The importance of software for the stereometric analysis of artworks. A case study with Pieter Saenredam

This paper, which is a development of a part of our Ph.D. project, completed in 2008, tells us the importance of using computer-aided design software in the analysis of works of art and particularly in easel paintings that show compositions with geometric perspective. In the text we emphasize the importance of the computer-aided drawing tool that allows for verifying the coherence of the construction of the perspective representation of certain paintings from a certain historical period. In this way, we try to demonstrate that computer-aided design has the advantage of being able to signal with great accuracy the coherence of the geometric lines of certain painting compositions that represent real or illusory spaces and, in this way, classify the degree of erudition of certain workshops and artistic currents. In summary, we show in this article that computer-aided drawing can be an excellent tool for art historians, allowing them to conclude results in relation to the preparatory drawing that was practiced in the pictorial workshops of one or several historical times. Finally, in the text we present a case study with the analysis of a painting by the Dutch artist Pieter Saenredam. This artist is a paradigmatic example of the exhaustive use of linear perspective drawing that organises the composition of his paintings. For this reason, we present him here as a case study.


INTRODUCTION
The analysis of works of art, and more specifically those in which the use and visualization of geometries and one of its branches, that is drawing in linear perspective, is more evident, becomes more effective with the aid of computer-aided design software.The analyses carried out with the aid of vector drawing programs allow faster and more rigorous analysis of the perspective constructions of the authors of the works, allowing the verification of the degree of erudition and knowledge, in the field of geometric and perspective design, of certain workshops and certain authors.Such is the case of the painting by the Dutch artist Pieter Saenredam, where an analysis carried out by us with the aid of computer software allowed us to verify that this artist knew the theories and certain geometric methods that were disseminated in the Netherlands in his time.This text presents a case study and the application of computer technologies in the study and analysis of works of art, in this case, in the work of the aforementioned Dutch artist.

THE HISTORIOGRAPHICAL TRADITION AND THE FORTUNE OF COMPUTER-AIDED DESIGN
We know that since the fortune of the discovery of linear perspective in the middle of the Renaissance period and even before, when some painters were already experimenting with linear perspectives, there were many authors who, through various tricks or artifices, tried to represent the illusion of three-dimensional spaces, real or illusory.From the paintings that survive to our days, from the frescoes of Pompeii to representations by Giotto, Masaccio, Raphael, Leonardo, Domenico Veneziano, Piero della Francesca, Antonello da Messina, Bellini, Vermeer, among other authors, we know that the respective authors used preparatory drawing that preceded and arranged the pictorial layers.In these preparatory drawings, in an attempt to represent spatial stereometry, in the linear perspective applied to represent the illusion of living spaces, sacred or profane, the artists used more artisanal methods than more scientific methods.In fact, some used herringbone layouts, others applied mathematical relationships based on proportions, some of them based on the theorem attributed to Thales, especially for the construction of floors with quadrilateral mosaics.This seems to have happened to Giotto and Pietro and Ambrosio Lorenzetti and other artists even later, who assimilated and applied the rules of Brunelleschi's and/or Alberti's perspective in their work.In the same historical times, especially in the Renaissance and Baroque periods, we often see painting workshops that mastered the science of linear geometric perspective, with its precise rules, from the application of the costruzione legittima method to the application of the points from distance method.By contrast and in these same historical times, other workshops did not have the same degree of erudition and contrasted with each other in the use and effectiveness of the preparatory conceptual design [1].In this sense and follow-up, we know that there were several historians who addressed the theme of perspective in Renaissance painting and in other periods.
From Panofsky in 1927 with his Perspective as a Symbolic Form, to Miriam Schild Bunim, with Space in Medieval Painting and the Forerunners of Perspective in 1940, to John White in 1957, with The Birth and Rebirth of Pictorial Space, Alessandro Parronchi in 1969, in the work Studi Sulla Dolce Prospecttiva, or later with Martin Kemp in his 1994 work [8], The Science of Art.Optical themes in western art from Brunelleschi to Seurat, among others, that we found that the authors when carrying out the study of linear perspective, even in the analysis of works of art and paintings, most of the time used drawings made manually or analogically, many of them even designed by outside authors.Only more recently, some authors, such as Lino Cabezas Gelabert [2], Joaquim Garriga I Riera [3] and Andrès de Mesa Gisbert [4] in our neighbouring Spain, among other authors in other geographical areas, carried out more detailed and careful studies, using computer-assisted design to study linear perspective and for the analysis of paintings that contain geometry and linear perspective.There is an enormous advantage in the use of this digital drawing software for the analysis of paintings, as they allow faster and more precise tracing to carry out studies by superimposing lines on the images of the digitized paintings themselves.In certain cases, the advantage of these computer-assisted drawings also allows the restitution and reconstitution of environments or spaces simulated three-dimensionally by the painters, in the form of plans, sections and elevations in multiple orthogonal projections.
Obviously, these case studies of paintings with linear perspective can be carried out analogically, however, with computer-assisted drawing there is the advantage of being able to eliminate and improve lines, being able to build new ones without any traces or signs of amendments remaining in the drawings, in addition to making cleaner drawings and also with the easy advantage of extracting measurements that can be very useful to reach conclusions.There are several computer-aided design software programs in the market, from AutoCAD, to AutoSketch, to Rhinoceros, among others.Just turn on the software and use the commands to insert a pictorial image from a file into the drawing page and use other commands to quickly draw points, lines, straight lines and curves on the image, thus proceeding with the geometric analysis of the selected works.However, there is always the advantage of the authors who carry out the analytical studies of the pictorial works, having some knowledge of geometry, or of linear perspective, because without this knowledge it is difficult to make and understand the drawings of the analyses carried out.In addition, some geometric operations of spatial resolution and others, even if drawn with computer-aided design, are not resolved by the former, requiring knowledge, logic and geometric technical methods.This idea is also supported by Riccardo Migliari [5,6].

