Abstract
In this article, we present a point cloud-based geometric analysis method applied to the 19th-century stellar vaults of the Saint Elisabeth Cathedral of Košice (SK). We based our analysis method on the geometric typology systems we worked out for net vaults during our earlier research. After elaborating on the method’s application to stellar vaults, we present the results of the analysis. This includes the exact geometric description of the vaults’ webbing and rib system, which allowed for deductions about their original construction techniques. Then, we present the re-modelling of the rib systems of the cathedral’s stellar vaults based on the most influential theoretical works of the 19th century (B. Ranisch, F. Hoffstadt, G. G. Ungewitter), such as the principle of the longest route or projecting the junction points to a spherical surface. Afterwards, the comparative analysis of the real vault geometries and the re-modelled rib systems was carried out. Based on this, we discussed the potential reason behind the differences detected. This gives a valuable insight into whether the 19th-century theoretical works and their contemporary building practices differed. Additionally, we elaborated on the implications of the multiple possible three-dimensional geometries of the same rib pattern regarding the authenticity of theoretically reconstructed stellar vault structures.
INTRODUCTION
General introduction and aims
The majority of the written sources about the construction and building techniques of Gothic vaults are of 18–19th century origin, representing the most influential ideas on the topic that have persisted to the present day. These sources are contemporary with Neo-Gothic vault construction, which sought to revive the Gothic style. However, Gothic sources about vault constructions are scarce,1 and the survey techniques of the 18th–19th centuries were inadequate in determining the precise geometry of above-head structures. Contemporary analyses, conducted based on the real geometry of Gothic structures, have revealed that Gothic net vault geometry exhibits a far wider typology concerning both the connection of the ribs and the webs2 and the global geometry of the rib system.3 Presumably, the various construction and building strategies were a means of overcoming the challenges posed by the uneven outlines of the spaces that were to be vaulted.4 These solutions, however, remain undocumented in written sources from the Gothic era or the 18th and 19th centuries. Consequently, while it is reasonable to infer that Neo-Gothic vault structures were influenced by their contemporary ideas, a thorough analysis of the individual structures is warranted. More unique solutions presumably evolved from constraints of e.g. previous building periods earlier,5 thus not recording certain methods in the era does not necessarily mean that they were not used.
In the present article, an attempt is made to carry out an exact geometry-based analysis of the 19th-century stellar vaults in the Saint Elisabeth Cathedral of Košice (SK) (Fig. 1). The church itself is of Gothic origin, and it had an original, Gothic stellar vault system as well. However, the building underwent a thorough reconstruction directed by Imre Steindl. During this phase, it got its new stellar vaults in the crossing and the transept, and the vaults of the main nave that can be also regarded as stellar vaults, although they are built above decidedly rectangular plans.
The inner space of the church with the vaults of the crossing and the main nave.
Courtesy of László Daragó
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
The analysis method employed in this study is fundamentally similar to the one previously established for the analysis of Gothic net vaults (with suitable modifications for stellar vaults, as outlined in the Methodology section).
In the case of Gothic vaults, it is rare to find plans and preliminary drawings of a given example. However, from the 19th century onwards, these sources became more abundant, although they are not necessarily easily accessible for research. The question of the alignment between a given vault and its plans could also be a fruitful research topic. However, the present article is concerned with theoretical vault construction methods of the 19th century and their connections to a real structure. Therefore, the re-modelled version of the examined vaults is presented, based on several different vault construction methods while retaining the outlines and rib patterns. The subsequent phase involves conducting a comparative analysis of the remodelled vaults and the results of the real vaults’ analysis. This phase also involves a discussion of potential reasons for any observed differences.
Short history of the vault system
The building of the Saint Elisabeth Cathedral that partially still exists today started in the second half of the 14th century, after the previous Gothic church was destroyed in a conflagration. (According to some researchers, the first Gothic church had a Romanesque antecedent as well.)6 The building of the new church was a long process, during which, supposedly, plan alterations also occurred.7 Based on the tombstones that appeared in the foundations of its main pillars, the main nave was likely built after 1378,8 and written documents support that the building works were still going on in 1402.9 The building works halted in the middle of the 15th century, however, during the reign of Matthias I, they recommenced.10
Most researchers agree that the nave of the church was built previous to its apse.11 The dating of the apse is a controversial issue. Some researchers date it to the beginning of the 16th century, however, others claim that it was carried out in the second half of the 15th century.12
After its completion, the church went through numerous phases of destruction and restoration: In 1490–1491, during the armed conflict between Vladislaus II and Prince John Albert cannonballs damaged the building (restoration: 1492–1496).13 In 1556, during the conflagration of Košice, most of the church’s altars and its roof were destroyed – the restoration works started immediately.14 From the 16th century, the church was the subject of continuous conflicts between denominations and served alternately for Catholics and Calvinists.15 From the 17th century, multiple written sources describe that the church’s vaults and roof were in bad condition.16 In 1706, during Rákóczi’s War of Independence, the church was damaged again,17 however, the next big restoration only started in the second half of the 18th century.18 In 1934, an earthquake hit Košice that damaged the walls and vaults of the church.19 In 1845, because of a flood, its floor gave way above the sepulchral vaults.20 In 1846, in a conflagration, its roof caught on fire, however, in this case, the damages were not severe.21 Restoration was carried out between 1857 and 1863,22 how-ever, these did not prove to be professionally done and on the whole, they damaged the building even further.23 This was found out when, owing to a windstorm in 1875, the church suffered further damage. By this point in time, the main pillars had undergone a lateral displacement of 18–20 cm from the vertical plane, while the ribs of the vaults had been displaced from their original positions by 10cm.24 In 1877, restoration works commenced, based on the plans of Frigyes Schmidt, led by Imre Steindl25 (who also carried out the detailed survey of the building).26 In his report written in 1880, Steindl describes that the stellar vaults of the main nave and the transepts are not coherently formed with the pillars, therefore he thinks that these vaults are either the results of a plan alteration or the remnants of a later building period than the pillars. Also, the vaults of the aisles were quite irregular, Steindl claims that their ribs within one vault were not uniform, and the nave arches between the aisles and the naves were also different: compressed pointed arches, circular arches and three-centered arches. He also accentuates that in the case of the apse, the wall pillars and the ribs of the vaults belong to one period.27
