# Rampant Arch and Its Optimum Geometrical Generation

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## Abstract

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## 1. Introduction

## 2. Rampant Arches of Two Arches Graphically Constructed

#### 2.1. Rampant Arch 1: Known A and C (Starting of the Two Arches)

#### 2.1.1. Method 1

- The ABCD rectangle is drawn;
- With center in A, circumference of R = AD, obtaining E;
- Mediatrix m of BE is drawn;
- The intersection of m with the rectangle determines centers O and O’.

#### 2.1.2. Method 2

- The ABCD rectangle is drawn;
- Mediatrix m of AB determines M;
- With center in M, circumference of diameter AC is drawn;
- The intersection of this circumference with m is E;
- The perpendicular by E to AC determines the centers O and O’.

#### 2.2. Rampant Arch 2: The Radii of the Two Arcs (R1 and R2) Are Known

- R1 + R2 = horizontal distance between both points (span);
- In A and B, perpendicular to the pillars where they are taken, R1 and R2 are used to obtain the centers O and O’, respectively, of the arches.

## 3. The Optimal Rampant Arch

#### 3.1. Graphical Design

- Once the gradient “d” is defined, which indicates the slope of a flying buttress or, where appropriate, of a staircase, the start of the arches is determined by points A and B, which are on a line parallel to “d”.
- Assuming the problem is solved, centers O and O’ are on the straight perpendicular to the gradient line “d”. In addition, O and O’ are on the horizontal straight lines from A and B.
- If from B, the line BC is drawn perpendicular to “d”, where logically BC = OO’.
- At this point, the question is reduced to place a segment OO’ on the bisectors of the angles in M and N.

- Any PQ line was drawn parallel to “d”.
- The bisectors b
_{1}and b_{2}of the angles were in P and Q. - BC was traced perpendicular to “d”.
- BC was moved (by parallelism) until the TS = BC segment was obtained on the bisectors.
- TS was moved (by parallelism on the perpendicular to AC) to points O and O’, which are the centers of the arches.