THE IMPORTANCE OF USING SOFTWARE COMBINED WITH GEOMETRIC KNOWLEDGE IN THE ANALYSIS OF THE PICTORIAL WORK. THE CASE OF PIETER SAENREDAM
The paintings of the dutchman artist Pieter Saenredam (1597-1665) reveals that the author had geometry knowledge and particularly in the linear perspective science.In fact, the author masterfully represents the illusory architectures represented in the paintings and even the scale of the costumes that inhabit them.The painter seems to have used the point of distance method in his work.The analysis carried out by us with the aid of computer-assisted design with the AutoSketch 10 software, confirms the coherent geometric design of this author in his volumes resolution of the illusory representation of paintings of church interiors, as was usual with the author.In Saenredam's geometry we glimpse treatise sources where the painter may well have acquired geometric knowledge, such as the distance-point method, with the distance points construction.We know that in 1583 Egnatio Danti published the Due Regole della Prospettiva Pratica de Vignola, where he once again describes the costruzione legittima rule and method, and the point of distance rule, the latter most likely also emerging from the construction based on the third points and the main point of Jean Pelérin Viator who published his perspective treatise in the year of 1505 (Figure 1, Figure 2).This author's treatise was known to some artists not only in Italy but also in other countries in Europe.To do this, consider several lines passing through the vertices of the geometric figures and proceed to the respective intersection of these lines directions, determining the points of figures [7].One of the most interesting aspects of this author's treatise arises in the scientific understanding of the point of distance concept, described by him for the first time.
Once the scientific concept of the costruzione legittima rule was definitively assimilated, but especially the distance point rule, simpler and less time-consuming, other treatises quickly appeared with applications of perspective constructions using the main point and the distance point.This is the case of the perspective treatise of Jean Vredeman de Vries, published in 1604, where the author, despite some scientific errors, presents some architectural scenography constructions with perspectives of oblique objects in relation to the frontal plane of the perspective frame, but also other examples with barrel and cross vaults, built using those main and distance vanishing points.It is interesting to note and verify that in an example that illustrates the intersection lines of two secant cylinders that model an edge vault, the author considers, in one of the examples, the two tangents of the respective level, at the apex of the vault, which corresponds to the highest point of the vaults, and the highest point of their section lines (Figure 3).Following the publication of Jean Vredeman de Vries perspective treatise, other treatises with representations made with the aid of the main point and the distance point, all of them centred on the interiors representation and architectural spaces and whose echoes and theories, it seems, were in tune with the painting of some plastic operators such as Pieter Saenredam (Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8).Pieter Saenredam (1597-1665) presents architectural compositions with unmatched coherence in this Dutch painting period, in interior and landscape scenes [8], also making use of the main point P and the distance points D e D1 (Figure 9).In one of several paintings by the author, more specifically the one that represents the interior of the church of Sant Anne in Haarlem, the painter proved to be familiar and in perfect harmony with the rule and use of the point of distance, in addition to the geometric skeleton of the painting being in perfect harmony also with the examples of perspectives described by Jean Vredeman de Vries, a countryman of the author (Figure 9, Figure 10 and Figure 11).Before analysing in particular this painting by Peter Saenredam, that represents the interior of the church of Sant Anne in Haarlem, it should be noted that in order to analyse the geometry and the stereometry of any paintings that contain drawings with linear perspective with the aid of