In 1884, as the vaults started to crack, to save the church, the arches between the aisles and the nave were walled up and the vaults were supported with temporary structures.28 Imre Henszl mann claimed that these structures were not built professionally and that the ratios of the widths of the aisles and the nave to each other were incorrect, causing stability problems. He states that the nave must have been planned originally as a five-aisle system, however, due to a change of the master builders it was carried out as a three-aisle system, meaning that the aisles were far too wide.29 Eventually, the decision was made to rebuild the nave of the church, based on the plans of Imre Steindl, using a five-aisle system.30 The demolition works were carried out between 1887 and 1888, only the apse, the walls of the aisles and the towers remained from the Gothic structure.31 The rebuilding of the Saint Elisabeth Cathedral was concluded with the nave vaults in 1894.32 As József Sisa highlighted in his monography about Steindl, the 19th-century plans of the Saint Elisabeth Cathedral used more uniform vaults instead of the original, more diverse solutions.33
LITERARY FRAMEWORK AND ITS EVALUATION
The majority of extant written sources about the construction and building techniques of Late Gothic vaults originate from the 18–19th centuries, thus, they can be considered contemporary to the building of Neogothic vault structures. These sources have exerted a profound and enduring influence on even contemporary thinking on the subject. The most widely cited of these are the works of Bartel Ranisch’s,34 Friedrich Hoffstadt’s35 and Georg Gottlob Ungewitter’s36 books. The stellar-net constructing methods presented in these works are elaborated on in the ‘Results’ section.
Examining these works, we found that a universal assumption is present in every case study: that the outlines of the vaulted spaces and the rib system’s pattern projected to the plan are perfectly regular. However, in the case of the real net and stellar vault structures, this premise is not valid. This means that accepting it as an axiom limits the variety of plausible construction methods, because with a regular plan, the simplest method of constructing a net or stellar vault is to start with the longitudinal and cross-directed coordinates of the rib junctions and add the vertical coordinate as a second step, as presented in these books. However, based on our previous research, this was not the only method used in the Gothic period: Other construction strategies can be identified from the geometric analysis of real structures, even if they are not recorded in written sources.37
In this article, we use the term ‘stellar vault’ (‘Sterngewölbe’ in German) for the vaults we have studied. In most early and contemporary sources, stellar vaults are defined as ribbed vaults that have a rib system with a ‘star-like’ pattern on plan view. They are often described as ‘spherical’ vaults, but as we discuss in the ‘Results’ section, their global geometry does not necessarily have spherical qualities. Stellar vaults and net vaults are often discussed together and distinguished by their rib patterns.
METHODOLOGY
Survey technique
We used a point cloud generated by terrestrial laser scanning to perform an accurate geometry-based analysis of the stellar vaults in the Košice Cathedral. We used a Leica BLK 360 scanner and processed the data using Leica 360 Cyclon Register. With a scanner accuracy of 8 mm at 20 m,38 potential inaccuracies of 1–2 cm are expected based on the dimensions of the church spaces. Therefore, given the expected accuracy of masonry techniques, we maintain that these point clouds are suitable for the intended analyses.
The analysis methods we have applied to the point clouds are based on the analysis methods we have developed for the study of Late Gothic net vaults,39 with appropriate modifications for stellar vaults. Our analytical approach is based on the premise that the geometry of the vaults reflects the original construction techniques, including the use of temporary supporting structures. This enables the application of a ‘reverse-engineering’ methodology. Consequently, any irregularities in the current vault geometry can provide valuable insights into the original construction methods, as various techniques could have been used to achieve regular geometry. However, the presence of ‘regular irregularities’ limits the range of possible construction approaches. (For this, the post-building deformations of the vaults must be considered as well. In this case, such deformations could not be detected.)
About ‘mapping’ stellar vaults
The initial phase of the analysis entails the mapping of the vaults. This process involves the horizontal division of the vaults into multiple layers, followed by the projection of these sections into the plan view. The result of this process may supply information on the overall geometry of the webbing and the rib system, as well as the relationship between the ribs and the webs.
In the context of stellar vaults, the overall geometry of the webbing may manifest as a spherical surface – a dome –, indicating that the ribs are added solely for decorative purposes (‘pseudo stellar vaults’). (This outcome is analogous to that of the ‘pseudo net vaults’, where the webbing is presumably structurally a barrel vault).40 Alternatively, a ‘real ribbed vault’ may be evidenced, signifying that the rib system was built prior to the webbing and served as its form-work.41 In this scenario, the webs could be built on the ribs either with42 or without43 formworks. However, within a given vault’s webs, the combination of the two web-building methods was possible.44
These different building strategies result in different mapped plans. Pseudo-stellar vaults’ webbings exhibit section lines that maintain a perfect circularity upon crossing the ribs, deforming not at all. In contrast, real ribbed vaults’ web section lines curve towards the ribs before crossing them. In the case of webs built with formwork, horizontal web sections are nearly straight, potentially slightly convex or concave, while when formworks were used, web section lines curve significantly.
