geometric interpretation of the albertian model

Leonardo

Geometric Interpretation of the Albertian ModelAuthor(s): Tomás García SalgadoSource: Leonardo, Vol. 31, No. 2 (1998), pp. 119-123Published by: The MIT PressStable URL: http://www.jstor.org/stable/1576514 .

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GENERAL ARTICLE

Geometric Interpretation

of the Albertian Model

Tomds Garcia Salgado

Ma priego mi perdonino se, dove io in prima volli essere inteso, ebbi riguardo affare il nostro dire chiaro molto piu che ornato.

-Alberti, Della Pittura (1436)

The purpose of this essay is to demonstrate the universality of the Albertian model of perspective. I will do this using the method of comparative geometric analysis, which consists of

correlating the Albertian model with another model, in order to analyze by comparison the principles, definitions and

geometric properties of the elements of Alberti's procedure. For this analysis, I have chosen to use my own system, which I call the "Modular Perspective" model, which affords not

only the geometric reconstruction of the Renaissance

principia, but also the traditional methods of vanishing points and the numerical procedures that are currently popular with the use of computers.

Due to the brevity and specificity of this essay, I do not refer to a critical analysis of other interpretations [1], linguistic issues or the history of perspective itself [2]. These topics are no doubt very important, but they have been widely covered in other studies.

8-STEP DESCRIPTION OF THE ALBERTIAN MODEL

Throughout this paper, each of the "Steps" will begin with three texts. The first paragraph in each case contains the

original Italian version from Alberti's text [3], the second

paragraph contains my English translation of the original Ital- ian [4] and the third paragraph my interpretation of the text, based on the system I have developed, called the Modular model [5]. The texts describe the procedure step by step, and each step number correlates with one of the figures (Figs 1-

8). Figures 1, 2, 3, 6, 7 and 8 are frontal views; Figs 4 and 5 en-

compass lateral views; and Figs 3', 5', 7' and 8' correlate the Albertian model with the Modular model. All these figures are presented in Plate A. In order to facilitate the construction of the Albertian model, the geometric elements and definitions of the model and the complementary comments that determine approximately the Albertian conception of picto- rial space are introduced one by one.

Step One Principio dove io debbo dipigniere. Scrivo uno quadrangolo di retti

angoli quanto grande io voglio, el quale reputo essere una finestra aperta per donde io miri quello che quivi sard dipinto; e quivi determino quanto mi piaccino nella mia pictura huomini grandi e

divido la lunghezza di questo huomo in tre parti, quali amme ciascuna sia

proportionale ad quella misura si chiama braccio . . . et con queste braccia segnio la linea di sotto qual giace nel quadrangolo in tanti parti quanto ne riceva et emmi questa linea medesima proportionale a quella ul- tima quantitd quale prima mi si traverse inanzz.

I commence where I wish to paint. I draw a quadrangle consisting of right angles. I make it as large as I wish and consider it to be an open window

ABSTRACT

A Iberti's treatise Della Pittura has always been controversial because it does not contain a single drawing. Theoreticians and historians have offered different interpretations of Alberti's method of perspective, with some substantial differences between them. The interpretation presented here is based on a geometric analysis of the method described by Alberti, starting from the premise that all the geometric elements and principles employed by him have one and only one meaning- that is, there is no ambivalence, inconsistency or ambiguity between them. Thus, it is possible to define the geometric structure of what I refer to here as the Albertian model. The order in which Alberti introduced the definitions (14 in all) as he required each one is notable, and this made it possible to reconsider and apply the definitions, in the sense of associating some of them with the description of the Albertian model presented here.

through which I observe what is to be painted. I also determine here the height I wish the men in my painting to have and divide this height into three parts [Fig. 1], each one proportional to that measure called an arm. Using this measure, I mark the base line of the quadrangle into as many modules as may fit. This base line is proportional to the last transversal quantity, which previously was closest to me.

The visual field of the observer is demarcated as if it were a window. This element of the model makes it possible to in- fer the current notion of a limit of the visual field, since the window represents the natural limit of peripheral vision, while at the same time geometrically defining the projective plane of the visual field. The height of this plane is then modulated in three parts according to the height of the observer, and the same module is used to divide its base line in as many parts as desired.