software such as, for example, AutoSketch 10, we enunciate the following steps: 1. Insert the image of the painting on the software page screen or layout, importing it from a computer file; 2. With the help of the specific commands, firstly draw the horizon line and the main vanishing points seen in the painting; 3.After that and finally, with the marking of the referred horizon line and the vanishing points, we proceeded to other geometric calculations allowing to complete the drawings and thus to evaluate the drawing rigour of the authors of the paintings.Of course, for the drawing geometric analysis operation, it is essential that the researchers have knowledge in the geometric drawing and in linear perspective drawing areas.Only with this geometric knowledge it will be possible to carry out good analyses with results.The computer alone is not enough, it is just a work tool or a strong instrument work.
Back to the painting, that represents the interior of the church of Sant Anne in Haarlem, after an analysis carried out by us (Figure 10 and Figure 11), we verified that all the orthogonal and level lines that are respectively 90ºand 45ºwith the frontal plane of the perspective frame are rigorously drawn, as well as the very low Horizon Line, which allows the reader that obtain a more global reading of the upper part of the represented building.In this painting, at the same time, we verify that all the shapes modelled by the upper quadrilaterals present the respective two diagonal directions converging directly to the distance points D and D1, which proves that they are square in shape, as is the case of the quadrilateral ABCD that models the lower part of a roof located on the right side of the composition; as well the quadrilateral 1234, delimited by the upper and closest points to each other of the pilasters, located in the connection of those next to the capitals that delimit the crossing and that support the respective cover modelled by the intersection of two orthogonal secant cylinders and that thus form the known cross vault.It should be noted that in this sector of the cross vault, despite the respective two front arches and the two opening profile arches that are being correctly designed, however, the intersection lines of the two secant cylinders that model the respective four internal divisions do not coincide with the study carried out by us.In fact, they have a slight flaw in representation accuracy.Indeed, when we carried out an analysis using level planes, where we did not represent the lines, which intersect both secant cylinders according to pairs of mutually orthogonal generatrix, we found the points 5, 6, 7 and 8 of the two intersection lines which, with the point R, which corresponds to the upper apex of the vault, allows us to accurately trace the lines of intersection.If we look closely, the lines determined by us do not coincide well with the curvature of the intersection lines drawn by the artist, although at point R the two-level tangents t and t1 considered by us allow us to verify the accuracy of the upper part of these intersection lines.

CONCLUSIONS
The use of computer-aided design, with the use of certain software programs, also allows us to discover procedures and methods used in the composition geometric drawings of certain artists, as we saw with the Pieter Saenredam example, which is why it constitutes a strong research instrument, even for art historians, precisely to verify the geometric knowledge in the area of geometry and perspective, in the preparatory drawing analysis of certain paintings and pictorial schools.Added to this is the fact that the same software is also useful in the present, because in conceptual terms it can also be an auxiliary medium in the realization of compositional scenes, not only for architecture, where they are abundantly used, such as AutoCAD, but also for the visual arts in general.

Figure 8 :
Figure 8: Pieter Saenredam, painting representing the interior of the Choir of St Bavo's Church in Harlem, Holland, 1660, oil on wood, 79.3x100.5cm,Worcester Art Museum.Author's perspective analysis.On the figure we superimpose a drawing where we point out the dominant directions and their vanishing points.Image source: Web Gallery of Art, 1996, http://www.wga.hu/framese.html?/html/s/saenreda/index.html