About analysing the global geometry of the rib systems
The second step of the analysis is the examination of the spatial positioning of the junction points. This is carried out in a visionary coordinate system positioned along the longitudinal, cross, and vertical directions of the vault.
As previously discussed in the ‘Literary framework’ section, the construction of ribbed vaults is subject to certain axioms that limit the possible variations. However, based on our aforementioned research, in Gothic practice, such methods were also employed that have no known written records – most notably, there were instances where the net vault’s construction did not start with its plan view, but rather its cross-section directed projection.45 This being the case with net vaults implies that stellar vaults may have also been built according to more diverse principles than initially believed. The potential for these methods to have been employed in the 19th century remains a viable hypothesis, given the absence of extant data on the subject.
Therefore, it can be posited that, per the typology that applies to net vaults’ rib systems,46 a typology may be established for stellar vaults as well. In this case, three possible outcomes may be considered. The first one is when the spatial positions of the junction points’ three coordinates can be independently determined, i.e. the construction method for any one coordinate can be reconstructed without relying on knowledge of the others.
The second one is when the rib system’s geometry suggests the primacy of the plan view over the vertical dimension. Here, determining the vertical coordinate of a rib junction point requires knowledge of the position of its plan projection. In practice, examples of this method may include the determination of the vertical coordinate of the junction points by projecting their plan position onto a spherical surface,47 or the application of the longest route principle48 (see later).
In the context of stellar vaults, a third theoretical outcome emerges: the construction of the vault commences on a vertical plane, presumably a plane parallel to the walls of the vaulted space. This analogy is substantiated in the case of net vaults;49 however, no known stellar vault example has been identified to date.
It is noteworthy that theoretical depictions in literature often illustrate rib systems with outlines, resulting in a single point representing a rib junction element’s ‘position’. This point presumably lies on the rib system’s lower surface, as that can be most easily positioned with the help of the temporary supporting structures. (The drawing of Hontañon supports this idea as well.)
About the individual geometry of the ribs
In the literature on late Gothic net and stellar vaults, the ‘Prinzipalbogen’ principle has been the subject of much discussion.50 This principle posits that each rib within a given net or stellar vault has identical curvature, with the objective of standardising ribs for prefabrication, thereby facilitating the construction process.51 While some studies have cast doubts on this theory,52 empirical evidence from actual cases supports both its presence53 and absence54 in Late Gothic architecture. Consequently, the verification or refutation of this principle in the context of Neogothic stellar vaults (as case studies) emerges as a pivotal undertaking, thus the measurement of the curvature of the radius of individual ribs constitutes the third step in the analysis method.55
Overall, this point-cloud-based analysis results in a precise geometric description of the given net vault. Based on that, further deductions may be made about the original construction and building methods.
RESULTS
Re-constructing the vault patterns on a theoretical basis
After reviewing the construction strategies that have been presented in the literature, we applied the most widespread methods to the rib patterns represented in the Saint Elisabeth Cathedral of Košice. For the ‘re-modelled’ vaults, the rib patterns of the transept and the main nave were used: the former as an example of a rectangular plan that is close to a square, and the latter as an example of an elongated rectangular plan. This resulted in theoretical rib systems. As previously discussed, multiple different web-building techniques could be applied to these rib systems resulting in a great variation of theoretical vaults of the same pattern.
Ranisch
In Beschreibung Aller Kirchengebäude Der Stadt Dantzig,56 Bartel Ranisch gave his slightly varying methods for constructing the spatial geometry of stellar vaults. The principle of these methods is very close to the ‘Prinzipalbogen’ principle. However, as Elena Pliego de Andrés has noted, the precision of the accompanying drawings is not consistent.57 For the vault over the altar in St. Catarinen, which is pictured in figure No. 3 on page 53 in Ranisch’s book,58 in her article, Pliego provides a transcription and a translation, as well as a step-by-step construction method.59 For re-modelling Ranisch’s method following the patterns of the Saint Elisabeth Cathedral, we adhered to the instructions provided for this particular structure.
The rib pattern of the vault presented by Ranisch and Pliego is more complex than the rib pattern of the vaults in the Cathedral of Košice. Therefore, the steps taken for the construction of the simpler vaults of the Saint Elisabeth Cathedral are detailed herein.
For the rib pattern of the crossing’s vault, we assigned letters (a–e) to the rib junction points. (For simplification, we assigned the same height [c] to all the junctions adjacent to the crown point [a].) We draw a circle quadrant with a radius of ab length (Fig. 2a).
Then, a vertical line was drawn in an ac distance from point a’, and ac distance was measured from point a. Where the latter intersects the original circle quadrant, the height of point c is given. This height was then projected to the vertical line. Then, from midpoints a and c, circles were drawn with a radius of ab distance, that is, our ‘Prinzipalbogen’. Their intersection resulted in the midpoint of the circle segment that determines the curvature of rib ac (Fig. 2b).
Then, a vertical line was drawn in a cd distance from point c’, and ad distance was measured from point a. Where the latter intersects the original circle quadrant, the height of point d is given. This height was then projected to the vertical line (Fig. 2c).
Repeating the same steps, point e and the adjacent rib-curvatures’ midpoints can be constructed (Fig. 2d).
The same method was carried out with the rib pattern of the main nave’s vaults (Fig. 3a–b).
Finally, the results of the 2D construction steps were applied to visualise the 3D rib systems (Fig. 4a–b).
The resulting rib systems’ junction points and ribs do not fit on the surface of a sphere.