Definitions * Quadrangle (quadrangolo) is the geometric envelope of

the visual field. * Window (finestra) defines the visual field of the observer.

The plane of the window itself is the surface [6], that is to

say, the projective plane in the Modular model. * "Mark refers to anything which lies on the surface and

may be seen by the eye" [7]. * Arm's length (braccio) is a unit of measure, proportion or

modulation (? 58 cm). * "Quantity refers to each space on the surface which runs

from one border to the other. The eye measures the

quantity with the visual rays as if they were the legs of an

open sextant. Hence, it is common to say that a triangle is

Tomas Garcia Salgado (architect, scientist), Faculty of Architecture, Universidad Nacional Aut6noma de Mexico (UNAM), Palacio de Versalles #200, Lomas Reforma, c.p. 11020, Mexico, D.F., Mexico. E-mail: <[email protected]>.

LEONARDO, Vol. 31, No. 2, pp. 119-123, 1998 119 ? 1998 1SAST



formed when one views. The base of the triangle is the quantity and its sides are the visual rays which, in turn, travel from the extreme points of the quantity to the eye. It would therefore be impossible to view any quantity without seeing the triangle" [8].

Discussion. Alberti begins the description of the procedure explaining what he does when he paints: "I commence where I wish to paint"-that is, to a certain degree he makes reference to a

practical situation in which the painter is standing before the wall he is about to

paint, and he therefore modulates the

height of the window based on the

height of the standing observer (3 arms), thus relating his scale to the pic- torial scene. The result is that the window is modulated on a natural scale;

geometrically this means that the projective plane is situated at zero depth [9].

Step Two Poi dentro a questo quadrangolo, dove amme

paia, fermo un punto il quale occupi quello luogo dove il razzo centrico ferisce et per questo il chiamo punto centrico. Sara bene posto questo punto, alto dalla linea che sotto giace nel quadrangolo non pizu che sia I'altezza del huomo quale ivi io abbia a dipigniere, pero che cosi et chi vede et le dipinte cose vedute

paiono medesimo in su uno piano. I then mark within this quadrangle a point

where I consider it appropriate to define the place where the central ray should arrive, so I call this point the central point [Fig. 2]. This point shall be well-situated, at a height above the base line of the quadrangle no higher than that of the man I have drawn, so that seen as such, things appear to be on the same floor as that of the observer and painter.

Plate A. This plate shows, step by step, the author's interpretation of the Albertian model; the figures are numbered 1 to 8'. Fig. 1: The visual field of the observer is demarcated as if it were a window. The black squares indicate 90? angles. Fig. 2: The projection of the central

vanishing point on the projective plane. Fig. 3: From the modulation of the lower border of the projective plane, the visual rays leading to the central vanishing point are drawn one by one. Fig. 3': Using the Modular model, the three remaining planes are made to vanish at in-

finity in order to cover the entirety of the observer's visual field. Fig. 4: The observer is located laterally at the same height as the central vanishing point. Fig. 5: The distance between the observer and the projective plane is defined in a lateral view. The black square indicates a 90? angle. The gaps in the visual rays emphasize the cutting points (in the lateral view) of the visual pyramid. Fig. 5': A comparison between the distance point method and the Albertian model. Fig. 6: The depth of the transversal lines is drawn. Fig. 7: The floor must be legitimized in perspective by the diagonal line of the reticulate. Fig. 7': The extension of the transversal modulation to the limits of the projective plane defines the observer's amplitude of the visual field. Fig. 8: The visual horizon is drawn on the projective plane. Fig. 8': The visual horizon establishes the symmetry between the floor and the ceiling.

SALGADO'S AL LgE D L LLA ? ! ITT i A INTERPRETATION nlLBERTF DE ) &L1LA M TT11 URly A

.FIG. ...DIST E (. FIG.5 DISTANCE (IL

FIG.6 TRANSVERSAL LINES

FIG3' VISUAL FIELD FIG.3 IOOULATION F16. 7 DIAGORAL

F . .. ........ ..............................

