Rendering of the Gallery (E. Garbin, LAR, DPA, IUAV)

The Borromini Gallery in Palazzo Spada, Rome
Ideal regular model and deformed real model
Abacus of the perspective deformations

Camillo Trevisan
trevisan@iuav.it

Paper in PDF format (980 kB)
Versione italiana

Programme of the conference held May 19th 1999
Centro Svizzero di Cultura di Roma
part of the cycle "Prospettiva e Prospettive"
edited by prof. Rocco Sinisgalli
September 1999
Best view 1280x1024

Any attentive observer visiting and walking through the Gallery of Palazzo Spada, in Rome, will be struck by a real and deliberate contradiction: he witnesses a deformed architectural structure but perceives – due to rational decodification – an ideal regular gallery in which the columns are all of the same height and equidistant from one another.
In reality, a similar experience occurs every time we observe a perspective or a photograph. Nevertheless, in this case the scenic effect created by entering into a perspective, enabling us to run it through in its entirety even to an uncertain and unfamiliar threshold, projects us into a new dimension. This resulting light dizziness is then accentuated by the continuous modification of both what we can see and what we perceive. Once this immediate visual wonder has been overcome, the desire to understand remains. How was the Gallery created? Which rules does it follow? How should it be followed if we want to recreate the evident correspondence with a regular gallery? And with which regular gallery?
The aim of this study is to present the answers to some of these questions.
The first section investigates and studies in detail the configuration of the real gallery, suggesting the existence of a method which is not perspective of the arrangement in depth of the columns and the floor panels.
Since the disposition of the columns does not follow a perspective – and there is therefore no projective mechanism which places the Gallery in a close and biunique relationship to a family of regular models – it is justified to ask if a regular interpolating model exists, preferable to all the other, infinite, possible regular models. The second section studies the ideal regular model under the guidance of proportional research in the absence of indications of a distinctively projective type. Finally, the last section examines in detail the geometrical analysis, subdividing and illustrating the effects of the variations of some constructive parameters of a solid perspective.
The appendix includes some methods of geometrical construction of a solid perspective, illustrating the use of a software for the generation of solid perspectives and their counter deformation. Finally, it includes the definitions of the nomenclature used in the article.  