Construction process based on Ranisch’s method, using the rib pattern of the crossing’s vault. The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
Construction process based on Ranisch’s method, using the rib pattern of the main nave’s vaults. The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
Remodelled 3D rib systems based on Ranisch’s method, a) using the rib pattern of the crossing’s vault b) using the rib pattern of the main nave’s vaults. The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
Hoffstadt
In Gotisches ABC-Buch,60 and the figures accompanying the written work,61 Hoffstadt defined multiple methods for constructing Gothic vaults. On sheet XIV.A., figures 1–4 present a similar method to Ranisch. In figure 5,62 the ‘principle of the longest route’ is shown. We applied this method to the rib patterns we examined:
Construction process based on Hoffstadt’s method (principle of the longest route), using the rib pattern of the crossing’s vault. The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
First, on the plan of the rib system, the longest rib-route between the impost and the crown point (without ‘turning back’) must be found. Then, the lengths of the segments of this line were measured along a straight line. Next, a circle quadrant was drawn with a radius equal to the sum of the segments’ lengths (Fig. 5).
Then, the individual segments’ endpoints were projected to the circle quadrant. This results in the height values of the junction points. The radius of the circle quadrant is used as the ‘Prinzipalbogen’ for the vault. It must be noted that the arches of the formerets often protrude from the compact form of a stellar vault. Hoffstadt shows a solution for a polygonal space, where the height of the formerets is determined by a regular triangle drawn over the length of the polygon’s side. However, in our case, this method is not suitable, as it would result in transverse arches that are higher than the adjacent junctions, which are closer to the crown point – thus in a non-tectonic solution. Instead, we constructed these junctions the same way as any other junction in the vault (Fig. 5).
The same method was carried out with the rib pattern of the main nave’s vaults (Fig. 6).
Finally, the results of the 2D construction steps were applied to visualise the 3D rib systems (Fig. 7a–b).
The resulting rib systems’ junction points and ribs do not fit on the surface of a sphere.
Construction process based on Hoffstadt’s method (principle of the longest route), using the rib pattern of the main nave’s vaults. The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
Remodelled 3D rib systems based on Hoffstadt’s method (principle of the longest route), a) using the rib pattern of the crossing’s vault b) using the rib pattern of the main nave’s vaults. The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
Ungewitter
In Lehrbuch der gotischen Konstruktionen,63 Ungewitter also presents his interpretation of Ranisch’s method64 and the principle of the longest route65 (for each rib junction he calculates with a route that goes through that junction). He notes that in the case of stellar vaults, the latter method is particularly useful if there are no ribs along the whole of the diagonal.66
On Tafel XIX, figure 146, Ungewitter shows how to construct a stellar vault whose ribs fit on a spherical surface, however, he also notes, that a vault like this is unlikely in practice, as the differently directed ribs would have different curvatures.67 Nonetheless, we applied this method to the examined rib patterns. The results are presented in Fig. 8a–b.
Remodelled 3D rib systems based on Ungewitter’s Fig. 146, a) using the rib pattern of the crossing’s vault b) using the rib pattern of the main nave’s vaults. The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
On Tafel XIX, figure 147, Ungewitter presents a stellar vault that is constructed with a ‘Prinzipalbogen’ equal to the curvature of the ribs on the diagonal. As among my examined vaults, only the vaults of the main nave have continuous diagonal ribs, we applied the method on that rib pattern. The construction steps were as follows:
I defined the curvature of the diagonal rib as a semi-circle. Then distances between the midpoint of the vault’s plan and the junction points (on plan) were measured on the horizontal axis (that represents the height of the imposts). The endpoints of these segments were projected to the semi-circle resulting in the heights of the junctions (Fig. 9a).
Then, we carried out the same basic steps in a case where the ribs of the diagonal form a pointed arch, as Tafel XIX, figure 149 shows.68 In this case, for the ratios of the pointed arch we used an approximation from the real vaults (see later) (Fig. 9b).
The results of the 2D construction steps were applied to visualise the 3D rib systems (Fig. 10a–b).
Construction process based on Ungewitter’s a) Fig. 147 and b) Fig. 149, using the rib pattern of the main nave’s vaults. The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
Remodelled 3D rib systems based on Ungewitter’s a) Fig. 147 and b) Fig. 149. The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
Although Ungewitter shows how to construct the midpoints of the individual ribs’ curvatures (the same method as described in the section about Ranisch’s method), it must be noted that in both cases, these midpoints have no practical use: Knowing only the ribs’ curvatures (from the ‘Prinzipalbogen’ principle) and the height of their end-junctions, centrings can be built and the ribs can be carried out.
Point cloud-based vault analysis
The vault of the crossing
‘Mapping’
Examining the mapping of the crossing’s vault (Fig. 11), we found that the horizontal section lines of the webs conform to the ribs’ positions. These section lines are clearly curved, suggesting that in the case of this vault, the rib system was built first, acting as centrings for the webs that were presumably built without formworks. Regarding the webs that are adjacent to the neighbouring vaults (‘lunettes’), the building technique of the webs was found to be identical to that of the other webs. Furthermore, we found that certain adjacent webs’ section lines appear to be continuous, suggesting that these webs may have been built simultaneously and are structurally interconnected. However, it is also evident from the mapping, that this continuity is not detectable in the case of all adjacent webs, and – as previously emphasised – the tracing of the section lines is not independent of the positions of the ribs. In summary, the vault is a real ribbed vault, with webs built on the ribs, and not a dome structure.
Mapping of the crossing’s vault. The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
Rib system’s global geometry
For an easier understanding of the construction steps presented below, alphanumeric signs have been assigned to the ribs of the vault, as presented in Fig. 12.