FIG. 4 DEPTHS IN LATERAL VIEW FIG. 8 VISUAL HORIZON

F16. 7' VISUL FIELD LMITS

FG.8' VISUAL FIELD SYMMETRY

The projection of the central vanish-

ing point on the projective plane-in other words, the point where the visual

symmetry intersects the projective plane-is defined. The observer and the central vanishing point are the extreme

points of the symmetrical line of sight. In the Albertian model, the central vanish-

ing point is given by the central point, which must correspond to the observer's

eye level so that what the observer paints is on the same floor as the painter.

Definitions * "The central ray shall be that which

makes the quantity have the same

angle, from top to bottom and from left to right" [10]. This is the symmetrical line of sight in the Modular model.

* The central point (punto centrico) is the central vanishing point in the Modular model.

* Floor or pavement (piano) is the reference plane for measuring space in both models.

Discussion. Alberti considered three

types of visual rays: extreme, medium and central. He called the central ray the "Prince of the rays" because it is the

only one that makes the visual triangle symmetrical. The extreme rays form the visual triangle, which is used to measure the size of the object. If the angle is acute, the object will be that much smaller, such that at a great distance the

quantity is no more than a point. In contrast with the Greek tradition, the appar- ent size of objects is not determined by the angle but rather by the distance: "Therefore the quantity, due to the distance, appears bigger or smaller" [11]. Alberti's visual triangle is interpreted from a lateral view, and even if the plane view is not explicit it is inferred when Alberti defines the visual pyramid [12].

Step Three Adunque, posto il punto centrico come dissi, segnio diritte linee da esso a ciascuna divisione posta nella linea del quadrangolo che giace. Quali segniate linee amme dimostrino in che modo, quasi per sino in

infinito, ciascuna traversa quantitd segua alterandosi . . . Onde loro succedono errori alla pictura non piccioli; adgiugni a qesto quanto la loro ragione sia vitiosa, ove il

punto centrico sia piu alto o piu basso che la lunghezza del dipinto huomo. Et sappi che cosa niuna dipinta mai parrd pari alle vere dove non sia certa distantia a vederle; . . .

Therefore, having situated the central point as mentioned before, I draw straight lines

120 Salgado, Geometric Interpretation of the Albertian Model



from this point to each of the modulations marked on the base line of the quadrangle [Fig. 3]. These lines show me, almost to

infinity, how any transversal quantity recedes. When the central point is higher or lower than the height of the man drawn, significant errors arise in the painting in addition to those made from the outset. Never shall anything which is painted appear real if it is not seen from a certain distance [vantage point].

From the modulation of the lower border of the projective plane, the visual rays leading to the central vanishing point are drawn one by one-in other words, the

pavement plane vanishes at infinity. Us-

ing the Modular model (Fig. 3'), I have made the three remaining planes also vanish at infinity in order to cover the en-

tirety of the observer's visual field.

Discussion. This passage makes evident the conception of infinite space in the Albertian model, given that when he takes the orthogonal lines to the central

point, Alberti states explicitly, "almost to

infinity." Of course, Alberti did not call the vanishing lines "orthogonal lines," but their geometric interpretation is un- mistakable. In Step Two, reference was made to the visual triangle as seen later-

ally; in this step Alberti considers the visual triangle as viewed frontally when he refers to "any transversal quantity." In

Fig. 3, it is possible to associate the idea of infinite space with the classic example of standing on railway tracks and run-

ning one's sight along them to the horizon at infinity.

The remark that the central vanish-

ing point is higher or lower than the

height of the observer is emphatic. It is derived from an error in observation, not from an error in drawing, given that the painting must be seen from a certain distance or vantage point in order to look real. This reasoning was ar- rived at through observation, and its

geometric interpretation establishes the distance point principle: "I have demonstrated that by changing the distance and position of the central ray, suddenly the surface will appear al- tered. Hence the distance and position of the central rays are of great impor- tance to the certainty of viewing." [13]

Step Four Ma, nelle quantitt transverse, come l'una

seguiti l'altra cosi seguito: prendo uno picciolo spatio nel quale scrivo una diritta linea et

questa divido in simile parte in quale divisi la linea che giace nel quadrangolo; poi pongo di

sopra uno punto, alto da questa linea quanto nel quadrangolo posi el punto centrico alto

dalla linea che giace nel quadrangolo; et da

questo punto tiro linee a ciascuna divisione

segniata in quella prima linea. With respect to the transversal quantities,

given that they are in succession, I proceed as follows: I take a small portion of space, in which I draw a straight line, I modulate the line into equal parts as I did with the base line of the quadrangle, and I then place a point above this line at a distance equal to that at which I placed the central point above the base line in the quadrangle, and from this point I draw lines to each of the modulations marked on that first line [Fig. 4].