1. The characteristics of the arrangement of the columns and panels of the floor in the real Gallery
Rocco Sinisgalli [1] has pointed out, after careful research, that the Gallery in Palazzo Spada is not really a true and characteristic solid perspective. Or rather, of the two fundamental characteristics of a solid perspective – but also of a linear perspective, that is the convergence of orthogonals of the projection plane towards a single vanishing point and the arrangement in depth of the objects in function of their distance from the viewpoint – only the former is rigorously respected and verified. In fact, if a single vanishing point [2] exists, nevertheless the vertical axis of the 12 pairs of columns of the real Gallery are not placed planimetrically following a rigorous perspective arrangement [3] (cf. figure 1.1 and 1.2).
A generic regular model [4], deformed in solid perspective so that its first and last columns coincide with the first and last column of the real Gallery, does therefore not maintain the coincidence of the other intermediate columns too with the same of the solid perspective: the greater the distance at the beginning or end of the Gallery, the greater their displacement will be with respect to the foreseen position in a true solid perspective.[5]
In other words, although they do not refer to the scheme of deformation of a regular model scanned by identical intercolumns, the columns of the Gallery Spada do not refer to the scheme of deformation of a regular model (cf. fig. 1.2). The attempt to answer the following question is therefore of great interest: why did Borromini choose precisely that arrangement in depth instead of that perspective? Does a clear and connecting rule exist between the planimetrical positions of the axis of the columns or were they positioned following inscrutable and indefinable subjective mechanisms, simply liberally modifying a basic perspective arrangement?
An accurate analysis of the data of the measurements of the inter-axis and of the original floor panels leads us to the conclusion that a rule does exist and it is defined by a geometrical series.
Let us assume, in fact, that the series begins with a seed of five roman palms (60 ounces). The distance between the axis of the first two columns will thus be equivalent to 60 ounces and the length of the first floor panel, including the first space, equivalent to 12 ounces, a fifth of sixty). To define the successive inter-axis (and the successive panel), a reduction of the proportion 5/6 is adopted; the second inter-axis (and panel) will therefore be 50 ounces (10 ounces for the second space, a fifth of fifty). The third length of the series will be equivalent to five sixths of fifty, that means 41 ounces and two thirds.
From the fourth element of the series being constructed the coefficient of reduction changes, going from 5/6 (10/12) to 11/12 and from 1/5 to 1/4 for the space between panels. The fourth length of the series will therefore be equivalent to eleven twelfths of 41 and 2/3 and so on to the last inter-axis and the last floor panel.
Even though it has nothing to do with the perspective, this form of detailed arrangement in depth produces a sequence that is much more modulated, without any effect of muddle of the columns towards the end of the Gallery and reducing the length of the first lacuna (cf. fig. 1.1).
Before discussing this hypothesis very closely, two preliminary considerations are necessary.
The first concerns the transformation in roman palms of the measurements taken in metres. Not only do different measurements exist in reference to the palm [6], but also the instruments themselves, which are used, could be slightly different, depending on the reference sample. Furthermore, ignoring minor errors of construction and survey, it is therefore not possible to define the length of the Gallery or its width of the various inter-axis in palms with absolute precision. To complicate the problem further, the columns are deformed and slightly off axis with respect to the plinths.[7]
There is another aspect to be considered – although the successive reductions (5/6, 5/6, 11/12, 11/12, 11/12, etc.) produce rational numbers, which can be easily expressed as fractions amongst small wholes, they soon produce fractional values, which are difficult to identify. For example, the last value of the series is equivalent to 434.00218599… ounces. We therefore round off to the nearest ounce, or to the fifth of an ounce (a minute). In the latter case one obtains better results than rounding off to the ounce (cf. table 1 and 2, column 4 and 7); however, considering that a minute is less than 4 millimeters, it seems excessive, if not impossible, to think that so much extra calculation was carried out.
To verify the hypothesis, a computer program has been developed. It is able to reduce the mean least square between the data produced by the proposed geometrical series and those measured, also modifying the coefficient of transformation from meters to palms apart from a translation factor which moves the grid calculated with respect to the one measured together, to reduce the difference. As can be seen upon close examination of tables 1 and 2 and figure 1.3, the variation of the rounding off modifies only minimally both the mean least square and the conversion coefficients (the final models differ by not more than 1.8 mm); basing the first around a centimeter, the second on the values of 0,22445 for the calculation of the inter-axis and of 0.22285 for the floor panels.[8]
There is therefore a difference of around a millimeter and a half between the measurements of the two roman palm samples; this data could lead us to believe the technicians who placed the columns and those who constructed the floor used different instruments.
The theoretical distance between the first and last axis is therefore equivalent to 434 ounces (36 palms and 1/6); even if the same tables show that the first inter-axis seems to have been moved back by one ounce, ideally bringing the sum of the inter-axis to 435 ounces (36 palms and 1/4) and that of the panels to 430 ounces (35 palms and 5/6).
This scheme is therefore very close to the real Gallery with respect to a solid canonical perspective, which differs from the Gallery with average and maximum deviations, which are five times greater.
Finally, it should be noted that geometrical series, which are completely analogue to those proposals, can also be obtain via graphics and with great ease and proportional accuracy (cf. figure 1.4).[9]
If this was the method used to position the columns and the floor panels, it naturally leads to the use of the 12 viewpoints to deform them, one column at a time.[10]
Indeed, once the positions of the axis were found, it was necessary in each case to deform each column perspectively both on the plane and in height. A combination of the two mechanisms which have been illustrated is always valid for the deformation: the convergence of the orthogonals to the frontal plane towards the common vanishing point and the reduction in depth, this time not of the inter-axis but of the plinths, of the bases, of the shafts and the capitals of the columns.
The adoption of a centre of deformation for each column – placed in a sequence that reproduces the arrangement in depth from the axis – means that the deformation is contained within acceptable limits from the viewpoint of an observer walking along the Gallery.
Thus, the Gallery seems to have been built on the basis of separation and accumulation of the effects derived from different mechanisms: the perspective pyramid that converges to the vanishing point; the geometrical arrangement in depth of the principal elements; the individual deformation, once again perspectively, of each column of the Gallery.

Figure 1.1   Comparison of the plane of the real Gallery (in red) and the regular deformed model with perspective method (in blue), so as to make the axis of the first and last columns coincide with the same columns of the real Gallery. To the left the deviations between the axes are highlighted, in meters. To the right, the deviations between the floor panels. The mean least square for the axis is equivalent to 8 centimeters; for the floor panels it is 10 centimeters.