Legend to the crossing’s vault. The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
Analysing the plan view of the crossing’s vault, we found that the outlines of the vault form a regular rectangle, but not a square, as the length in the east-west direction is shorter than in the north-south direction. The ‘rib pattern’ projected onto the plan view displays highly regular features, exhibiting symmetry to both the north-south and the east-west axes. As previously explained, highly regular geometries possess the characteristic of constructibility through multiple methods. The following section presents one possible approach for constructing the plan, although it is imperative to emphasise that alternative methods are equally valid:
First, the mid-points of the rectangle’s sides and the axes of the outline rectangle can be defined (Fig. 13a).
Then, Thales circles can be drawn to the east-west directed sides. The intersection of the axis lines and the Thales circles results in the junction points next to the crown point on the south-west axis. Connecting this junction point to the correspondent corner points, the plan of the ‘diagonal’ ribs can be drawn (B and H ribs) (Fig. 13a).
Then, by dividing the vault into 6 segments along the east-west axis, the north-south directed coordinates of the junctions can be defined. The intersection of the segmenting lines closer to the crown of the vault and the rib defined in the previous step (B and H ribs) results in two further junction points (Fig. 13b).
Then, forming a regular square on the east-west axis (using the ribs and junctions resulting from the previous steps) results in ribs G. The junction points formed by the intersection of the adjacent ribs Gs can be connected to the corner points and the mid-points of the east-west directed sides. This step results in ribs L and K (Fig. 13c).
Then, mirroring ribs L to ribs B, the line of ribs C and E can be constructed. The intersection of this line and the previously constructed axes and dividing lines define the remaining junction points, thus the construction of the rib system’s plan is finished (Fig. 13d).
All in all, we found that the plan view of the vault’s rib system is highly regular and constructible.
Construction steps for the plan of the crossing’s vault. The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
Analysis of the sections of the crossing’s vault demonstrated that the rib junction points fit on the surface of a sphere. The midpoint of this sphere is in the midpoint of the bay’s outlines and it is at the height of the imposts. The radius of this sphere corresponds to the distance of the bay’s midpoint and the innermost points69 of the imposts on plan view (Fig. 14a–b).
Construction of the vertical coordinates of the junctions in the crossing’s vault. The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
Individual rib geometry
Examining the curvatures’ radii of the ribs in the crossing’s vault, we found that the curvatures are not uniform in the entire rib system. However, orderliness can be observed in this regard: in the case of the ‘B’ and ‘H’ ribs (as defined in Fig. 12), the curvatures are uniform and equal to the sphere’s radius. The remaining ribs (except for the formerets) demonstrate curvatures that can be accepted as uniform (notwithstanding certain deviating values), however, their standard deviation is still notable (5.86m average with 17.2cm standard deviation). Nonetheless, a significantly lower standard deviation was observed in ribs A, C, E and L within each group, and in groups C, E and L combined (5.635, average and 9.7cm standard deviation).
Therefore, in this vault, curvatures may be considered as a value that belongs to rib types as per their functions in the rib system. The groups exhibiting significantly lower standard deviation may indicate ribs of primary importance and less significant ribs within the system. However, it also must be noted that rib groups with larger standard deviations correspond to shorter ribs, where measuring the rib curvatures is notably more uncertain.
Regarding the chord lengths and arch heights of the ribs, we found that these may be seen as uniform as per their functions in the rib system. It is also notable that in the case of ribs B and H, and ribs C and E, the standard deviation is significantly smaller when these values are examined as if the ribs that are the continuation of each other were one rib, than when they are regarded as separate ribs. This suggests that these ribs (B and H, and ribs C and E) may have been constructed and built as one longer rib rather than two shorter ones.
All in all, based on the individual characteristics of the ribs, and their function in the rib system, ribs B and H (as one), ribs C and E (as one) and ribs L may be the primary ribs in the system (Fig. 15).
Primary ribs in the crossing’s vault. The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
The vault of the transepts
For an easier understanding of the construction steps presented below, alphanumeric signs were assigned to the ribs of the vault, as presented in Fig. 16. The figures illustrating the analysis of the transept vaults use the point cloud of the northern transept. As the two transept vaults are identical, the statements apply to the southern transept vault as well.
Legend to the transept’s vault. The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
‘Mapping’
Examining the mapping of the transept vaults (Fig. 17), we found that the horizontal section lines of the webs are curved, and they are conforming to the ribs’ positions. Therefore, it is likely that the rib system was built first, and it served as centring for building the webs. Because of the significant curvatures of the webs, they could be built without formworks. In this vault, the adjacent webs’ section lines seem to be continuous only in the webs that are connected to the crown point. In the case of the lower webs that are on the south-north and east-west axes, the section lines show that these webs have slight edges in their symmetry axes. Regarding the webs that are adjacent to the neighbouring vaults (‘lunettes’), the construction technique of the webs was identical to the other webs. Based on the mapping we see it proved that the vault is a real ribbed vault, with webs built on the ribs, and not a dome structure.
Mapping of the transept’s vault. The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
Rib system’s global geometry
An analysis of the plan view of the crossing’s vault reveals that the outlines of the vault form a regular rectangle, rather than a square, due to the length in the east-west direction being shorter than in the north-south direction. The ‘rib pattern’ projected to the plan view displays highly regular features, exhibiting symmetry to both the north-south and the east-west axes. As previously mentioned, highly regular geometries possess the characteristic of being constructible in multiple ways. Below, we present one possible example for constructing the plan, however, it must be emphasised that other methods may be equally valid.
In the first step, the mid-points of the rectangle’s sides and the diagonals of the rectangle can be found.
Then, the ribs framing the lunettes (ribs B and C) can be constructed by trisecting the 90° angles of the outline’s corners. Thus, the upper junction points of the ‘lunettes’ can be found (Fig. 18a).