The observer's eye is located at the same height as the central vanishing point, and the depth visuals are drawn from the observer to the floor base line modulation. This modulation is the same one used in the frontal view (see Fig. 3).

Discussion. This passage of Alberti's work has been considered obscure because he uses the expression, "I take a small portion of space," when he refers to the way in which the pavement transversal lines are obtained. This is the only passage in which Alberti uses this ex-

pression, but it is substantial for under-

standing the relationship between the

procedure's frontal and lateral outlines. As assumed in Step One, the painter is before the wall he is about to paint, and therefore when he says, "I take a small

portion of space," he is indicating that the lateral outline should occupy a small

(picciolo) space (spatio), but that it should be at the side of the wall to be painted, since the derivation of the pavement transversal lines requires a small amount of space. In other words, he was describ-

ing a practical situation in a general sense, and this is why Alberti used this

unspecific, but not obscure, expression. Panofsky, Ivins, and Klein [14-16] all in-

terpret this issue in a similar manner: as a drawing executed on a separate sheet, or even on a smaller scale and a different piece of paper-that is to say, with a

table-drawing situation in mind, instead of a mural outline. This interpretation differs from my own.

Step Five Poi constituisco quanto io voglia distantia dall'ochio alla pictura et ivi segnio, quanto dicono i mathematici, una perpendiculare linea tagliando qualunque truovi linea . . . Questa cosi perpendiculare linea dove dall'altre sard talliata cosi mi dard la successione di tutte le traverse quantitd.

I then establish a distance of my choosing from the observer's eye to the painting, I mark it, and there I draw a perpendicular line that-

as mathematicians say-cuts each of the lines it encounters [Fig. 5]. Thus, the points where this perpendicular line is cut by the others gives me the succession of all the transversal quantities.

The distance between the observer and the projective plane is defined in a lateral view. Where the depth visual rays intersect the projective plane, they define the succession of the transversal lines on the plane. In Fig. 5, the projective plane is the perpendicular line and, in turn, this is the Albertian window but in lateral view.

Definitions "A perpendicular line is a straight line which, upon cutting another

straight line, forms a right angle on either side" [17].

Discussion. The establishment of the distance with respect to the projective plane is the condition for deriving the transversal lines. The distance between the eye and the painting defines the distance point with geometric rigor. Alberti

clearly establishes this distance in the lateral view. As shown in Fig. 5', it would be confusing to do this in the frontal view [18]. From examination of the

original text, there is no doubt that Alberti was aware of the distance point in the spectator's real three-dimensional

space. Perhaps Alberti chose the lateral view as the clearest view to explain the distance point and avoid confusion. This has caused some contemporary critics and historians to misinterpret this passage [19] and claim that Alberti did not take into account the distance point principle. According to Malle, "Alberti, concerned with the question of distance, sought a partition to infinity of the foreshortened quadrangle-which is the projection of the space to be repre- sented on the picture plane. The manner of this search is its true conquest" [20]. Malle's observation summarizes the universal essence of the Albertian model: the representation of the observer's visual space in the projective plane [21].

Time confirmed the usefulness of the distance point for geometric designs, and therefore it is thought that its practical application required a more ad- vanced method than Alberti's. How- ever, this popularity is currently opaqued by computerized mathemati- cal drawing methods in which, ironi-

cally, it is more feasible to apply math- ematics to the Albertian model than to the distance point principle because

Salgado, Geometric Interpretation of the Albertian Model 121



the plane of the Albertian pavement can be understood to be a reference

system in which points are defined by their coordinates and their perspective is obtained by means of geometric transformation equations. In other words, the Albertian pavement can be

interpreted by applying numerical procedures [22]. With respect to geometric drawing procedures, the Albertian

pavement can be applied as a method for measurement and direct deduction in the projective plane, contrary to the

descriptive methods that rely entirely on projective geometry.

