The Crypt of Colonia Güell (18981914), 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.
RESUMEN
La Cripta de la Colonia Güell (18981914) diseñada por el arquitecto Antoni Gaudí, es un bien de interés cultural y patrimonio histórico de España. Según algunos historiadores y arquitectos, las primeras bóvedas de la historia de la arquitectura con forma de paraboloide hiperbólico se encuentran en el techo del pórtico de acceso a la cripta. El presente estudio pretende comprobar la anterior afirmación con un método geométrico objetivo. Con este proceso determinamos: 1) Cuáles son los 19 paraboloides hiperbólicos que mejor se ajustan a las superficies generadas por los 19 fragmentos de bóvedas que conforman el pórtico; 2) Ofrecemos una medida objetiva de tal ajuste y, 3) Presentamos una clasificación de tales paraboloides mediante sus parámetros geométricos.
The Colònia Güell was a purposebuilt 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 (12), 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 doublecurvature vaults, which were very common amongst the Catalan architects at the time (45). Despite that, (6) claims that these are the first vaults in the history of architecture having the shape of a hyperbolic paraboloid.
Entrance porch to the crypt of Colònia Güell by Antoni Gaudí. [Picture taken by the authors.]
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 34).
Threedimensional model of the 19 vaults forming the entrance porch to the crypt. [Image generated by the authors].Orthogonal projection on the ground plane of the 19 vaults forming the entrance porch to the crypt. In the right image, the 13 vaults which bear a coloured ceramic cross on their surfaces are highlighted in red. The triangle pinpoints the location of the entrance to the crypt. [Image generated by the authors.]Threedimensional model of vault number 5, as generated with the software PhotoScan; its cloud <inlinegraphic xlink:href="ic_63691_form38b"/> is made up by <italic>n</italic> = 148622 points. [Image generated by the authors.]
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.
Figures 2 and 3 below show the aforementioned 19 vaults in their entirety. For this reconstruction, we have used photogrammetrical techniques and 3D modelling with commercial software PhotoScan and MeshLab.
METHOD USED TO DETERMINE THE HYPERBOLIC PARABOLOID AND ITS GEOMETRIC ELEMENTSGeometric regression
Let 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 of the scanning device (Figures 4 and 5). It is to be noted that the spatial position of this reference ℛ′ is geometrically unknown at the start of calculations.
Number <italic>n</italic> of points in each of the clouds <inlinegraphic xlink:href="ic_63691_form38a"/> forming the studied vaults.
Vault #
Number of points n in cloud
Vault #
Number of points n in cloud
1
50046
11
415198
2
125451
12
371637
3
125645
13
98293
4
179999
14
350632
5
148622
15
212260
6
31180
16
245032
7
320018
17
364235
8
380264
18
293295
9
438630
19
228404
10
281882


Threedimensional renderings of the point clouds <inlinegraphic xlink:href="ic_63691_form38b"/> which define the studied vaults (except for cloud 5, which is displayed in more detail in <xref reftype="fig" rid="F4">Figure 4</xref> above). [Image generated by the authors.]
The geometric determination process begins by changing from the initial coordinates to the coordinates (x, y, z)_{
φθα
} of the following orthonormal reference system which is mobile depending on φ, θ and α, as follows [1,2,3,4,5,6]:
where φ ∈ [0, π), θ ∈ [0, π).
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 in any position. Second (using formula [2] and angular amplitude θ), we place unit vector in such a way that it is perpendicular to and lies within linear subspace . Third (using formula [1]), by means of cross product we generate vector such that the basis is orthonormal and direct. Fourth (using formulas [45] and angular amplitude α), we rotate vectors around vector (formula [6]), thus obtaining the rotated vectors such that the basis is orthonormal and direct.
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 .
Next we calculate Г_{
φθα
}, which is the surface of the regression hyperbolic paraboloid which fits to cloud , and we obtain the following equation Eq. [7] in the reference system ℛ_{φθα}:
This regression surface, Г_{
φθα
}, described by Eq. [7] in the reference system ℛ_{φθα}
, is the one which best fits the point cloud , minimizing the sum of the quadratic residues , being B_{φθα}C_{φθα}
< 0. 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, , and .
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 generated by the vault in system ℛ′ takes the form 0 = Bx^{′2} + Cy^{′2} + Dz^{′2} + Ex′y^{′} + Fx′z^{′} + Gy′z^{′} + Hx^{′} + Iy^{′} + Jz^{′} + 1, where 0 = −CF^{2} + FGE − BG^{2} − DE^{2} + 4BCD and where 0 < F^{2}I^{2} − 4CF^{2} − 2FGHI + 4FGE + 4CFHJ − 2FJEI + G^{2}H^{2} − − 4BG^{2} − 2GHJE + 4BGJI − 4CDH^{2} + 4DHEI + J^{2}E^{2} − 4BCJ^{2} − − 4DE^{2} − 4BDI^{2} + 16BCD. 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 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 .