Figure 1.2   To the left, the comparison of the ideal regular model (in blue) and the counter deformation of the real Gallery (in red). To the right, the verification of the presence of more than one viewpoint and dimensions of the ideal regular model (cf. section 2), expressed in roman palms.

Figure 1.3   Comparison of the plan of the real Gallery (in red) and the arrangement in depth generated by the geometrical series illustrated in section 1. To the left the deviations between the axes can be seen in centimeters. To the right, the deviations of the floor panels. Mean least square is approximately 1 centimeter, both for the axis and for the floor panels.

Figure 1.4   Graphic method for the construction of the geometrical series of reduction. Once the segments are defined – AB (60 ounces, first inter-axis), BC (26.666 ounces), BD (40 ounces), BE (45 ounces) and BF (53.333 ounces), pointing the compass on A with opening AB, trace the first arc to cross the segment AD. By lowering the perpendicular to AB, the length of the second inter-axis is defined and so on. The line AD corresponds to a reduction equivalent to 0.8321 (5/6 = 0.8333, difference of 0.0012); the line AC to 0.9138 (11/12 = 0.91666, difference of 0.0029; AE to 0.8 (4/5 = 0.8) and finally, AF to 0.7474 (3/4 = 0.75, difference of 0.0026).
Figures A. 1-3   Graphic method of construction of solid perspective (cf. appendix A).

Table 1 – Column inter-axes (cf. figure 1.3)

Table 2 – Floor panels (cf. figure 1.3)

Explanation of tables 1 and 2 (cf. figure 1.3)

Column 1
Here we find the measurements calculated on the inclined plane of the Gallery and projected on the horizontal plane (cf. Rocco Sinisgalli, Una storia della scena prospettica …, op. cit., p.17). To facilitate the reading and comparison of the data, the first value of each table is zero. In the first table the positions of the axis of the columns measured are noted (the intersecting points of the diagonals of the quadrilaterals of the bases are considered axis of the columns, even if other reference points could have been considered; see, for example, the first columns with the torus that protrudes from the base). On the other hand, in the second table we have a list of the initial and final positions of the fourteen original floor panels. Unit of measurement: meter.
It should be noted in table 1 that rows 4, 8, 12 correspond to the lacuna placed between the four groups of three columns: for this reason there are no data since there are no columns in those positions. However, the geometrical series does include the lacunas (columns 3, 6, 9 of table 1). 

Columns 2, 5, 8
In these columns we can see the equivalent measurements in meters of the first column of the table in ounces. Since the precise characteristics of the instruments used to construct the Gallery are not known, various hypothesis have been formed (cf. columns 3, 6, 9), each of which shows a slightly different result in function of the coefficient of conversion between the meter and the roman palm.

Columns 3, 6, 9
To verify the hypothesis of the geometrical series (with reference to the disposition of the axis of the columns and the floor panels), above all, it is necessary to convert the measurements from meters to roman palms or rather to ounces (1 palm = 12 ounces). The hypothesis foresees a starting point (both for the first inter-axis and for the first floor panel, equivalent to 60 ounces (5 palms) and a progressive reduction of the intervals, equivalent to 5/6 for the first two successive intervals and to 11/12 for all the others (the spaces between the floor panels are easily 1/5 the size of the interval in the first two cases and 1/4 in all the others). The sequence of reduction produces, however, values which have not always been rounded off (cf. column 9). However, other two hypotheses are considered: the first foresees the rounding off to the ounce (column 3), the second to a fifth of an ounce (the minute). Such hypothesis considers the coefficients of conversion from meter to palm slightly differently to let the proposed hypothesis coincide more accurately with the hypothesis proposed with the calculation. However, the difference is minimal.

Columns 4, 7, 10
Deviations expressed in centimeters between the hypothesis and calculated values and converted to ounces (cf. columns 2, 5, 8).
Therefore, the columns 2, 3, 4 refer to the hypothesis of rounding off to the ounce of the value calculated for the geometrical series; columns 5, 6 and 7 to the rounding off to a fifth of an ounce; columns 8, 9, 10 to the values which have not been rounded off. The hypothesis regarding columns 2, 3, 4 seems more likely for both tables even if it is not the best result of the deviations.
In figure 1.3 the deviations are illustrated in 2 tables (column 4) between the axis and the real and calculated floor panels.

