Determinación geométrica de los paraboloides hiperbólicos de las bóvedas que configuran el pórtico de acceso a la Cripta de la Colonia Güell de Antoni Gaudí ; Geometric determination of the hyperbolic paraboloids of the vaults in the entrance porch to the Crypt of Colònia Güell by Antoni Ga

The Crypt of Colonia Güell (1898-1914), designed by the architect Antoni Gaudí, is an asset of cultural interest and historical heritage of Spain. According to some historians and architects, the first vaults with the form of a hyperbolic paraboloid that have been described are found on the roof of the entrance porch to the Crypt. The present study aims to verify the previous statement using an objective geometric method. With this process we determine: 1) What are the 19 hyperbolic paraboloids that best fit the surfaces generated by the 19 fragments of vaults at the entrance porch; 2) We offer an objective measure of this fit and, 3) We classify those paraboloids based on their geometric parameters.


IntroduCtIon
The Colònia Güell was a purpose-built industrial district in Santa Coloma de Cervelló (Barcelona). Its construction began in 1890 on the initiative of Eusebi Güell y Bacigalupi, a Spanish entrepreneur. The crypt of the church, commissioned by Güell and designed by Antoni Gaudí (1)(2), is the most remarkable building in this complex. Despite remaining unfinished following the death of its owner in 1918, the crypt has been the subject of many studies because of its structural and geometric innovation. According to (1), hyperbolic paraboloids were used here for the first time in the history of architecture. For this reason, and also because of the architectural prowess of the indoor space and the entrance porch, this building is considered to be one of the key pieces of the twentieth century architecture.
The porch giving access to the crypt comprises a set of pillars, arches and vaults. These vaults fill the triangular and trapezoidal voids of the arch structure ( Figure 1). According to (1,3), the normal solution would have been to build positive double-curvature vaults, which were very common amongst the Catalan architects at the time (4)(5). Despite that, (6) claims that these are the first vaults in the history of architecture having the shape of a hyperbolic paraboloid.
When talking about the architectural composition and morphology of the entrance porch, it is claimed that all of its 19 vaults were designed in the shape of hyperbolic paraboloids -in this regard, readers may turn to the following bibliographic references: (1,2,3,6,7,8,9,10,11). Besides, 13 of these surfaces are decorated with coloured ceramic crosses which, according to (1), follow the straight directrices of these hyperbolic paraboloids (Figures 3-4).
This purpose of this paper is to check if those claims are true.
In order to achieve our results, we use a mathematically objective method to find the 19 hyperbolic paraboloids which best fit the surfaces of the 19 vaults forming the entrance porch to the crypt of Colònia Güell (Figures 2 and 3), 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. Lastly, using these techniques we provide the geometric parameters of these paraboloids.
i=n be the point cloud outlining the surface contour of a vault in the entrance porch to the crypt of Colònia Güell. These points were obtained using photogrammetrical techniques and the software PhotoScan. Table 1 shows the number n of points in each of the clouds forming the studied vaults. For these points, we use 3D coordinates (x ′ , y ′ , z ′ ) according to the 3D orthonormal coordinate system It is to be noted that the spatial position of this reference ℛ ′ is geometrically unknown at the start of calculations.
The geometric determination process begins by changing from the initial coordinates to the coordinates (x, y, z) φθα of the follow- which is mobile depending on φ, θ and α, as follows [1-2-3-4-5-6]:   Therefore, after defining parameters φ, θ and α, we have the coordinates (x i , y i , z i ) φθα in the reference system ℛ φθα for each of the points P i in the cloud N = P i Next we calculate Г φθα , which is the surface of the regression hyperbolic paraboloid which fits to cloud N, and we obtain the following equation Eq. [7] in the reference system ℛ φθα : where α ∈ [0, π).
We said that the orthonormal reference system ℛ φθα is mobile depending on φ, θ and α because ℛ φθα is the rotation of system { } when φ, θ and α change. A progressive way of generating that rotation is as follows: First (using formula [3] and angular amplitudes φ, θ), we place a unit vector  v 3ϕθ in any position. Second (using formula [2] and angular amplitude θ), we place unit vector  v 2ϕθ in such a way that it is η adj 2 * 100 = d ϕθα is the proportion in which the variable (z i ) φθα of cloud N is statistically explained by the least-squares correlation between (z i ) φθα and (x i , y i ) φθα . In other words, this value d φθα indicates the percentage of the variable (z i ) φθα of cloud N which is statistically explained by the corresponding regression surface Γ φθα . Namely, d φθα is a statistical measurement of how well the regression paraboloid Γ φθα fits N.