Then, four circles can be drawn: their midpoints are the upper junction points of the ‘lunettes’ and their radius is the fourth of the diagonal of the circle that can be drawn on the innermost points70 of the imposts on plan view. (The latter circle is significant in constructing the vertical coordinates of the junctions as well – see below.) The intersections of these and the previously defined lines result in all the junction points in the rib system (Fig. 18b). The plan view of the vault’s rib system was found to be highly regular and constructible.
Construction steps for the plan of the transept’s vault. The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
Upon analysis of the sections of the crossing’s vault, it was determined that the rib junction points fit on the surface of a sphere. This sphere has its midpoint in the middle of the vault’s outline and its diagonal is the same as that of the circle defined in the last step of the plan view’s construction (see above) (Fig. 19). This sphere that determines the geometry of the transept vault’s geometry is congruent to the sphere that determines the crossing vault’s geometry.
Construction principle of the vertical coordinates of the junctions in the crossing’s vault. The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
Individual rib geometry
Examining the curvatures’ radii of the ribs in the vault of the transept, we found that they are not uniform regarding the whole rib system. However, groups with approximately uniform curvatures can be defined in the rib system. These groups correspond to the ribs’ function in the rib system: ribs D, F and ribs E1, E2, E5, E6, G2, G4 (thus, those ribs among E and G ribs that are on the north-south axes) form the first such group (average: 5.22 m, standard deviation: 10cm), ribs B, C, G2 and G4 form the second one (average: 5.76m, standard deviation 3.1cm). Ribs E3, E4, E7 and E8 display an average curvature of 4.82m with a standard deviation of 8cm.
Regarding the chord lengths and arch heights of the ribs, we found that these may be seen as uniform as per their functions in the rib system. (However, a certain rib belonging to the shorter [east-west] or the longer [south-north] axis plays a role in this – sometimes ribs of the same ‘function’ form sub-groups, in the same cases that were highlighted in connection with the curvatures.)
The vaults of the main nave
Mapping
Examining the mapping of the vaults in the main nave (Fig. 20), we found that the horizontal section lines of the webs are curved, and they are conforming to the ribs’ positions. Therefore, it is likely that the rib system was built first, and it served as a centring for building the webs. Because of the significant curvatures of the webs, they could be built without formworks. In this vault, the adjacent webs’ section lines seem to be continuous in the adjacent webs that are connected to the same impost. In the case of the webs that are on the east-west axis, the section lines show that these webs have slight edges in their symmetry axes. Regarding the webs that are adjacent to the neighbouring vaults, the construction technique of the webs was identical to the other webs. However, in the case of the lunettes that are connected to the east-west directed walls, the webs’ section lines clearly show that the surfaces are close to perpendicular to the walls – these webs clearly emerge from the overall shape of the vault. Based on the mapping we see it proved that the vault is a real ribbed vault, with webs built on the ribs, and not a dome structure (not even disregarding the lunettes).
Mapping of the main nave’s vaults – presented in the figure the easternmost three bays. The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
Rib system’s global geometry
For an easier understanding of the construction steps presented below, alphanumeric signs were assigned to the ribs of the vault, as presented in Fig. 21.
Legend to the main nave’s vaults – presented in the figure the easternmost three bays. The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
Regarding the plan of the main nave’s vaults, we found that the outline of the easternmost one is not congruent with the other five: it is longer in the east-west direction. The rib pattern of the vaults on the plan view is quite simple. It may be constructed by drawing the diagonals of the bays and the diagonals of the bays’ northern and southern sides (separately) too (Fig. 22a). This construction results in a quite accurate replication of the vault’s plan. However, in some cases, we found that the ribs that are supposedly on the same line, in reality are not (e.g. the western-most bay on the eastern side of the crossing, or the easternmost bay on the western side of the crossing) (Fig. 22b). The position of the junction on the plan may be determined in another way as well. The cross-directed coordinates of the junctions can be defined by dividing the vaults’ southern and northern sides by three along the north-south axes (in the case of the junctions closer to the crown), and by two (the upper junction of the lunettes). The longitudinally-directed coordinates can be found by defining the north-south axes of symmetry of the bay and dividing the bay’s eastern and western sides by three (Fig. 23). This method results in the exact positions of the junction points. Therefore, we think that as the practical method that was employed for constructing the vault, the latter solution is the most likely, however, as the former one also resulted in a quite exact picture of the real vault, that method cannot be excluded from the options. All in all, the plan view of these vaults was likely constructed, and – as they have quite regular geometries – the exact method of their construction cannot be established with certainty.
Construction steps for the plan of the main nave’s vault based on diagonals – presented in the figure the easternmost three bays. The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
Construction steps for the plan of the main nave’s vault based on axes that adjust to the nave’s walls – represented in the figure the easternmost three bays. The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
Examining the vertical coordinates of the junction points, we found that the vaults above the uniform five bays are identical. The position of the crown point can be constructed by intersecting two identical circles that also determine the form of the diagonal ribs (ribs D). The midpoints of these circles fall in the height of the imposts. Their distance from the vertical mid-axis of the vault is 2/9 of the distance between the vault’s mid-axis and the imposts. The circles’ diagonal is likely determined by the height of the main nave, as it is equal to the height of the lowest level of the main nave’s inner facade (Fig. 24).
Construction principle of the crown point in the main nave’s vault (general). The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
The easternmost bay’s vault is higher than the other vaults in the main nave. Furthermore, the two diagonal ribs on the eastern side and the two on the western side are not congruent. Although the crown point can be constructed as the intersection of their circle segment arches in this case as well. The eastern ones have a radius that is equal to the length (east-west direction) of the easternmost bay. The midpoint of the circle is in the height of the imposts and its distance from the midaxis is 1/5 of the distance between the midaxis and the imposts. The western two diagonal ribs of the easternmost bay also follow circle segments. The radius of these circles equals the distance between the bay’s midpoint and imposts on the plan view. However, the midpoint of the circle segment is not in the height of the imposts, but higher. Thus, the construction of the vertical dimensions of the easternmost bay differs from the other bays in the values it presumably used for the construction (Fig. 25).