The mean least square (MLS) is expressed in centimeters and indicates the “quality” of interpolation with a single statistical value. In fact, in this data, the major deviations “weigh” more heavily than the others, since the deviations themselves are squared.
The translation, in ounces, shows the displacement of the data together shown in columns 3, 6, 9 to reduce the deviation between this data and the data present in columns 2, 5, 8 to a minimum. The translation corresponds to the deviation of the first values of columns 4, 7, 10 (expressed here in centimeters).
The values of conversion from meters to roman palms (all different but taken with a close approximation around the average value of 0.224 meters per palm), take the values calculated in columns 2, 5, 8 into account. The slight differences between the values found may be explained – apart from their constructive imprecision – also by the different instruments used by the builders.
Finally, it should be noted that, as has already been said, the calculated measurements were projected on a horizontal plane with a reduction equivalent to 0.99556 (a cosine of 5.4° inclination of the floor of the Gallery). If the measurements of the geometrical series found had been used directly on the inclined plane – as seems probable – the values of conversion from meters to roman palms would undergo a slight increase (1.00446), rising respectively to 0.2255 (for the axis, table 1) and 0.2255 (for the floor panels, table 2); keeping the values presented in both tables unchanged and taking into consideration all things said so far.

2. The regular models and the ideal regular model:  dimensions and proportions from a constructive point of view
Even though it has been verified that there is no regular model at the basis of the real Gallery (cf. figure 1.1 and section 1), even if there may be an infinite number of regular models, which, once deformed, interpolate the Gallery itself, it is still evident that the architect needs to consider an ideal regular model as reference. The definition of the arrangement in depth of the columns and of the floor panels is not at all sufficient for the completion of the project: for example, how many paneled ceilings should be taken into consideration each time? What proportions should be given to the columns, to the trabeation and to the bases? Which regular inter-column should be adopted? To be able to answer these and other questions it is necessary to consider the measurements of the calculation (in roman palms):

	palms	fractions	modules
Diameter of first column at base (module)	1	2/3	1
Distance in breadth between columns (internal face of frontal first couple)	14	-	8.4
Distance between axis of 2 internal frontal columns	15	2/3	9.4
Distance between axis of 2 external frontal columns	20	1/3	12.2
Distance in breadth between bases	13	-	7.8
Distance in breadth between axis of first two pairs of columns	2	1/3	1.4
Total height of columns (with base, capital excluding plinth and abacus)	13	1/3	8
Diameter of facade arch	13	1/3	8
Total height of trabeation	3	-	1.8

As has already been noted, when looking at the front of the colonnade, there are an infinite number of regular models – that is, an infinite possibility of inter-columns – that satisfies the question: finding an ideal model in which the axis of the first and last columns are superimposed upon the axis of the first and last column of the real calculated model. Using the measurements of the paneled ceilings placed on the vault, it is, however, possible to define a regular model - that fulfils the requirement of maintaining the paneled ceilings themselves squares - better than the others. Accordingly, the paneled ceilings of the vault are the only useful evidence in defining the dimensions of the regular model which we would define “ideal”.
To define the inter-columns of such a model, the paneled ceilings are to be considered as squares.
The arch contains seven paneled ceilings (width 8 intervals), placed in rays with an additional belt on the right and left of the base of the arch itself. Since the diameter of the arch facade is equivalent to 13 1/3 palms, its development corresponds to approximately 21 palms. Taking into consideration that the two belts at the impost of the arch have a width of 1.2 palms each, 18.6 palms remain. Thus, if one paneled ceiling is 2.6 palms: 2.2 p. + 0.4 p. of space – we get 2.2 x 7 + 0.4 x 8 = 18.6 palms.
In depth the arcades contain four paneled ceilings with three intervals, equivalent to two inter-columns plus a module (a diameter of a column). Thus, two inter-columns plus a module are equivalent to: 2.2 x 4 + 0.4 x 3 = 10 palms (6 modules).
Therefore, the inter-column is equivalent to 4 1/6 palms (2 1/2 modules: the elegant and solid Eustilo), whereas the inter-axis is equivalent to 3.5 modules = 5 5/6 p. With these measurements, the paneled ceilings turn out to be squares and the overall dimensions sufficiently “round”, with a total height of the columns corresponding to the diameter of the arch facade (8 modules) and to the length of the three groups of columns (2.5 x 2 + 3), measured on the external surface of the three columns.
Each group of three columns therefore contains a perfect cube.
In short, the proposed hypothesis produces the following ulterior measurements (cf. figure 1.2):

	palms	fractions	modules
Length of ideal colonnade [11]	84	1/3	50.6
Inter-axis of columns	5	5/6	3.5
Length of floor panels (with space)	5	5/6	3.5
Space between panels (equal length/depth)	1	-	0.6
Breadth of floor panels	3	-	1.8
Length of bases (3 columns)	14	1/3	8.6
Distance between bases (lacuna)	9	-	5.4
Distance between axis of two final internal columns	11	-	6.6
Distance between column plinths	3	1/2	2.1
Length and depth of column plinths	2	1/3	1.4
Breadth of base (one column)	2	2/3	1.6
Breadth of base (two columns)	5	-	3

