circle, ellipse, parabola and hyperbola) and the two types of hyperbolic-cosine curves (Rankine and catenary).…”

“…One of the two major contributions of the paper is the demonstration that multifocal curves (Weberian multi focal/directorial curves) are a good approximation for the curve fitting process in the case of some architectural objects' geometry analysis. They are equally applicable to curves which have been used so far for such purposes, including circle, ellipse, parabola, hyperbola and catenary geometry in the works of González et al (2018), Samper et al (2017) and Huylebrouck (2007). The second major contribution is the determination of the most suitable Weberian curve, with four collinear foci, for the approximation of the contour in the case of the monumental Dome of San Carlo alle Quatro Fontane based on the data points (Table 1) from the Fig.…”