Construction principle of the crown point in the main nave’s vault (easternmost bay). The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
As we described above, the circle segments on the diagonals are not only important because their intersection results in the crown point, but also because the diagonal ribs’ lower surfaces follow the form of these segments. Therefore, by defining the vertical positions of those junction points that are on these diagonals, the junction’s plan view can be projected up to the diagonal arch.
The apex of the transverse arches between the main nave’s bays can be constructed based on the same principle as the crown points. It is the intersection of two congruent arches that defines the correspondent ribs’ lower surfaces as well. The circle segment’s radius is equal to the radius of the circumscribed circle of the main nave’s bays (except the easternmost ones). The midpoints are in the height of the imposts and its distance from the midaxis is 2/7 of the distance between the midaxis and the imposts.
The vertical coordinate of the upper junction of the lunettes may be constructed using circles that have a radius that equals the (east-west) length of the main nave’s bays (except for the easternmost one). The midpoint of the circles is at the height of the imposts, and their directions adjust to the direction of the ribs that were constructed on the plan view (Fig. 26).
Construction principle of the vertical coordinates of the junctions in the main nave’s vault – presented on the first bay east from the crossing. The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
Individual rib geometry
For the examination of the individual ribs’ geometry in the main nave’s vaults, we carried out our measurements on the vault that is next to the crossing to the east. Examining the curvatures’ radii of the ribs we found that the ‘Prinzipalbogen’ principle does not apply to the structure, however, the curvatures in the case of ribs A, B, C and D are quite uniform within each group. This cannot be stated about ribs E and F, although their chord lengths and arch heights are just as uniform as those of the other rib types in the vault.
DISCUSSION
Comparative analysis of the construction methods of the written sources and the real rib systems
In the case of the crossings’ and the transepts’ vaults, we concluded that their junction points fit on the surfaces of identical circles, thus these vaults may be regarded as examples of those stellar vaults whose construction process started with their plan view. Using spheres for determining a stellar vault’s geometry is an idea well presented in literature. However, as Ungewitter also highlighted, even if the junctions fit on a spherical surface, the ribs likely do not.71 On the other hand, unlike the construction methods presented by Ranisch, Hoffstadt and Ungewitter, the ‘Prinzipalbogen’ principle does not apply to the stellar vaults in the Saint Elisabeth Cathedral of Košice. Rather, there seem to be primary ribs in the system that are coordinated in their curvature and secondary ribs that are more randomly formed. Interestingly, this technique is proved to be around in the Late Gothic period as well, as measured and proven by Clemens Voigts in the case of the Georgskirche in Augsburg.72 However, this method was not recorded in the influential 19th-century works cited in this article.
In the case of the main nave’s vaults, the rib systems’ geometries are clearly determined by the diagonal arches, which are pointed arches. This underlying principle is demonstrated by Ungewitter.73 However, in the case of the cathedral of Košice, the following steps are not those presented by him,74 rather, in this case as well, it seems that there are primary ribs that determine the general geometry of the rib system. Thus, these vaults may be regarded as examples of the second theoretical outcome described in the Methodology section.
All in all, it may be said that the general concepts for constructing the main geometry of the examined vaults’ rib systems are presented in the literature of the era. However, not using “Prinzipalbogen” is scarcely considered in these works, and even then, is regarded as the impractical consequence of an idea (e.g. stellar vault fitting on a sphere),75 rather than a practical solution. This seems to be a definite difference between the theoretical models and the real vaults. A possible reason for this could be that – accepting the results as proof for using primary and secondary ribs in the systems – building practices overwrote the less crucial points of the theoretical considerations. In this context, the proof for the use of the same ‘less perfect’ method in the Gothic era76 may suggest that the practices lived through even though the leading theoretical works did not record them. However, deciding on this question requires further research.
Evaluating the implications of the remodelled vaults
Regarding the remodelled rib system geometries and the real vaults, another implication must be discussed. Using the same rib pattern and the widely accepted literary ideas of vault construction, numerous rib systems may be constructed. Adding to this the apparent uncertainty of the invariable use of the ‘Prinzipalbogen’ principle, we may find that knowing the rib pattern and the curvature of certain ribs still allows for multiple outcomes. Moreover, to each rib system, multiple webbing construction methods (thus, webbing geometries) may be applied. To illustrate the differences between the models, Fig. 27 shows the models based on the crossing’s rib pattern projected together.
Remodelled 3D rib systems based on Ranisch’s method, using the rib pattern of the crossing’s vault projected together. Continuous line: longest route principle; dashed line: Ranisch’s method; dotted line: projection to a sphere. The author’s work
Citation: Építés – Építészettudomány 53, 1-2; 10.1556/096.2025.00140
In our opinion, these considerations suggest that for understanding the structure of stellar vaults, both Gothic and Neogothic, the study of the exact geometry of the individual examples is crucial. Furthermore, they also indicate that in the case of stellar vaults that are only known from their remnants, creating a fully authentic (theoretical) reconstruction in terms of geometry and structure may not be possible.
CONCLUSION
In this article, 19th-century stellar vault construction techniques and the case study of the stellar vaults in the Saint Elisabeth Cathedral of Košice were discussed.
I presented 2D construction steps and 3D models based on the literature’s most widely cited core works applied to the rib patterns of the Košice Cathedral’s stellar vaults. This resulted in a variety of different rib systems, suggesting the necessity to examine individual case studies if structural data is required about a certain vault.