This “ideal” model can be “projected” to define a solid perspective so that the axis of the first and last column are superimposed on the axis of the first and last column of the real Gallery (cf. figures 1.1 and 1.2).
The parameters of transformation are:
Plane of the traces:             placed on the axis of the first column
Viewpoint:                         distance from plane of traces is 5 2/3 palms; height: 6 2/3 palms (approx. 1.5 meters)[12]
Vanishing point:                 distance from the plane of traces is 69 2/3 palms.

3. Study of the mechanism of perspective deformation: definition of the limits, the extreme results and implications and links that exist between visual construction and fruition
The observations that follow refer to a construction of the solid perspective corrected from a geometrical point of view: without, therefore, the proportional alterations, which are characteristic of the Gallery in Palazzo Spada (cf. section 1).
The main objective of these examples is to highlight the role of each parameter of visual construction and fruition: the viewpoints and vanishing points, the plane of traces and the eye of the observer.[13]
Indeed, it is not only the choice of constructive parameters that greatly influences the deformed model – very often difficult to understand and classify even if geometrically inevitable – but the same visual fruition of the solid perspective follows a logic which apparently seems unexplainable.
Two series of examples have therefore been chosen: the first – paragraphs 3.1.1-5 – analyses the modifications undertaken in the solid perspective in the variation of one or more constructive parameters (the viewpoint, the vanishing point or the plane of the traces); the second – paragraphs 3.2.1-3 – studies the phase of ideal reconstruction of the regular model varying the position of the eye of the observer who explores the solid perspective.
Moving along the solid perspective, for example, the height of the observer with respect to the floor remains constant. Walking along the inside of a solid perspective at a constant speed therefore corresponds to – in the interior of an ideal regular model – a displacement also in height and with an accelerated or decelerated movement, according to the direction.
If both are observed from the viewpoint used for the construction of the solid perspective the initial regular model and the solid perspective coincide.
This superimposition is maintained even when the perspective plane is rotated and inclined: the two models remain superimposed – in perspective – precisely because the lines that join each point of the regular model with the viewpoint also pass along the same points of the solid perspective, regardless of the position of the perspective plane. 
Since all the semi lines which irradiate from the viewpoint and pass along any point of the solid perspective, also pass along the same point of the regular model, the observer, with his eye on the constructive viewpoint need not, however, necessarily direct his eye towards the vanishing point.
In other words, if we place the focus of the lens of a camera on the constructive viewpoint, in the photograph the solid perspective and the initial regular model are superimposed regardless of the rotation of the lens.

3.1 Modification of the solid perspective to vary the parameters (the regular model remains constant) 
In the first five examples the regular model is kept constant whereas the constructive parameters (the viewpoint and vanishing point and plane of the traces) are either varied one at a time or in pairs. The resulting solid perspective will therefore have characteristics, which will closely depend on the variations. The derived characteristics can be combined amongst each other generating infinite possibilities of overall variations.

3.1.1 Variation of the solid perspective with displacement of the viewpoint  (cf. figure 3.1.1)
If the regular model, the vanishing point and the plane of the traces are kept constant, the displacement of the viewpoint along the axis of symmetry, the solid perspective is deformed. Such a model will constrict in length if the viewpoint moves away from the vanishing point; it will expand – towards the vanishing point -  if the opposite is the case.

3.1.2 Variation of the solid perspective by displacing the vanishing point (cf. figure 3.1.2)
Keeping the regular model, the viewpoint and the plane of the traces constant, the displacement of the vanishing point along the axis of symmetry deforms the solid perspective. Such a model will decrease in length if the vanishing point moves towards the viewpoint; it will increase – towards the vanishing point – if the opposite is the case.

3.1.3 Variations of the solid perspective by displacing the plane of the traces (cf. figure 3.1.3)
Keeping the regular model, the vanishing point and the viewpoint constant, the displacement of the plane of the traces along the axis of symmetry deforms the solid perspective. Such a model will increase if the plane of the traces moves away from the viewpoint; it will decrease – towards the vanishing point – if the opposite is the case.