Best fit
Using a calculation software that we created with C++, we repeat all the above calculations for each triplet (φ, θ, α) The three amplitudes vary in a discrete manner -not in a continuous manner-, with successive increments of 0.005 radians. Therefore, we repeat the calculation 625 3 = 244140625 times. Each triplet (φ, θ, α) corresponds to one of the 625 3 references ℛ φθα , which are the rotations of the initial reference ℛ ′ . For each rotation we obtain the surface Γ φθα , which has its own statistical fit measurement d φθα . We calculate the maximum value of the 625 3 fit measurements d φθα . This maximum value is obtained for certain values φ δ , θ δ and α δ , and we call it δ = d ϕ δ θ δ α δ .
Having determined this maximum value δ, we obtain the hyperbolic paraboloid which in reference system and which corresponds to this maximum δ. This is the hyperbolic paraboloid which statistically best fits the cloud N; we will refer to this hyperbolic paraboloid as Γ δ , we will refer to this reference system as R δ = R ϕ δ θ δ α δ , and the coordinates determined by ℛ δ are noted as (x, y, z) δ . The results for vaults 5 and 7 are displayed graphically in Figures 6 and 7, respectively.
However, these coordinates (x, y, z) δ of the system are not intrinsic parameters of Γ δ because they depend on the initial 3D orthonormal coordinate The origin O of the coordinate system ℛ δ is translated to the vertex V δ of Γ δ . Thus, we obtain the orthonormal coordinate In order to do this translation, it must be borne in mind that the coordinates of V δ are [12]: In the system surface Γ d is either [13]: [13] With regard to calculations, the reader must take into ac- This regression surface, Г φθα , described by Eq. [7] in the reference system ℛ φθα , is the one which best fits the point cloud i = n , minimizing the sum of the quadratic residues Eqs. [8] below are the Gauss normal equations which provide the solution to the calculation problem of Г φθα . This equations have a range of variation 1-n in Einstein summation convention, being 1 i = 1. For example, Remark: Formula [7] is the reason why we rotate the system ℛ ′ when φ, θ and α change, thus obtaining the system ℛ φθα . It is not true that the equation of any hyperbolic paraboloid in system ℛ ′ takes the form 0 = Bx ′2 + Cy ′2 + Hx ′ + Iy ′ + Jz ′ + 1. Actually, the equation of the hyperbolic paraboloid Γ δ which best fits the cloud N generated by the vault in system ℛ ′ takes the form Nonetheless, what matters here is not the equation for Γ δ , but the paraboloid Γ δ itself. Therefore, instead of finding Γ δ in the system ℛ ′ , we do as follows: First, we find the hyperbolic paraboloid Γ φθα which fits cloud N such that the equation of this hyperbolic paraboloid in the system ℛ φθα takes the form [7]; and then, as we will see in sections 2.2 and 2.3, we vary the angle distances φ, θ and α until we find the hyperbolic paraboloid Γ φθα which best fits the cloud N.

