The Palau Güell, by Antoni Gaudí, is a UNESCO World Heritage Site. Architects and historians commonly claim that the surface of the dome in Palau Güell’s Central Hall is a paraboloid. In this study, using photogrammetrical and geometrical techniques, we prove this claim to be wrong and we define the quadric surface which best fits the dome. Besides, we provide an objective measure of that fit and we state the geometric parameters defining this surface.

El Palau Güell de Antoni Gaudí está catalogado por la UNESCO como World Heritage. Es comúnmente afirmado por arquitectos e historiadores que la superficie del diseño de la cúpula que cubre el Salón Central del Palau Güell es un paraboloide. Mediante técnicas fotogramétricas y geométricas mostramos que tal afirmación no es cierta. Esta investigación determina cual es la superficie cuádrica que mejor ajusta a la cúpula. Además, damos una medida de ese ajuste, y mostramos cuáles son los parámetros geométricos que configuran esa superficie.

The Palau Güell (1885-1890) is amongst the first important projects by Antoni Gaudí (1852-1926). The building was commissioned by the Barcelona businessman Eusebi Güell Bacigalupi, who gave the architect total freedom in design. The dome of the Central Hall is one of the most characteristic geometric shapes of this building

Several relevant literature references have tried to describe this dome from a geometric and architectural point of view. Some of these references do not specify the surface which best fits the dome, while others claim that this surface is an elliptical paraboloid, though they do not provide any geometric arguments which would substantiate that claim

Perhaps the most important literature references are those from the scientific journal

The authors of the abovementioned papers make different claims regarding the geometric type of this dome, but those claims are not supported by any physical or geometric reasoning. On page 19 of the paper by Antoni González and Pablo Carbó

In addition to the abovementioned monographic issue, among all publications reviewed by us we wish to highlight two books.

First, a book by Daniel Giralt-Miracle

Having examined the most relevant literature references

This paper intends to determine the geometry of this dome with a dual purpose:

To show that the the surface which best fits the dome is not a paraboloid and not even a surface of revolution, thus correcting the widespread misconception about its geometric type.

To provide the equation and all the geometric parameters of the surface which best fits the dome designed by Gaudí. This information may be useful for several reasons, including:

In the field of architectural renovation, it is important to have an accurate knowledge of the object to be restored. Having geometric control of the dome would enable the technician to anticipate possible problems which might arise during reconstruction, or even to obtain an accurate estimate of the material costs and the amount of time needed for repairs. The fact that Palau Güell has been listed as World Heritage by the UNESCO makes it important to know the precise geometry of each element, should an intervention become necessary in the future

In addition to the relevance in the field of architecture and architectural heritage, knowing the geometric parameters and the analytical equation of the dome can help to study it from other points of view. For instance, this equation could help monitor mechanical behaviour and come up with structural hypotheses; it could even help conduct a more detailed analysis of the acoustic impact on the space covered by the dome. It is well known that parabolic and elliptical shapes direct the incident sound waves towards their respective focal points; therefore, these shapes affect the acoustic comfort in spaces

In the next section, we briefly describe the geometric method used to determine the quadric surface which best fits the dome in Palau Güell by Antoni Gaudí. This objective method provides a specific measure of that fit and does not involve mechanical, constructive or structural processes; it only involves standard geometric processes, numerical processes, computing, statistics and 3D data acquisition. We also use this method to find the best-fitting paraboloid. Lastly, using these techniques, we show the geometric parameters of the best-fitting paraboloid and the best-fitting quadric surface. We use the same methodology showed in

Let

We calculate Γ, which is the regression quadatric surface for

This regression surface Γ, the equation of which is _{i} = 1. For example:

The solution of

we find that _{00} < 0, _{00} < 0 and

Next we calculate the orthonormal reference system _{00}, and the coordinates of θ are obtained as a solution of system

The coordinates of the points from cloud _{i}, y_{i}, z_{i}

In addition to

Next, we will calculate to what extent this ellipsoid Γ statistically accounts for point cloud ^{2}, see

where

The adjusted correlation ratio

We know that _{Γ} is the percentage by which the variables _{Γ} is a statistical measure of how well the regression ellipsoid Γ fits cloud

In the process described above,

This regression surface Δ, the equation of which is _{i} = 1.