3.1.4 Variations of the solid perspective by displacing both the vanishing point and the plane of the traces (cf. figure 3.1.4)
Keeping both the regular model and the viewpoint constant, the displacement of both the vanishing point and the plane of the traces together will modify the scale of the solid perspective. Such a model will be enlarged – maintaining the proportions of its parts – if the vanishing point and the plane of the traces move away from the viewpoint; it will be reduced – towards the vanishing point – if the opposite is true.

3.1.5 Variations of the solid perspective by displacing the viewpoint and the vanishing point with respect to the axis of the regular model (cf. figure 3.1.5)
If the viewpoint and the vanishing point define a line, which does not belong to the vertical plane of symmetry of the regular model, the resulting solid perspective will not keep its right-left symmetry but will generate an “oblique” model. However, by varying the height of the viewpoint with respect to the vanishing point (placed on the symmetrical plane of the regular model), the solid perspective will obviously maintain its right-left symmetry: the only differences regard the height. 

3.2 Modifications of the regular model varying the parameters (constant solid perspective)
In these examples the solid perspective remains constant while the observer places his eye in different positions to the point used to construct the solid perspective itself. He will therefore see – from a geometrical point of view – a regular model which differs from the initial one (it would be identical only if the observer’s eye were placed on the constructive viewpoint). Some “regularities” were maintained – such as the arrangement in depth of the inter-axis. Others, however, were changed.  

3.2.1 Variation of the regular model by the movement of the observer’s eye along the line for the viewpoint and the vanishing point (cf. figure 3.2.1)
By placing the eye in any other position of the line for the viewpoint and the vanishing point other than the viewpoint, the solid perspective and the regular model do not coincide. Nevertheless, for each point of observation along the line, there is a regular model – different to the original – and it too is regular: the only difference between these infinite regular models and the original regular model is due to a compression along the axis and therefore, in this example, due to the diverse inter-axis of the columns. If the eye moves closer to the vanishing point, the inter-axis becomes smaller; it expands, however, if the eye moves away from the vanishing point.

3.2.2 Variation of the regular model by the movement of the eye of the observer above or below the line for the viewpoint and the vanishing point (cf. figure 3.2.2)
If the eye is raised or lowered with respect to the viewpoint, the regular model is no longer completely regular: in fact, the floor is inclined towards the top or the bottom. It should be noted that the columns of the “regular model” restored in this manner still remain parallel to one another and are all of the same height: the observer who places his eye higher with respect to the viewpoint will still see a descending ramp which is regular; if the eye is lowered, the ramp will be ascending, but still regular.

3.2.3 Variation of the regular model by moving the eye of the observer to the right or to the left of the line for the viewpoint and vanishing point (cf. figure 3.2.3)
If the eye moves to the right or left with respect to the viewpoint, how does the regular model change? In this case the Gallery becomes oblique but the colonnades still remain parallel to one another. It is evident that the effects can add up: if the observer moves the eye further forward, above and, for example, to the left with respect to the viewpoint, he will see a regular gallery restored with the inter-axis of the columns compressed with respect to the standard regular model (eye moved further forward with respect to the viewpoint). Furthermore, the observer will see the Gallery as a descending ramp (since his eye is above the viewpoint) and finally, he will also see the ramp as an oblique on his right.

Figures 3.1.1-4.  Abacus of the perspective deformations (cf. section 3). The starting model is shown in red, constant and to be deformed (figure 3.1.5) or to be counter deformed (figure 3.2.1-3); the deformed or counter-deformed model is in blue. The viewpoint is at O; the vanishing point at F. 

Figures 3.1.5 e 3.2.1-3. Abacus of the perspective deformations (cf. section 3). The starting model is shown in red, constant and to be deformed (figure 3.1.5) or to be counter deformed (figure 3.2.1-3); the deformed or counter-deformed model is in blue. The viewpoint is at O; the vanishing point at F. 

Appendix A   Geometrical methods for the definition of the solid perspective

Method A [14], cf. figure A.1
The starting point is – on a plane view and in perspective – the object to be reproduced, in our case a box.
The position of the eye is fixed in O; the plane of the traces and the floor plane are assigned, inclined with respect to the horizontal plane. Point A will have as an image in the illusory space the point located by A’₁, on plane and A’₂ in height.
Between the plane and the elevation of each individual point, we can reconstruct the solid perspective of the given object in the space. If we trace the parallel to the plane p ₂ for O, F is the vanishing point of the solid perspective for where the plane of the vanishing points passes. The eye, placed on O will see the object box coincide with the deformed box.