Statistical fit measurement
Next we will calculate to what extent that surface statistically explains the cloud N. For these calculations, we will use correlation ratio η 2 , see Eq. [9]: ordinates of a point of the corresponding regression surface Г φθα ; that is [10]: Adjusted coefficient η adj 2 is given by Eq. [11]: [11] where d 1 = 5 is the number of parameters of the regression surface. We know that η adj 2 ∈ 0,1 ⎡ ⎣ ⎤ ⎦ in all cases, and the value we start with a director parabola p d (on a plane σ d , with focal point F d , major axis x d and straight directrix r d ) and a generating parabola p g (on a plane σ g , with focal point F g , major axis x g and straight directrix r g ). The results for vaults 5 and 7 are displayed graphically in Figures 6 and 7, respectively. Both parabolas have the same vertex V and the same major axis x d = x g , but the plane σ d is orthogonal to the plane σ g and the vertex V is located between both focal points F d − V − F g . We dynamically slide the generating parabola p g on the director parabola with a continuous movement, such that the sliding of the generating parabola p g keeps the vertex on the director parabola p d . This sliding is parallel to plane σ g . This movement geometrically generates the hyperbolic paraboloid H. The director parabola has the parameter k d and the generating parabola has the parameter k g . Parameter k d is the distance between the focal point F d and the straight directrix r d , and parameter k g is the distance between the focal point F g and the straight directrix r g .
Lastly, in order to understand the equation [15] it should be noted that, by normalizing the distances so that k d = 1 and k d ≤ k g , in the case of the hyperbolic paraboloid Γ δ we find that k g = b 2 . Furthermore, owing to the aforementioned changes of reference system, the equation of surface Γ δ reduces to Eq. [15].
However, there is another numerical parameter which is intrinsic to Γ d , since it is determined by the geometry of this surface.
In order to understand this parameter, the following has to be taken into account: After generating a hyperbolic paraboloid H, the vertex V of the director parabola p d and the generating parabola p g becomes the vertex of H. The major axis x d = x g of the parabolas is perpendicular to the plane tangent to H in V. We call this tangential plane σ V . The intersection of σ V and H consists of two straight lines which we call s 1 and s 2 (Figures 6  and 7). These straight lines s 1 and s 2 (both included in H, which is a ruled surface), plus the major axis x d = x g , determine the planes π 1 and π 2 , respectively. These planes π 1 and π 2 (which, as has just been said, intersect in the axis x d = x g of the hyperbolic paraboloid H) are the so-called director planes of H, and the straight lines s 1 and s 2 are the so-called straight directrices of H.
These director planes have the following geometric trait: through every point in H there are two straight lines which lie on H and generate the ruled surface H, such that one line of that pair is parallel to a director plane and the other line is parallel to the other director plane. The angles of the dihe- { } as the reference system; in this case the equation of the surface Γ d is { } as the reference system in order to ensure that (a δ ) 2 (where the equation of surface Γ d is the one described above) is called system Nonetheless, the parameters a δ , b δ , are not intrinsic parameters of the surface Γ d yet, because they still depend on the in- of the scanning device. Therefore, we create the orthonormal { } such that [14].
In this system nates (x, y, x), such that V δ = 0, 0, 0 ( ) and the equation of the surface Γ d is as follows [15]: The parameter b 2 = (b δ ) 2 (a δ ) 2 = B δ C δ is greater than or equal to 1, and now it is an intrinsic parameter of the surface Γ d . Equation [15] is an intrinsic equation of Γ d -it is the normalized equation of this surface.

geometric elements
In order to geometrically understand the normalized Eq. [15], the following has to be taken into account: In order to geometrically and dynamically generate any hyperbolic paraboloid H, We have applied this method to the point clouds outlining the surface contours of the 19 vaults in the entrance porch to the crypt of Colònia Güell.
We have applied the above described geometric process in order to provide a normalized intrinsic analytic equation [15] for the best-fitting hyperbolic paraboloid Γ δ , and we have determined its two intrinsic parameters α and b 2 . Table 2 below shows the results for the 19 vaults of the entrance porch (α is expressed in sexagesimal degrees).