In

In Next, we will calculate to what extent this elliptical paraboloid Δ statistically accounts for point cloud ^{2}, see

where

The adjusted correlation ratio

We know that _{Δ} is the percentage by which the variables _{Δ} is a statistical measure of how well the elliptic paraboloid Δ fits

The results of our calculations are displayed graphically in

After applying the method explained in subsection 2.1 above (Matrix _{0} = –0.0033, _{0} = –0.0028, _{0} = –0.0012, _{0} = –0.0008, _{0} = –0.0013, _{0} = –0.0027, _{0} = –0.1091, _{0} = –0.0531, _{0} = –0.0378.

Using C++ language, we ourselves have created all the computer programs needed for the numerical analysis methods used for calculation in this paper.

By means of the geometric transformations explained in the previous section, we find that the normalized general _{1} = –0.9456, _{1} = –0.9362, _{1} = –0.0759, _{1} = –0.0087, _{0} = –0.3181.

After applying the method explained in subsection 2.3 above (Matrix _{2} = –0.8640, _{2} = –0.8526, _{2} = –0.0022, _{2} = –0.4577.

Using these values in the _{Γ} = 99.79 %. Similarly, the elliptical paraboloid which best fits _{Δ} = 99.19 %. This difference in fit measure can be visualized in

As stated before, we start from the reference system _{00}. The coordinates of θ are obtained as a solution of system

The reader may now repeat our calculations and check that, in the system

With all the above, we can consider

But the canonical

By means of calculation, we find that, in order to position the reference system

The intersection of the ellipsoid Γ to the plane of equation _{ac}. The eccentricity of this ellipse is ε_{ac} = 0.9591. The intersection of the ellipsoid Γ to the plane of equation _{bc}. The eccentricity of this ellipse is ε_{ac} = 0.9585.

The focal points _{ac} are located on the axis of the direction vector _{bc} are located on the axis of the direction vector

As for the geometric parameter related to surface fracture in case of deformation (that is, the Gaussian curvature) and the geometric parameter related to the minimum possible surface area (that is, the mean curvature),

As already mentioned in the introduction, there is no known original document from Gaudí explaining how this dome was designed or built, and there are no quotes from Gaudí describing its geometric type. There is, however, a graphic document of that period which includes this architectural element. It is a cross section of the Central Hall in Palau Güell

Therefore, we believe it is important to incorporate into our paper a geometrical analysis of the dome appearing in the cross section drawn by Joan Alsina i Arús. This analysis will enable us to identify the type of arc used to represent the dome, and thus we will infer the type of surface that Joan Alsina i Arús wanted to depict on the document. For this geometrical analysis we have used a method which is described in full detail in

Thus, despite all the literature consulted by us claims that the dome in Palau Güell corresponds to a paraboloid, the geometric analysis of the arc drawn by Alsina i Arús shows that the conical curve which best fits the dome’s cross section is an ellipse, and the statistical measure of that fit is 99.85%.

After the calculations in sections 3 and 4, we have ascertained that, contrary to what is claimed in the specialised literature, the surface which best fits the dome in Palau Güell is an ellipsoid, and Joan Alsina i Arús also expressed this graphically. In mathematical terms: In view of the statistical measure _{Γ} = 99.79 % for the fit of the ellipsoid Γ (best-fitting quadric surface), we can claim there is sufficient statistical evidence that the dome was designed based on the geometry of an ellipsoid. This fit can be visualized in _{Δ} = 99.19 % of its fit. This fit is substantially lower and does not provide sufficient statistical evidence that the dome was designed based on the geometry of a paraboloid. The reader may visually perceive this difference in fit through

As already stated, it is commonly claimed that the dome in Palau Güell is a paraboloid and, specifically, a paraboloid of revolution. However, we have proved that the ellipsoid Γ is not an ellipsoid of revolution. More specifically:

In the intrinsic canonical

we see that the difference in length between the two semi-minor axes is 1.0069 – 1 = 0.0069 length units. If Γ were to be an ellipsoid of revolution, this difference would have to be 0. The length unit of the real dome is more or less 3 meters, and the difference in length between the two real semi-minor axes is 0.0069 * 3 = 0.0207

Despite there is no single document indicating whether Gaudí intentionally designed a surface a revolution or not, based on our investigation we can claim that quadric surface which best fits the dome of the Central Hall in Palau Güell is an ellipsoid, and not a paraboloid.