Method B, cf. figure A.2
This method, as do all the others, maintains the horizontality of the segments parallel to the plane of the traces and the verticality of all the segments (the segments parallel to the plane of the traces remain as such during the transformation).
The coordinates of O and two lines must be defined: r₃ which indicates the course of the real model, r₄ which indicates the progress of the deformed model.
For each point A of the model (in the example, A belongs to the ideal not deformed model; but the same algorithm can also be applied the other way round to find the ideal model starting from the deformed one):
- Define the line O-A (r₁)
- Define the horizontal line for A.
- Find the intersection A1 with r₃ (course of the real model).
- Define the line for A1-O (r₂).
- Find the intersection A₂ with r₄ (course of the deformed model).
- Define the horizontal line for A₂.
- Find the intersection A₃ with r1. A₃ is the point you are looking for.
Knowing the height of A with respect to O (A-A’), one can find the height A₃-A” (always with respect to O), through the two similar triangles AA’O and AA₃A”.

Method C, cf. figure A.3
Once again this method is based on the properties of similar triangles and, from an algorithmical point of view, the application of a factor of scale, homogeneous and variable, is translated to the coordinates X, Y, Z of point A. The scale factor will have the value OF/(OF+AC) with the pole in O where OF is the distance between the viewpoint and the vanishing point and AC is the distance of point A of the plane of the traces. Segment AC is considered of positive value if A, with respect to the plane of the traces, is placed towards F; negative if placed towards O. Therefore, the scale factor will be less than 1 for all points placed on the semi-plane, defined by the plane of the traces which contains F (OF+AC>OF, if AC is positive), greater than 1 for all the points placed on the other semi-plane (OF+AC<OF if AC is negative) and equal to one for all points placed on the plane of the traces. Segment OF is constant for all the transformations of the points. When segment AC is zero the scale factor will therefore be equal to one: indeed, on the plane of the traces the two models coincide and there is no deformation. If AC is equal, for example, to the half of OF – with A towards F – the scale factor will be equivalent to 2/3; if, on the other hand, if AC is congruent with OF, the scale factor will be 0.5 and so on. In figure A.3, AC is equivalent to 0.18 times OF and therefore the scale factor is 0.8475.
A particular case can be noted: if segment AC it congruent to OF and point A is placed on the semi-plane containing O (AC negative), the scale factor cannot be calculated. In fact, in algebra the scale factor is OF/O and from a geometrical point of view this configuration foresees that the line for CF and AO are parallel to each other, therefore not being able to define A1 in their point of intersection, if not as an inappropriate point.
If – in that same configuration with A in the semi-plane of O – segment AC were actually greater than OF, one would obtain a symmetrical inversion of point A₁ with respect to the line for OF (scale factor negative). However, these methods can only be applied to configurations that foresee – if A is placed towards O – a segment AC which is smaller than OF.

Figure 1.4   Graphic method for the construction of the geometrical series of reduction. Once the segments are defined – AB (60 ounces, first inter-axis), BC (26.666 ounces), BD (40 ounces), BE (45 ounces) and BF (53.333 ounces), pointing the compass on A with opening AB, trace the first arc to cross the segment AD. By lowering the perpendicular to AB, the length of the second inter-axis is defined and so on. The line AD corresponds to a reduction equivalent to 0.8321 (5/6 = 0.8333, difference of 0.0012); the line AC to 0.9138 (11/12 = 0.91666, difference of 0.0029; AE to 0.8 (4/5 = 0.8) and finally, AF to 0.7474 (3/4 = 0.75, difference of 0.0026).
Figures A. 1-3   Graphic method of construction of solid perspective (cf. appendix A).