I also presented an analysis method for the exact, point cloud-based geometric analysis of stellar vaults. This method is based on the one we developed for net vaults during our previous works and considers the overall geometry of the webbing and that of the rib system, as well as the connection between the webs and the ribs.
Then, the presented analysis method was applied to the stellar vaults in the Saint Elisabeth Cathedral of Košice. This resulted in the exact geometric description of these vaults. Based on that, we attempted to deduce the original construction techniques that were used. Apparently, the crossing’s and the transepts’ rib junction points fit on the surface of identical spheres, while those of the main nave were constructed based on a pointed arch that determines the diagonal ribs’ geometry. However, we also found that the ‘Prinzipalbogen’ principle does not apply to these vaults, rather ribs of primary importance and secondary importance can be differentiated. This may be highlighted as a difference between the theory and practice of Neogothic net construction.
Afterwards, we carried out a comparative analysis of the models and the real vaults and considered the plausible reason for their differences.
Finally, we evaluated the implications of the remodelled vaults, in terms of the importance of examining the individual vault’s geometry, and the theoretical possibility of fully authentic (theoretical) reconstructions with regard to geometry and structure may not be possible.
ACKNOWLEDGEMENTS
I would like to express my gratitude to my supervisor, János Krähling PhD for his support, as well as to László Daragó DLA for his advice during the research, and to Balázs Halmos PhD for his help with the point cloud.
The research was supported by the ‘Egyetemi Kutatói Ösztöndíj Program’ (EKÖP), the Ministry of Culture and Innovation (Kulturális és Innovációs Minisztérium) and the National Research, Development, and Innovation Fund of Hungary (Nemzeti Kutatási, Fejlesztési és Innovációs Alap).
Examples of such sources include the plan documentations of the Wiener Bauhütte (Böker 2005. 15), the 16th-century sketchbook from Dresden (Bucher 1972), the drawing of Rodrigo Gil de Hontañon (Huerta 2012), the Manuscript Ms. 12686 of the National Library of Spain (Baño and Glera 2020), and the sketchbook of Master WG (Bucher 1979).
Weiss 1857. 238; Wick 1936. 20, footnote 3.
Václav Mencl, as quoted by Marosi 1967. 582.
Marosi 1967. 575–576.
Dragóner 1880. 100; Wick 1936. 163–164.
Dragóner 1880. 104.
Henszlmann 1884. 195; Wick 1936. 166.
Henszlmann 1884. 198.
Sisa 2005. 95; Ungewitter 1901. Tafel X. Fig. 83. also shows the plan of the original vaults, described as ‘Netzgewölbe’ (net vaults), not as ‘Sterngewölbe’ (stellar vaults).
https://shop.leica-geosystems.com/leica-blk/blk360/blog/blk360-frequently-asked-questions (Accessed 30 October 2023)
Roth 1905. 36; Fabini 1999; Harsányi 2001. 302.
This method was first described by Saunders in 1814 and specified later by Willis in 1842 (as quoted by Wendland 2007. 342.) and mentioned by Ungewitter 1901. 37.
In this case, full-surface formworks were placed on the ribs, on which the webs were built (Voigts 2021. 78) resulting in quite flat web surfaces (Voigts 2021. 78; Wendland 2007. 342) and the occasional sagging of the formwork may cause the introflexion of the webs (Schuller 2016. 474).
In this case, either only a centring (such device is pictured e.g. in Viollet-le-Duc (1854–68. Tome 4. 106. Fig 58); Ungewitter (1901. 117); Fitchen (1961. 101, Fig. 40)) was used or not even that (e.g. Fitchen (1961. 69) – based on Lassaulx (1831) – pictures how a stone-weighted rope device can stabilize the unfinished courses), and each course of the web worked as a self-supporting arch after completion (Voigts 2021. 79).
Voigts 2014. 250; Voigts 2021. 80–81.
Presented in detail in Jobbik–Krähling 2024b.
Ungewitter 1901, 66.
Bucher 1972. 47; the treatise of Lorenz Lechler as quoted by Shelby–Mark 1979. 125.; Hoffstadt 1840b. XIV.A/5.; Ungewitter 1901. 67–68; Meckel 1933. 108.
It first appeared in written sources in the 16th century, in Lorenz Lechler’s treatise for his son (as quoted by Shelby–Mark 1979. 125) and in the manuscript of Jacob von Andernach (as quoted by Müller 1974. 65–66); then appeared in Ranisch 1695; Hoffstadt 1840b; Meckel 1933; Müller 1990; Tomlow 1991, etc.
Renn et al. 2014. 71; Vidal 2017. 1007.
Lassaulx 1835, as quoted by Wendland 2012. 106.
Voigts 2015. 56–57; Jobbik–Krähling 2023b.
In this article, a rib is defined as the structure between two junction elements, and the elements composing a rib are referred to as ‘rib elements’.
Pliego 2017. 413.
Ranisch 1695. 53.
Pliego 2017. 410–414.
Hoffstadt 1840b. XIV. A. Fig. 5.
Ungewitter 1901. Tafel X. Fig. 82.
Ungewitter 1901. Tafel X. Fig. 87; Tafel XIX. Fig. 151.
Ungewitter 1901. 68.
Ungewitter 1901. 66.
Description to the figure: Ungewitter 1901. 67.
The innermost points of the imposts are defined at the lowest point of the rib profiles.
The innermost points of the imposts are defined at the lowest point of the rib profiles.
Ungewitter 1901. 66.
Voigts 2015. 56–57.
Ungewitter 1901. Tafel XIX. Fig. 149.
Ungewitter 1901. 67.
Ungewitter 1901. 66.
Voigts 2015. 56–57.
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