Determination of the geometric shape which best fits an architectural arch within each of the conical curve types and hyperbolic-cosine curve types: The case of Palau Güell by Antoni Gaudí

We provide a method to objectively determine which is the geometric shape which best fits an arch of a heritage building within each of the conical curve types - ellipse, hyperbola, parabola - and hyperbolic-cosine curve types - catenary, hyperbolic cosine, Rankine -, and we also provide an objective measurement of that fit. This method does not involve mechanical, constructive or structural processes; it only involves standard geometric processes, numerical processes, computing, statistics and 3D data acquisition. Using these techniques, we generate a method, which allows an objective determination of an arch's geometry in a heritage building. For architectural and historical reasons, and also due to discrepancies regarding the arch's geometry, as an application case for this method we have chosen the arch on the façade of Palau Güell (1886-1890) in Barcelona, a heritage building designed by Antoni Gaudí.