Research Article
Comparative Study of Gaudí's Sinusoidal Conoid Arches in the Original and Replica Sagrada Familia Schools
https://doi.org/10.21203/rs.3.rs-7268713/v1
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The current study aims to compare the designs of the sinusoidal conoid arches of the Gaudí Schools with the replica built in 2002 as a training exercise for the relocation of the original schools next to the Sagrada Familia Expiatory Temple. Photogrammetry techniques are employed to create point clouds and compare the curves and angles of both buildings. This method allows for effective, reliable, and affordable comparative analyses for similar structures that deteriorate over time.
Gaudi
Gaudí schools
conoid arches
sinusoidal
The necessity of relocating the Gaudí Schools adjacent to the Sagrada Familia Expiatory Temple in 2002, to facilitate the Temple's construction, raised concerns regarding the already damaged structure, which had previously suffered a fire and subsequent reconstruction. It was decided to construct a replica of the original structure to gain firsthand knowledge of the construction criteria and to utilize this experience for a successful relocation. The copy was built in Badalona; however, despite expectations of an identical reproduction, some voices claimed that certain details were modified during its construction (Adell, 2005). The application of photogrammetry techniques to compare the original and its replica allows for a deeper understanding of the curves and characteristics of the sinusoidal conoid arches, as well as enabling a detailed comparison between the two constructions to identify any potential differences.
The application of point clouds and photogrammetry in architecture has revolutionized the documentation, analysis, and preservation of built heritage, offering precise, non-invasive methods to capture complex geometries and material properties. These technologies, particularly Structure from Motion (SfM) photogrammetry, Multi-View Stereo (MVS), and laser scanning, have become indispensable for architectural surveying, conservation, and replication, especially for historically significant structures like those designed by Antoni Gaudí. This chapter reviews the current state of research and practice in these areas, focusing on their relevance to the comparative analysis of architectural structures, such as the sinusoidal conoid arches of the Gaudí Schools.
Photogrammetry, particularly SfM and MVS techniques, has emerged as a cornerstone for generating high-fidelity 3D models of architectural structures. SfM-MVS leverages multiple overlapping photographs to reconstruct three-dimensional geometries, producing detailed point clouds that capture both metric and textural data (Pàmies et al., 2021; Sánchez Riera et al., 2022a: Sánchez Riera et al 2022b). These methods are highly accessible, requiring only standard cameras and software like Reality Capture (or Metashape), making them viable for both professionals and students (Alonso Rodríguez & Calvo López 2010). The work by Alonso Rodríguez and Calvo López highlights multi-image photogrammetry as an efficient alternative to traditional methods, emphasizing its simplicity and precision in documenting complex geometries, such as the non-horizontal masonry joints in the Burgos Cathedral vault (Alonso Rodríguez & Calvo López 2010). This accessibility is further enhanced by low-cost photogrammetry, which uses standard cameras and open-source software to produce point clouds for archaeological and architectural applications, even in challenging conditions like poor lighting or reflective surfaces (Hernández Cordero 2016).
Photogrammetric rectification, another key technique, transforms single images into orthographic, metric representations, ideal for documenting planar architectural elements like façades (Martín Talaverano 2012). This method, combined with SfM for complex 3D structures, has been successfully applied to heritage sites like the Iglesia de Benicalaf, where drones and terrestrial photography addressed access limitations to produce precise dihedral plans and textured 3D models (Rodríguez-Navarro et al. 2014, Martín Talaverano 2012). These advancements underscore photogrammetry’s versatility in capturing both 2D and 3D data, making it a critical tool for the preservation and analysis of architectural heritage.