Step Six Et a questo modo mi truovo descripto tutti e paralelli cioe le braccia quadrate del

pavimento nella dipintura . .. In this way, I obtain all the parallel lines of

the space-that is, the modulation of the pavement of the painting [Fig. 6].

The depth of the transversal lines is drawn. In order to do this, lines are drawn parallel to the floor base line from the intersections of the depth visuals with the projective plane. The floor becomes modulated when this design is

superimposed on Step Three.

Definitions * The transversal quantity (traverse

quantitd) is the decrease in the floor frontal modules according to the

depth of each transversal line (refer to Step Three).

* (Braccia quadrate) refers to each pavement module in perspective.

Discussion. In this drawing, the lateral view (picciolo spatio) is switched to a frontal view, which may also be understood as a superposition of both views. Geometri-

cally, the passage from one drawing to the other presupposes a correlation between two mutually perpendicular planes or, equivalently, the three-dimensional reduction of the procedure by means of two planes of two dimensions.

Step Seven ... quali quanto sieno dirittamente descripti ad me ne sard inditio se una medesima ritta linea continoverd diamitro di piu quadrangoli descritti alla pictura.

I shall have verified the modulation of the pavement-which has been drawn directly-if a straight line itself contains the diagonal of the modules drawn in the painting (Fig. 7).

Given that the floor modulation is the same and proportional in both direc- tions, it must be legitimized in perspective by the diagonal line of the reticulate.

Definitions * "Mathematicians refer to the diam-

eter of a quadrangle as the straight line which runs from one angle to the other, dividing the figure in two, such that the quadrangle has only two triangles" [23].

Discussion. Given that the floor is the main geometric element of the Albertian model, the diagonal line is the

simplest and most elegant demonstra- tion of its exact construction. If this geometric truth holds in real space, why should it not be true in artificial space? This is the reasoning that makes the Albertian model universal, since it makes possible the exact measurement of the depth of the visual space in the

projective plane. Current standard methods for extend-

ing the floor grid are based on the use of diagonal lines, given that they vanish at the distance point(s) [24]. In the Modular model the extension of the transversal modulation to the limits of the projective plane defines the observer's amplitude of the visual field, as shown in Fig. 7'.

Step Eight Fatto questo, io descrivo nel quadrangolo della pictura, ad traverso, una dritta linea dalle inferiori equedistante quale dal uno lato all'altro, passando su pel centrico

punto, divida il quadrangolo. Questa linea amme tiene uno termine quale niuna veduta quantita, non piu alta che l'occhio che vede, puo sopra giudicare; et questa perche passa pel punto centrico dicesi linea centrica. Di qui interviene che li huomini

dipinti posti nell'ultimo braccio quadro della dipintura sono minori che gli altri; qual cosa cosi essere la natura medesima ad noi dimostra.

Having done this, in the quadrangle of the painting I draw from side to side a straight line, equidistant from the lower line, that, running through the central point, divides the quadrangle [Fig. 8]. This line defines a limit for me. Therefore, no quantity seen-which is not higher than the observer's eye-may exceed it. Since this line runs through the central point, it is called the central line. Hence, the men painted in the last module of the pavement of the painting will be smaller than those painted before, as demonstrated by nature itself.

The visual horizon is drawn on the projective plane, running necessarily through the central vanishing point. This line is parallel to the floor line and runs along the entire width of the visual field.

Definitions * The central line (linea centrica) is the

visual horizon in the Modular model.

Discussion. In Step Three, the width (transversal quantity) was related to the

depth, and in this last step, height is related to depth. In this manner, the Albertian model completes the description of the three dimensions repre- sented on the surface. The geometric interpretation under the Modular model is the same; the width and depth coordinates (X, P) and the height and

depth coordinates (Y, P) are drawn di-

rectly on the projective plane using the Modular scale [25]. Alberti conceived the central line to regulate the depth of the visual space. In the Modular model, this line (the visual horizon) establishes the symmetry between the floor and the

ceiling, as shown in Fig. 8'.