ConCluSIonS
In the introduction to this paper we said that, when talking about the architectural composition and morphology of the entrance porch to the crypt of Colònia Güell, it is commonly dral formed by the director planes π 1 and π 2 have the same width as the angles formed by the straight directrices s 1 and s 2 . The results for vaults 5 and 7 are displayed graphically in Figures 6 and 7, respectively. If the width of the acute angles is known, then the width of the obtuse angles is also known. Hereinafter, the width of the acute angles is called α.
The numerical parameter α is intrinsic to H. In the case of the hyperbolic paraboloid Γ δ , we have the following analytic formulas for the numerical calculation of α [16][17][18][19]: The statistical concepts used in this paper, such as the correlation ratio in [9] and the adjusted coefficient in [11], can be found in several books, for instance, in the well-known references (12, pp: 264-265; 13, pp: 584-585). The numerical concepts of the Gauss normal equations can be found, for instance, in the famous reference (14, pp: 671-675). The classification and analysis of quadratic surfaces can be found in many modern books, such as (15, pp: 559-592), and also in old books, such as the classic reference (16, pp: 200-205).

reSultS
In the previous section we have provided a mathematical process to objectively determine which is the hyperbolic paraboloid Γ δ which best fits a point cloud N = P i i=n , and we have also provided an objective statistical measurement δ of that fit.  The above paragraph has the following meaning: Even if the architect had wanted to create paraboloids, the fact is that none of the surfaces generated by the 19 fragments of vaults actually are paraboloids. This is so for several reasons: building errors, materials deformations, mechanical settlements, etc. Nevertheless, maybe some of them were designed in the shape of hyperbolic paraboloids. In other words: the vault may have been built using a centering made up of straight beams which are parallel to a given director plane and rest on another straight beam (i.e., the definition of a hyperbolic paraboloid using its generatrices). In this case, we say that the vault was designed in the shape of a hyperbolic paraboloid. But it is also possible that the vault was not built in that way. Namely: perhaps the vault was not built using a centering as described above. In this case, we say that the vault was not designed in the shape of a hyperbolic paraboloid. We do not know how each of these vaults was built, so we do not know if each vault was or wasn't designed in the shape of a hyperbolic paraboloid. Therefore, we only say whether we have or we don't have enough statistical evidence of that.
With regard to the decorative crosses that are present in 13 of the 19 vaults, we find that they do not follow the straight directrices of the hyperbolic paraboloids generated by these vaults. As an example, we show the location of the cross and the straight directrices for vault number 5 ( Figure 8).
It is clear that our working method -a mixture of statistics, geometry and numerical calculation-was never in the mind of Gaudí. However, it is not our intention to present a historical topic. Using the mathematical tools available to us, we intend to show the readers another point of view for studying this historical heritage.
claimed that all of its 19 vaults were designed in the shape of hyperbolic paraboloids. Besides, 13 of these surfaces are decorated with coloured ceramic crosses which allegedly follow the straight directrices of these hyperbolic paraboloids.
Using photogrammetrical techniques, we obtained the point cloud outlining the surface contour of a vault in the entrance porch to the crypt of Colònia Güell. In section 2 of this paper we outline the method used to find the hyperbolic paraboloid Γ δ which best fits this point cloud N = P i i=n . Besides, we have developed a fit measurement model and found a measurement δ of how well the hyperbolic paraboloid Γ δ fits the cloud N. Also, we have laid down the necessary calculations to find the intrinsic normalized equation and the intrinsic geometric parameters α and b 2 of the hyperbolic paraboloid Γ δ .
After carrying out these calculations, we present the intrinsic normalized equations of the 19 hyperbolic paraboloids Γ δ which best fit the 19 vaults of the porch (see Section 3 Results). We also provide the corresponding statistical fit measurement δ of each hyperbolic paraboloid Γ δ for each vault; and in Table 2 we display the geometric parameters α and b 2 which are intrinsic to each of the 19 hyperbolic paraboloids Γ δ .
As a consequence of all this geometric calculation process, we conclude that 1) there is insufficient statistical evidence that vaults number 8, 9, 10, 11 and 17 were designed based on hyperbolic paraboloids (since their corresponding fit measurements δ are below 99%); and 2) there is sufficient statistical evidence (fit measurement above 99%) that the remaining fourteen vaults were designed based on hyperbolic paraboloids.