Point clouds, whether generated through photogrammetry or laser scanning, provide a dense, accurate representation of architectural surfaces, enabling detailed geometric and structural analysis. Terrestrial Laser Scanning (TLS) offers high-precision point clouds but is limited by line-of-sight constraints, often resulting in data gaps in complex or elevated areas (Pu et al. 2022). Hand-held Laser Scanning (HLS) and Unmanned Aerial System (UAS) photogrammetry complement TLS by capturing data from multiple perspectives, improving model completeness (Pu et al. 2022). For instance, Pu et al. demonstrate how combining TLS, HLS, and UAS data enhances the quality of 3D models by addressing holes in point clouds through techniques like rasterized projection and seed point interpolation (Pu et al. 2022). This multi-source approach is particularly relevant for structures with intricate geometries, such as Gaudí’s sinusoidal conoid arches, where comprehensive data capture is essential for accurate comparison.
The analysis of point clouds extends beyond geometry to include material and structural diagnostics. The called “complex point cloud” integrates metric data, reflectivity, color, and thermal properties, enabling non-destructive analysis of material degradation and structural deformations (Aveta et al.2020). Similarly, a work applied point cloud analysis to the Faculty of Architecture and Urbanism (FAUUSP) at the University of São Paulo, using intensity data to detect biological degradation and guide restoration efforts (Bazani 2020). These studies highlight the potential of point clouds for multidisciplinary applications, combining metric precision with diagnostic capabilities to inform conservation strategies.
Comparative analysis of architectural structures, a key focus of the present study, relies on software like CloudCompare to align and measure differences between point clouds. CloudCompare employs algorithms such as Iterative Closest Point (ICP) and Hausdorff distance to quantify geometric discrepancies with millimetric precision (Mémoli & Sapiro, 2004). This capability is critical for comparing original structures with replicas or assessing changes over time due to wear or restoration. For example, a work used point clouds to verify geometric deviations in heritage buildings, identifying a 0.10 m discrepancy in the pulpit of the Church of the Company of Jesus in Quito (Moyano et al. 2022). Such precision is essential for ensuring the fidelity of replicas, as in the case of the Gaudí Schools’ relocation, where point cloud comparisons can validate structural accuracy.
The integration of point clouds into Historic Building Information Modeling (HBIM) represents a significant advancement in heritage management. HBIM combines point cloud data with parametric modeling to create semantically rich 3D models that support conservation, structural analysis, and project management (Moyano et al. 2022). Software like Revit and ArchiCAD, supported by plugins such as As-built for Revit and PointCab, facilitate the conversion of point clouds into BIM objects, though challenges remain in automating the “Scan-to-BIM” process due to the complexity of heritage geometries (Moyano et al. 2022). The study of the Pavilion of Charles V in Seville illustrates how point clouds enhance HBIM by providing textured visualizations and structural diagnostics, improving the accuracy of restoration interventions (Moyano et al 2022.).
Efficient visualization of massive point clouds is another critical area of research, particularly for large-scale architectural projects. A study proposed a method for rendering point clouds without hierarchical acceleration structures, achieving up to ten times faster loading speeds compared to tools like CloudCompare or Potree (Otepka et al. 2020). This approach, which supports real-time attribute switching (e.g., RGB, intensity), is ideal for interactive applications like virtual reality (VR) and augmented reality (AR), enhancing stakeholder communication and public engagement (Otepka et al. 2020). Similarly, a work demonstrated how point clouds bridge landscape design and urban planning by enabling iterative design and environmental simulations, such as thermal comfort analysis in urban areas (Urech et al. 2020). These advancements suggest potential applications for visualizing and disseminating the complex geometries of Gaudí’s architecture to broader audiences.
A type of arch formed from a sinusoidal curve is a sinusoidal conoid arch. This curve can be described by a sine or cosine function, with a specific period and amplitude. Sinusoidal conoid arches are used for a variety of purposes, such as:
Architecture: Used to impart curved and organic forms to structures and buildings.
Engineering: Used in the design of bridges, tunnels, and other buildings.
Graphic Design: Used to create decorative shapes and patterns.