CONCLUSIONS

The reasoning behind the Albertian model is based on a cross-section of the visual pyramid-in other words, on the

projective plane of the observer's visual

space. It therefore represents space in

perspective; it does not represent objects in perspective as do other models. Alberti took from the cartographic tradition the reticular system, which basically refers to the form and its dimension in a

planar projection in order to define in

perspective a part of the observer's visual space. On the contrary, other methods such as the methods of descriptive geometry or the traditional vanishing point methods are not based on systems that represent space in perspective; instead they are based on geometric pro- jections of a body (plane and elevation) in order to deduce their projection in

perspective. In short, in the Albertian model it is possible to interpret a new

geometry that, in contrast with Euclid- ean geometry, establishes a finite plane (superficie) with at least one point at in-

finity and a line, also at infinity, that makes it symmetrical. After all, we can play with perspective, thanks to God's creation of the horizon: "I was there when he set the heavens in place. When he marked out the horizon on the face of the deep." (Proverbs 8:27.)

References and Notes

1. See Franco Borsi, Leon Battista Alberti (Milan, Italy: Electa, 1986) p. 201: "The controversy regard- ing Alberti is ... was Alberti aware of the point of distance? Steigmuller, Panofsky, Kern, and more

122 Salgado, Geometric Interpretation of the Albertian Model



recently, Klein, are of the opinion that he was not, whereas Parronchi believes he was. We agree with Parronchi." I agree with Borsi and Parronchi.

2. Kim Veltman, "The Sources of Perspective" (un- published manuscript given to me by Veltman in 1992). When this work is published, it will be essen- tial reference material because it covers all the im-

portant topics on the history of perspective in an orderly, broad and well-documented manner.

3. Leon Battista Alberti, Della Pittura, edizione critica a cura di Luigi Malle (Firenze, Italy: G.C. Sansoni, 1950; original publication 1547). The first paragraph of each of the eight steps has been taken from the original work and they are in the same se- quence as at the end of the first book of Alberti's treatise, pp. 70-74.

4. The translation from Italian to English presented here is ruled more by the geometric than by the lit- eral meaning of the text. With this in mind, I have updated the interpretation of certain terms. For example, I have replaced the word "divide" with the word "modulate," since the former term is more closely related to numerical calculations than to geometry and the latter term has broader meaning in geometry. On the other hand, in Step Six, it would be senseless to translate the term braccia quadrate as a square arm, given that the geometric meaning of the term refers to what we currently call a modulated or tessellated pavement or checkerboard. Therefore, I have rigorously analyzed each of the terms that plays a determinant role in the text. In addition, I have consulted the first Ital- ian edition of La Pittura, translated by M. Lodovico Domenichi (Venice, 1547), with a view to enriching this translation. Finally, readers wishing to compare with a philological translation may consult the En-

glish version of La Pittura in Cecil Grayson, Leon Battista Alberti, on Painting and on Sculpture (Lon- don, U.K.: Phaidon 1972).

5. Tomas G. Salgado, Perspectiva ModularAplicada al Diseno Arquitectdnico (Modular Perspective Applied to Architectural Design), Vol. 2 (Mexico City, Mexico: UNAM, 1982) pp. 828. See also Tomas G. Salgado, "A Modular Network Perspective Model vs. Vecto- rial Models," Leonardo 21, No. 3, 277-284 (1988) for a brief description of the Modular model and a glossary defining its elements.

6. ". .. et e superficie certa parte estrema del corpo quale si conosce no per sua alcuna profonditd ma solo per sua longitudine et latitudine et per sue ancora qualitd" (. .. surface is a certain external part of the body which is not perceived by its depth, but rather by its length and width, and also by its quality.) See Alberti [3] p. 56. According to Alberti, the surface is the section or base of the visual pyramid and it may have three qualities: flat, inwardly or outwardly concave, or spherical; a fourth quality may be added if two of these are combined. This suggests, in my opinion, curved and spherical perspectives in addition to linear perspective. See Alberti [3] p. 57.

7. "Segnio qui appello qualunque cosa stia alla superficie per modo che l'occhio possa vederla. " See Alberti [3] p. 55.

8. "Quantitc si chiama ogni spazio su per la superficie qual sia da uno punto dell'orlo al altro et misura l'occhio queste quantitc con i razzi visivi quasi come con un paro di seste. . . . Onde si suole dire che al vedere si fa triangolo, la basa del quale sia la veduta quantitd et i lati sono questi radii i quali da i punti della quantitd si extendono sino al occhio et e certissimo niuna quantitd potersi sanza triangolo vedere. "See Alberti [3] p. 59.