#### 3.2. Analytical Calculation

#### 3.3. Computer Software

## 4. Case Studies

#### 4.1. Church of Saint Urbain de Troyes (France)

#### 4.2. Cathedral of Palma de Mallorca (Spain)

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Warland, E.G. Modern Practical Masonry; Routledge: Abingdon-on-Thames, UK, 2006. [Google Scholar] [CrossRef]
- Theodossopoulos, D. Structural scheme of the Cathedral of Burgos. Struct. Anal. Hist. Constr.
**2004**, 4, 643–652. [Google Scholar] - Sánchez-Beitia, S. On-site Stress Measurements of a Flying Buttress in the Palma de Mallorca (Spain) Cathedral. Rock Mech. Rock Eng.
**2016**, 49, 315–319. [Google Scholar] [CrossRef] - Llopis-Pulido, V.; Durá, A.A.; Fenollosa, E.; Martínez, A. Analysis of the Structural Behavior of the Historical Constructions: Seismic Evaluation of the Cathedral of Valencia (Spain). Int. J. Archit. Herit.
**2019**, 13, 205–214. [Google Scholar] [CrossRef] - Viollet le Duc, E. Dictionnaire Raisonné de L’architecture Française du XIe au XVIe Siècle. B. Bance, París. 1854. Available online: https://fr.wikisource.org/wiki/Livre:Viollet-le-Duc_-_Dictionnaire_raisonn%C3%A9_de_l%E2%80%99architecture_fran%C3%A7aise_du_XIe_au_XVIe_si%C3%A8cle,_1854-1868,_tome_1.djvu (accessed on 15 April 2019).
- Tarrío, I. Los arbotantes en el sistema de contrarresto de construcciones medievales: Teorías sobre su comportamiento estructural. In Actas del IX Congreso Nacional y I Internacional Hispanoamericano de Historia de la Construcción; Instituto Juan de Herrera: Madrid, Spain, 2015; Volume 3, p. 1677. [Google Scholar]
- Deio, G. Durham Cathedral Plans. Available online: https://www.medart.pitt.edu/image/england/Durham/Cathedral/Plans/dc149dur-b.jpg (accessed on 15 April 2019).
- Choisy, A. Histoire de L’architecture Tome II; Beránger, G., Ed.; Gauthier-Villars: Paris, France, 1899. [Google Scholar]
- Courtenay, L.T. The Engineering of Medieval Cathedrals; Routledge: Abingdon-on-Thames, UK, 2016. [Google Scholar] [CrossRef]
- Moya, D. El Origen de los Arbotantes en la Historia de la Arquitectura, X Certamen Arquímedes de Introducción a la Investigación Científica; Ministerio de Educación y CSIC: Madrid, Spain, 2011; p. 2. [Google Scholar]
- Tempesta, G.; Galassi, S. Safety evaluation of masonry arches. A numerical procedure based on the thrust line closest to the geometrical axis. Int. J. Mech. Sci.
**2019**, 155, 206–221. [Google Scholar] [CrossRef] - Panofsky, E. Gothic and Architecture and Scholasticism; Archabbey Press: New York, NY, USA, 1957; ISBN 0529020920. [Google Scholar]
- Davis, M.T.; Neagley, L.E. Mechanics and Meaning: Plan Design at Saint-Urbain, Troyes and Saint-Ouen, Rouen. Gesta
**2000**, 39, 161–182. [Google Scholar] [CrossRef] - Heyman, J. Beauvais cathedral. Trans. Newcom. Soc.
**1967**, 40, 15–35. [Google Scholar] [CrossRef] - Pelà, L.; Bourgeois, J.; Roca, P.; Cervera, M.; Chiumenti, M. Analysis of the effect of provisional ties on the construction and current deformation of Mallorca Cathedral. Int. J. Archit. Herit.
**2016**, 10, 418–437. [Google Scholar] [CrossRef] - Derand, F. L’Architecture des Voutes, ou l’art, traits et coupes des voutes; traite tres-utile, méme nécessaire à tous les architectes, maitres-maçons, appareilleurs, tailleurs de pierres, et généralement à tous ceux qui se mélent de l’Architecture, meme militaire, París: S. Cramoysi, 1643. Available online: https://gallica.bnf.fr/ark:/12148/btv1b8626566f/f7.image (accessed on 24 April 2019).
- Blondel, F.N. Cours d’ Architecture enseigné dans l’Académie royale d’ architecture. Ed. P. Auboin et F. Clouzier (Paris). 1675. Available online: https://gallica.bnf.fr/ark:/12148/bpt6k85661p.r=cours%20architecture%20cours%20architecture?rk=21459;2 (accessed on 24 April 2019).
- Casado, E.A.; Sánchez, J.M.D. Geometría del arco carpanel. Suma: Revista sobre Enseñanza y Aprendizaje de las Matemáticas
**2015**, 79, 17–25. [Google Scholar] - Quintas, V. Structural analysis of flying buttresses. Eur. J. Environ. Civ. Eng.
**2017**, 21, 471–507. [Google Scholar] [CrossRef] - Portioli, F.; Cascini, L.; Casapulla, C.; D’Aniello, M. Limit analysis of masonry walls by rigid block modelling with cracking units and cohesive joints using linear programming. Eng. Struct.
**2013**, 57, 232–247. [Google Scholar] [CrossRef] - Frezier, A.F. La Théorie et la Pratique de la Coupe des Pierres et des Bois pour la Construction des Voûtes et Autres Parties des Bâtiments Civils & Militaires, ou Traité de Stéréotomie, à L’usage de L’architecture. Tome 3. Strasbourg and Paris (1737-9). Available online: https://gallica.bnf.fr/ark:/12148/bpt6k1040142z/f491.image (accessed on 15 April 2019).
- Breymann, G.U. Bau-Constructions-Lehre. Berlag von Gustav Beise. 1881. Available online: https://www.e-rara.ch/zut/content/pageview/8642945 (accessed on 15 April 2019).
- Heyman, J. The stone skeleton. Int. J. Solids Struct.
**1966**, 2, 249–279. [Google Scholar] [CrossRef] - Theodossopoulos, D.; Sinha, B.P.; Usmani, A.S. Case study of the failure of a cross vault: Church of Holyrood Abbey. J. Archit. Eng.
**2003**, 9, 109–117. [Google Scholar] [CrossRef] - Bruzelius, C. The Second Campaign at Saint-Urbain at Troyes. Speculum
**1987**, 62, 635–640. [Google Scholar] [CrossRef] - Davis, M.T. On the Threshold of the Flamboyant: The Second Campaign of Construction of Saint-Urbain, Troyes. Speculum
**1984**, 59, 847–884. [Google Scholar] [CrossRef] - Roca, P.; Cervera, M.; Pelà, L.; Clemente, R.; Chiumenti, M. Continuum FE models for the analysis of Mallorca Cathedral. Eng. Struct.
**2013**, 46, 653–670. [Google Scholar] [CrossRef] - Elyamani, A.; Roca, P.; Caselles, O.; Clapes, J. Seismic safety assessment of historical structures using updated numerical models: The case of Mallorca cathedral in Spain. Eng. Fail. Anal.
**2017**, 74, 54–79. [Google Scholar] [CrossRef] - Pérez-Gracia, V.; Caselles, J.O.; Clapés, J.; Martinez, G.; Osorio, R. Non-destructive analysis in cultural heritage buildings: Evaluating the Mallorca cathedral supporting structures. NDT E Int.
**2013**, 59, 40–47. [Google Scholar] [CrossRef] - Fitchen, J. The Construction of Gothic Cathedrals: A Study of Medieval Vault Erection; University of Chicago Press: Chicago, IL, USA, 1981. [Google Scholar]
- San-Antonio-Gómez, C.; Velilla, C.; Manzano-Agugliaro, F. Photogrammetric techniques and surveying applied to historical map analysis. Surv. Rev.
**2015**, 47, 115–128. [Google Scholar] [CrossRef] - Manzano-Agugliaro, F.; Montoya, F.G.; San-Antonio-Gómez, C.; López-Márquez, S.; Aguilera, M.J.; Gil, C. The assessment of evolutionary algorithms for analyzing the positional accuracy and uncertainty of maps. Expert Syst. Appl.
**2014**, 41, 6346–6360. [Google Scholar] [CrossRef] - San-Antonio-Gómez, C.; Velilla, C.; Manzano-Agugliaro, F. Urban and landscape changes through historical maps: The Real Sitio of Aranjuez (1775–2005), a case study. Comput. Environ. Urban Syst.
**2014**, 44, 47–58. [Google Scholar] [CrossRef] - Perea-Moreno, A.J.; Aguilera-Ureña, M.J.; Larriva, M.D.; Manzano-Agugliaro, F. Assessment of the potential of UAV video image analysis for planning irrigation needs of golf courses. Water
**2016**, 8, 584. [Google Scholar] [CrossRef] - Achille, C.; Adami, A.; Chiarini, S.; Cremonesi, S.; Fassi, F.; Fregonese, L.; Taffurelli, L. UAV-based photogrammetry and integrated technologies for architectural applications—Methodological strategies for the after-quake survey of vertical structures in Mantua (Italy). Sensors
**2015**, 15, 15520–15539. [Google Scholar] [CrossRef] [PubMed] - Karachaliou, E.; Georgiou, E.; Psaltis, D.; Stylianidis, E. Uav for Mapping Historic Buildings: From 3d Modelling to Bim. In International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Proceedings of the 8th International Workshop 3D-ARCH “3D Virtual Reconstruction and Visualization of Complex Architectures”, Bergamo, Italy, 6–8 February 2019; International Society of Photogrammetry and Remote Sensing (ISPRS): Hannover, Germany, 2019. [Google Scholar]
- Yilmaz, H.M.; Yakar, M.; Gulec, S.A.; Dulgerler, O.N. Importance of digital close-range photogrammetry in documentation of cultural heritage. J. Cult. Herit.
**2007**, 8, 428–433. [Google Scholar] [CrossRef] - Jiang, R.; Jáuregui, D.V.; White, K.R. Close-range photogrammetry applications in bridge measurement: Literature review. Measurement
**2008**, 41, 823–834. [Google Scholar] [CrossRef] - Grussenmeyer, P.; Landes, T.; Voegtle, T.; Ringle, K. Comparison methods of terrestrial laser scanning, photogrammetry and tacheometry data for recording of cultural heritage buildings. Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci.
**2008**, 37, 213–218. [Google Scholar] - Lerma, J.L.; Navarro, S.; Cabrelles, M.; Villaverde, V. Terrestrial laser scanning and close range photogrammetry for 3D archaeological documentation: The Upper Palaeolithic Cave of Parpalló as a case study. J. Archaeol. Sci.
**2010**, 37, 499–507. [Google Scholar] [CrossRef]