Appendix B    User manual for the software BURBON for the generation of a solid perspective
The software BURBON allows you to generate models of solid perspectives – or to counter-deform them into regular models – defining a three-dimensional model DXF, a viewpoint, a vanishing point and a plane of the traces.
The entities of the starting model are contained in a file type DXF of infinite size. The entities with points belonging to the group DXF 10..17, 20..27, 30..37 are modified. However, 3D faces, lines, points, 3D poly-lines, traces, texts etc. may be transformed. The blocks, the Mesh entities and the AME solids must be “exploded” repeatedly to obtain the individual primitive entities constituting them. In the case of the AME solids (only for the releases 11 and 12 of AutoCAD) it is advisable to apply the MESH command before “exploding” them; this enables you to obtain 3D faces and not simple lines. In this manner, once the deformed model has been brought back to AutoCAD, the hidden lines can be cancelled, applying the command HIDE. It should be noted that, in the “explosion” of the solids, AutoCAD (from version 13 on) produces entities of a Body type, which cannot be correctly transformed by the software rather than lines or 3D faces.
The software BURBON (given its name by Guidubaldo Burbon dal Monte) works in the following way:
- First the user constructs a regular three-dimensional model (which should be memorised in the DXF format, version 12), made up of faces (preferably), 3D poly-lines, points, lines etc. The model is then placed in space so that its axis (the axis that is to connect the viewpoint with the vanishing point) is parallel to the Y-axis.
- Selecting Proietta from the BURBON menu the command Proietta modello 3D is activated.
The name of the file of entry will have to be inserted (file of type DXF); the exit file (which will contain the model transformed from the software) and the coordinates of the viewpoint PV (X, Y, Z), the vanishing point PF (only the Y coordinate, since the coordinates X and Z will be the same as the PV) and of the plane of the traces (also in this case only the coordinate Y, since the plane is considered to be parallel to the plane XZ). The PV and the PF must neither coincide (amongst themselves) nor be placed on the plane of the traces. The line PV-PF is parallel to the Y-axis and orthogonal to the plane of the traces. On this plane the starting and end model coincide.
- If the signal Deformazione del modello is activated on the option (menu Proietta), the regular starting model will be deformed in the solid perspective of exit (direct deformation). In the opposite case (absence of signal, obtained by selecting the command itself), the deformed starting model will be counter-deformed into a regular model of exit (inverse deformation).
- If the signal on the option Scrivo punti su file (menu Proietta) is activated, the software will create a file in the ASCII format (with the same base name of the exit file and the suffix CDR) containing the initial and final coordinates of all the transformed points.
- Since the software automatically generates an ASCII file (with the same base name of file of exit and suffix PRM), containing the parameters of perspective transformation (coordinates PV, PF and plane of the traces), it will be possible to successively recall such a file using the command Carica parametri… (menu Proietta).
- The software also automatically generates or up-dates an ASCII file (with the same base name of file of entrance and suffix HST), containing the sequence of all the values used by the software, starting from the entrance file: in this manner it will be easily possible to reconstruct the various tests carried out on the initial model.
The software (including these instructions in HTM format and a DXF example file, containing a model of the ideal regular Gallery which has been simplified and illustrated in section 2) is found in the CD-Rom, in the file BURBON or near the site BURBON.
To obtain the solid perspective closest to the real Gallery of Palazzo Spada from the example model, the file BASE.DXF must be indicated as entry file; the following values must be used as parameters of transformation, already indicated in section 2:
X PV = 0.0; Y PV = -5.6666; Z PV = 6.6666; Y PF = 69.6666; Y perspective plane = 0.0.

Appendix C    Nomenclature

Regular model
Is the original three-dimensional regular model without perspective deformation. In the case of a colonnade similar to the one in Palazzo Spada, the intercolumns are all the same as are all the other repetitive elements (abaci, capitals, bases, etc.); the trabeations are horizontal and the one on the right is parallel to the one on the left. The gallery vault is semi-circular.

Solid perspective
Is the three dimensional model which has undergone the perspective deformation.

Viewpoint
Is the position that the eye of the observer must place itself to make the regular model coincide with the solid perspective: no other position allows such a superimposition. It is also used as a constructive parameter of the solid perspective.

Vanishing point
Is the side of the pyramid “containing” the solid perspective. This point is also used – together with the viewpoint and the plane of the traces – as a parameter to generate the solid perspective.

Eye
Is the position of the observer’s eye: it can coincide with the viewpoint but does not have to. If it does the regular model (constructed by the viewpoint) is superimposed on the solid perspective; if it does not, there is no superimposition.

Plane of projection
Is the plane that records the image of a perspective plane and is comparable to the film found in a camera.

Plane of the traces
Is the plane which is perpendicular to the line for the viewpoint and for the vanishing point, used to construct the solid perspective. All the elements of the regular model that lie on this plane undergo no perspective or scale deformation.

Direct deformation
Is the transformation that, using the parameters quoted previously, allows one to pass from the regular model to the solid perspective.

Inverse deformation
Is the transformation that, using the parameters quoted previously, allows one to pass from the solid perspective to the regular model.

English translation: Christina Cawthra