The equation of a sinusoidal conoid arch can be expressed as follows:
y = a * sin (b * x) + c
where:
y is the vertical coordinate of the point
x is the horizontal coordinate of the point
a is the amplitude of the curve
b is the period of the curve
c is the vertical displacement of the curve
The following are the characteristics of sinusoidal conoid arches:
They are closed curves that repeat frequently.
The height of the curve is determined by its amplitude.
The length of the curve is determined by its period.
The position of the curve in the plane is determined by its vertical displacement.
The Association of Devotees of Saint Joseph, established in 1866 by the bookseller Josep Maria Bocabella with the aim of constructing a church dedicated to the Sagrada Familia, commissioned the creation of the Sagrada Familia Parish Schools to Mosén Gil Parés, who was the chaplain custodian of the Sagrada Familia (at that time a parochial tenancy) and served as the school director until 1930 (Crippa 2007).
The Bishop of Barcelona, Juan José Laguarda y Fenollera, inaugurated the schools on November 15, 1909. The construction cost 9000 pesetas, funded by Gaudí (Giralt-Miracle 2012).
Gaudí developed a simple and effective structure, designed to maximize efficiency and cost reduction (Gómez 2006). The materials had to be the most appropriate for their purpose, and the shape and dimensions of the building had to be precise to ensure minimal cost and construction effort.
On July 20, 1936, a fire occurred at the School, and in 1938, Domènec Sugrañes reconstructed the school buildings with the limited resources of the CENU (Consejo de la Escuela Nueva Unificada) during the Civil War. Unfortunately, the work was set on fire again in 1939 (Estévez 2011). After several years, a second restoration was carried out under the direction of Francesc de Paula Quintana. During these repairs, modifications were made to the interior partitions, the upper cornices, and the roof, without adding drainage gargoyles at the ends (Ferrer & Gómez Serrano 2002).
In the year 2000, the decision was made to move to the current location due to progress in the Temple's construction. At the Institut Gaudí de la Construcció in Badalona, an example of the schools' reconstruction in the form of a replica was carried out. Mariona Bonet, daughter of the architect Jordi Bonet, and Cristina Agell were responsible for the work. Following this, the relocation took place, combining some parts of the original Schools with new ones.
In this study, we employed the SfM-MVS photogrammetry technique to create the elevations of both Gaudí Sagrada Familia Schools: the reconstructed original and the Badalona replica.
SfM (Structure from Motion) and MVS (Multi-View Stereo) photogrammetry are techniques that utilize multiple photographs to reconstruct the three-dimensional structure of an object or scene. To generate a 3D model, it relies on finding corresponding Image Points (IPs) in the images and calculating their relative position in space. The feature matching algorithm (such as SIFT or SURF) is used in SfM photogrammetry to find matching points in the images; the fundamental matrix estimation algorithm is used to calculate the epipolar geometry; and the bundle adjustment algorithm is used to estimate the three-dimensional position of the points in the images.
After obtaining the three-dimensional structure with SfM, point triangulation is used to generate point clouds. To form a surface, a polygonal mesh is then created that connects these points. Finally, the textures from the original images are projected onto the mesh to provide real color and detail.
The feature matching algorithm (such as SIFT or SURF) is used in SfM photogrammetry to find corresponding points in the images; the fundamental matrix estimation algorithm to calculate the epipolar geometry; and the bundle adjustment algorithm to estimate the three-dimensional position of the points in the images.
On March 10, 2022, 379 RAW photos of the School's surroundings at its Badalona location were taken. Subsequently, 147 aerial JPEG photos were captured, along with a series of video captures, which provided an additional 123 JPEG photos. In total, 649 photos were processed, yielding a point cloud and a mesh of 6 million polygons.
Due to ongoing work at the Expiatory Temple, the rear part of the Schools has been covered by a raised fence, preventing access and rear visibility. Furthermore, the lack of access facilities for elevation by the construction authorities has led to fragmented captures from various perspectives.