Statistical fit measurement
Next we will calculate to what extent that surface statistically explains the cloud . For these calculations, we will use correlation ratio η^{2}, see Eq. [9]:
where , and where (x_{i}, y_{i}, f(x_{i}, x_{i}))_{
φθα
} are the coordinates of a point of the corresponding regression surface Г_{
φθα
}; that is [10]:
Adjusted coefficient is given by Eq. [11]:
where d_{1} = 5 is the number of parameters of the regression surface. We know that in all cases, and the value is the proportion in which the variable (z_{i})_{
φθα
} of cloud is statistically explained by the leastsquares 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 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 .
Best fit
Using a calculation software that we created with C++, we repeat all the above calculations for each triplet (φ, θ, α) ∈ [0, π) × [0, π) × [0, π). 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 .
Having determined this maximum value δ, we obtain the hyperbolic paraboloid which in reference system is described by the equation and which corresponds to this maximum δ. This is the hyperbolic paraboloid which statistically best fits the cloud ; we will refer to this hyperbolic paraboloid as Γ_{
δ
}, we will refer to this reference system as , 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.
The cloud <inlinegraphic xlink:href="ic_63691_form38b"/> of vault number 5 is highlighted in orange, and its bestfitting hyperbolic paraboloid Γ<sub>
<italic>δ</italic>
</sub> is highlighted in yellow. The straight directrices <italic>s</italic>
<sub>1</sub> and <italic>s</italic>
<sub>2</sub> are shown in green; the axes are shown in red; the director parabola <italic>p<sub>d</sub>
</italic> and the generating parabola <italic>p<sub>g</sub>
</italic> are shown in blue. [Image generated by the authors.]The cloud <inlinegraphic xlink:href="ic_63691_form38b"/> of vault number 7 is highlighted in orange, and its bestfitting hyperbolic paraboloid Γ<sub>
<italic>δ</italic>
</sub> is highlighted in yellow. The straight directrices <italic>s</italic>
<sub>1</sub> and <italic>s</italic>
<sub>2</sub> are shown in green; the axes are shown in red; the director parabola <italic>p<sub>d</sub>
</italic> and the generating parabola <italic>p<sub>g</sub>
</italic> are shown in blue. [Image generated by the authors.]
However, these coordinates (x, y, z)_{
δ
} of the system , as well as the equation , are not intrinsic parameters of Γ_{
δ
} because they depend on the initial 3D orthonormal coordinate system of the scanning device.
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 system . In order to do this translation, it must be borne in mind that the coordinates of V_{
δ
} are [12]:
In the system we have the coordinates , such that and the equation for surface Γ_{
δ
} is either [13]:
With regard to calculations, the reader must take into account that and .
In the event that , we would use as the reference system; in this case the equation of the surface Γ_{
δ
} is . In the event that we would use as the reference system in order to ensure that in the equation . Hereinafter, the coordinate system (where the equation of surface Γ_{
δ
} is the one described above) is called system .
Nonetheless, the parameters , are not intrinsic parameters of the surface Γ_{
δ
} yet, because they still depend on the initial 3D orthonormal coordinate system of the scanning device. Therefore, we create the orthonormal coordinate system such that [14].
In this system , points have coordinates (x, y, x), such that V_{
δ
} = (0, 0, 0) and the equation of the surface Γ_{
δ
} is as follows [15]:
The parameter 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_{
δ
} –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 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_{
δ
} , 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 socalled director planes of H, and the straight lines s_{1} and s_{2} are the socalled 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 dihedral 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 wellknown references (12, pp: 264265; 13, pp: 584585). The numerical concepts of the Gauss normal equations can be found, for instance, in the famous reference (14, pp: 671675). The classification and analysis of quadratic surfaces can be found in many modern books, such as (15, pp: 559592), and also in old books, such as the classic reference (16, pp: 200205).
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 , and we have also provided an objective statistical measurement δ of that fit.
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 bestfitting 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).
Results for the 19 vaults of the entrance porch (<italic>α</italic> is expressed in sexagesimal degrees).
δ
α
b^{2}
1
99.67
0.39
88270.61
2
99.86
75.30
1.68
3
99.73
0.82
19700.29
4
99.85
0.90
16377.77
5
99.71
89.40
1.02
6
99.95
60.10
2.99
7
99.76
85.08
1.19
8
97.92
15.90
51.27
9
97.64
0.21
284071.32
10
98.85
1.37
7023.60
11
97.59
0.04
8108811.54
12
99.70
0.06
3462399.46
13
99.46
0.03
11083400.45
14
99.42
0.43
71687.29
15
99.16
0.16
531535.10
16
99.58
0.54
45211.87
17
98.36
7.89
210.34
18
99.86
0.21
307243.25
19
99.16
0.01
173306546.50
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 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 . Besides, we have developed a fit measurement model and found a measurement δ of how well the hyperbolic paraboloid Γ_{
δ
} fits the cloud . 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.
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).
The ceramic cross of vault number 5 and, in green color, the straight directrices of the bestfitting hyperbolic paraboloid Γ<sub>
<italic>δ</italic>
</sub>. [Image generated by the authors.]
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.
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