**Figure 3.**(

**A**) One- and two-arch circumference flying buttresses of Beauvais Cathedral (France). (

**B**) Flying buttresses of the Cathedral of Palma de Mallorca. (

**C**) Rampant arch in the staircase of the Monastery of Uclés (Cuenca). (

**D**) Rampant arch in the staircase of La Lonja (Valencia).

**Figure 5.**Rampant arches of two arches graphically constructed: (

**A**) known A and C (method 1); (

**B**) known A and C (method 2); (

**C**) known radii of the two arcs (R1 and R2).

**Figure 10.**Rampant arches with different geometries: (

**A**) numerical data input; (

**B**) graphical data input.

**Figure 11.**Calculation of the optimal rampant arch (in red) over a rampant arch of the flying buttress of the church of Saint Urbain de Troyes (France).

**Figure 12.**Calculation of the optimal rampant arch over a flying buttress (in red) of the Cathedral of Palma de Mallorca (Spain). Note that the yellow line highlights the initial and final points of the rampant arch.

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**MDPI and ACS Style**

Velilla, C.; Alcayde, A.; San-Antonio-Gómez, C.; Montoya, F.G.; Zavala, I.; Manzano-Agugliaro, F.
Rampant Arch and Its Optimum Geometrical Generation. *Symmetry* **2019**, *11*, 627.
https://doi.org/10.3390/sym11050627

**AMA Style**

Velilla C, Alcayde A, San-Antonio-Gómez C, Montoya FG, Zavala I, Manzano-Agugliaro F.
Rampant Arch and Its Optimum Geometrical Generation. *Symmetry*. 2019; 11(5):627.
https://doi.org/10.3390/sym11050627

**Chicago/Turabian Style**

Velilla, Cristina, Alfredo Alcayde, Carlos San-Antonio-Gómez, Francisco G. Montoya, Ignacio Zavala, and Francisco Manzano-Agugliaro.
2019. "Rampant Arch and Its Optimum Geometrical Generation" *Symmetry* 11, no. 5: 627.
https://doi.org/10.3390/sym11050627