On May 3, 2022, an elevated survey was conducted using a 360 camera, providing 9 photographs as a basis for initial photogrammetry. On December 1, 2022, a series of 281 elevated JPEG photographs were taken. On January 19, 2023, a series of 58 elevated RAW photographs of the roof were taken from the terrace of the building opposite. On March 22, a series of 212 JPEG photographs were taken from the fence surrounding the complex. On March 28, a series of 32 elevated JPEG photographs of the roof were taken. On March 29, a series of 72 JPEG photographs of the roof and eaves were taken.
Obtaining point clouds from both constructions allows for their superposition, enabling a detailed comparison of the construction differences between them. For this purpose, we used the specialized software CloudCompare.
CloudCompare is a 3D point cloud editing and processing program originally designed for comparing point cloud formats based on the octree structure. This program was used to measure the distance between the two input point clouds of the Gaudí schools using the C2C (Cloud to Cloud distance) algorithm. According to the program's basic documentation, this algorithm uses a variation of the Hausdorff distance calculation to obtain the distance for two equivalent points in two point clouds.
The Hausdorff distance is the greatest distance from a set of points to the closest point of the other set of points, as described in the figure. CloudCompare allows for simplified and approximate distance calculations using "local models." In our case, the basic two-step procedure was followed. First, we aligned the point clouds using the ICP (Iterative Closest Point) algorithm and ensured that the RMSE was acceptable and less than the unit to be measured. Afterward, the distance was calculated using the global model, as it offers greater precision. No differences were observed with the M3C2 plug-in.
Through this process, we obtain a series of parameters associated with the distance between two datasets, which allow us to ascertain the geometric difference for two theoretically identical or very similar buildings. This value is incorporated into a new data column as a "scalar value," enabling its representation using selectable color ramps as a result of the calculation.
As a result of the two generated point clouds, and once both were scaled equally, a comparison of the two models was performed using CloudCompare software, which allows for visualization of the result and provides millimetric distances between the two outcomes.
For the comparative analysis of both point clouds, Cloud Compare Software was used, with both samples containing 18M and 932K points, respectively. The methodology for the geometric comparison between the sculpture and the existing fountain consisted of aligning and scaling the sculptural element onto that of Plaza España. The algorithm used was ICP (Interactive Closest Point) with scaling and an RMS of 1times10 − 5 using Cloud-Compare software. For visual comparison, the real model was colored blue, and the distances were colored red.
The result of the alignment and scaling shows similarities in the base and a very similar geometry throughout the entire assembly, as well as in the main volumetry of the building.
Despite their advancements, point cloud and photogrammetry technologies face challenges. High data volumes require significant computational resources, and automated segmentation and classification remain limited by complex geometries (Urech et al. 2020; Moyano et al. 2022). Environmental factors, such as lighting or reflective surfaces, can affect data quality, necessitating multiple capture sessions or post-processing adjustments (Hernández Cordero 2016). Future research aims to address these issues through artificial intelligence (AI) and machine learning for automated feature extraction and integration with digital twins for real-time monitoring (Urech et al. 2020; Moyano et al. 2022). Additionally, standardizing workflows and improving interoperability between point cloud formats (e.g., LAS/LAZ) and BIM platforms will enhance accessibility and efficiency (Aveta et al. 2020; Otepka et al. 2020).
While the literature extensively covers photogrammetry and point cloud applications in heritage documentation, few studies focus on the direct comparison of original and replica architectural structures using these technologies. The work of Adell (2005) and Alsina (2002) provides qualitative insights into Gaudí’s sinusoidal conoid arches, but quantitative comparisons using point clouds are scarce (Adell 2005, Alsina 2002). The present study addresses this gap by employing SfM-MVS photogrammetry and CloudCompare to analyze the geometric fidelity of the Gaudí Schools’ replica against the original, contributing to the limited body of research on replica validation in heritage architecture. By leveraging methodologies from the cited works, such as precise point cloud alignment (Mémoli & Sapiro, 2004) and multi-source data integration (Pu et al. 2022), this study advances the application of digital technologies in ensuring the structural and historical integrity of replicated architectural works.
In conclusion, the works in point clouds and photogrammetry reflect a dynamic field that combines accessibility, precision, and multidisciplinary applications. These technologies enable detailed documentation and analysis of architectural heritage, offering tools to preserve and replicate complex structures like those of Gaudí. The present study builds on these advancements, using cutting-edge methods to provide a rigorous comparison of the Sagrada Familia Schools, thereby contributing to both the technical and cultural understanding of architectural heritage preservation.
The comparative research conducted between the sinusoidal conoid arches of the original Gaudí Schools and their replica, built in Badalona in 2002, offers a meticulous perspective on the analogies and geometric discrepancies between both architectural groups. The use of cutting-edge photogrammetry, specifically Structure from Motion (SfM) and Multi-View Stereo (MVS) techniques, has enabled the exact creation of point clouds, allowing for a detailed analysis of the geometries and angles of the arches in both buildings.
The results obtained reveal that, in general, the discrepancies between the arches of the original structure and its replica are minimal. CloudCompare software, when comparing the point clouds, shows that the geometry of the sinusoidal conoid arches in the two structures is almost identical. The observed differences are limited to minor alterations in a side wall of the replica, indicating that, in terms of structural fidelity, the replica maintains a high level of faithfulness to Gaudí's original design.
This accuracy in representing the geometric properties of the arches highlights the effectiveness of photogrammetry methods as a tool for the preservation and recovery of architectural heritage. Specifically, the ability to generate detailed three-dimensional models through the comparison of point clouds enables an exact assessment of modifications and wear in historical structures, favoring restoration interventions based on reliable information.
Additionally, this study offers a reflection on the challenges associated with the preservation of buildings that, due to their historical and architectural value, require a meticulous and technologically advanced approach to ensure the integrity of their original characteristics. The replica of the Gaudí Schools not only fulfills its purpose as a learning and reference tool for the relocation of the original constructions but also serves as a testament to the potential of photogrammetry in documenting highly complex architectural works.
Thus, the results of this research not only contribute to the field of architectural heritage preservation but also establish a solid foundation for future work seeking to utilize digital technologies in the restoration and replication of historical buildings. The precision and accuracy detected between the contrasted models underscore the relevance of employing techniques such as photogrammetry for the conservation and dissemination of architectural heritage, enabling its meticulous analysis without compromising its physical integrity.
Finally, it is important to note that while the differences between the replica and the original are almost imperceptible at a structural level, other factors such as the wear and tear of time and variations in the materials used could influence the visual and functional perception of the buildings over time. Therefore, continuous documentation and analysis using advanced technologies will be fundamental to ensure the long-term preservation of this valuable piece of Gaudí's architectural heritage.
3D Three dimensional
AI Artificial Intelligence
AR Augmented Reality
C2C Cloud to Cloud distance
CENU Consejo de la Escuela Nueva Unificada
FAUUSP Faculty of Architecture and Urbanism
HLS Hand-held Laser Scanning Unmanned Aerial System
IP Image Points
ICP Iterative Closest Point
HBIM Historic Building Information Modelling
JPEG Joint Photographic Experts Group
LAS Lidar Aerial Survey
LAZ Lidar Aerial Survey Zip
LiDAR Light Detection and Ranging
M3C2 Standard plugin of cloudComPy.
MVS Multi-View Stereo
NeRF Neural Radiance Fields
RAW Unprocessed digital image
SIFT Scale-Invariant Feature Transform
SfM Structure-from-Motion
SLA Stereolithography
SURF Speeded-Up Robust Features
TLS Terrestrial Laser Scanning
UAS Unmanned Aerial System
VR Virtual Reality
FUNDING DECLARATION
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
No competing interests reported.
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