9. In the Modular model, the projective plane is also situated at zero depth. This allows for controlling distance as a focal distance, while controlling the aperture of the visual field.

10. "Sara centrico razzo quello uno solo quale si cozza la quantita che di qua et di qua ciascuno angolo sia al altro equale." See Alberti [3] p. 61.

11. "Adunque le quantita per la distantia paiono maggiori et minori." See Alberti [3] p. 60.

12. "La Pirramide sara figura d'uno corpo ... i lati della pirramide sono quelli razzi i quali io chiamai extrinsici." (The pyramid shall be the figure of a body whose sides are formed by those rays I have termed external.) See Alberti [3] p. 60. Therefore, it is inferred that the visual triangle may be hori- zontal or vertical, depending on which side of the pyramid is considered.

13. "Parmi avere dimostrato assai che, mutato la distantia et mutato il porre del razzo centrico, subito la superficie parra alterata. Adunque la distantia et la positione del centrico razzo molto vale alla ciertezza del vedere." See Alberti [3] p. 62.

14. When Panofsky described the perspectival construction of the checkerboard according to Alberti (in Fig. 8 of his book) he established the following: "Above right: auxiliary drawing executed on a separate sheet (elevation of...)." See Erwin Panofsky, Perspective as Symbolic Form (New York, NY: Zone Books, 1991; originally published in Leipzig and Berlin, 1927) p. 64 (see also p. 136).

15. In my opinion, Ivins misinterprets this passage when he claims: "('uno picciolo spatio'-doubtless a piece of paper or board on which he is going to work out the measurements that he is going to carry over onto the drawing he has started to make on his slide). "See Will- iam M. Ivins, On the Rationalization of Sight (New York, NY: A Da Capo Press, 1975) p. 24.

16. Klein goes further than Ivins when he states, ". . . then, on a smaller scale and a different piece of paper, he traced the profile of the pyramid, situ- ating the taglio at a chosen distance; .. ." See Rob- ert Klein, Form and Meaning, Writings on the Renais- sance and Modern Art (Princeton, NJ: Princeton Univ. Press, 1981) p. 112.

17. "Dicesi linea perpendiculare quella linea dritta quale tagliando un'altra linea diritta fa, appresso di se di qua et di qua, angoli retti. "See Alberti [3] p. 73.

18. Alberti's procedure does not include the plan outline, as do the procedures of Pelerin and Piero della Francesca. If Alberti had included the plan outline in the frontal view, the result would have been confusing because two views would have been fitted into one. In order to clearly trace the distance point, it is necessary to obtain it first on the plan outline and then in the elevation. Another procedure to obtain the distance point which re- sembles that of the Albertian model is to superim- pose the frontal and lateral views, as shown in Fig. 5'. Applying this procedure, the succession of transversal lines becomes the same as in Fig. 5, thus showing the 1.5

19. "Alberti contrawise, seems to have been igno- rant of the real significance (almost of the exist- ence) of the distance point; but he knew and well understood the central vanishing point and the horizon." See Klein [16] p. 112.

20. "L'Alberti, preoccupato dalla questione della distanza, cerc6 la partizione ad infinitum del quadrangolo scorciato (che e la proiezione sul quadro dello spazio da rappresentare). II modo di questa ricerca e la sua vera conquista. "See Alberti [3] p. 25.

21. By defining the projective plane as a geometric space in itself, in which the fundamental axioms and theorems of perspective may be formulated, geometric perspective emerges as a geometry that is inde- pendent of Euclidean, projective, descriptive or hyperbolic geometry. In order to establish the basic distinction with respect to this new geometry, let us take the principle of parallelism, excluding the case of ultraparallel lines, and let us analyze its consis- tency with the Euclidean model of hyperbolic and perspective geometry. While in the Euclidean plane parallel lines do not intersect, in the hyperbolic plane these lines touch at their points of intersection with the circle that defines such plane. In the perspective plane, parallel lines come together at a common point but do not intersect. In the Modular system, the postulate of parallelism is expressed as follows: every bundle of parallel lines in space (whose orientation remains the same regardless of

direction) has one and only one vanishing point, which may be within or without the projective plane.

22. La Caille, in 1750, used algebra to establish the fundamentals of perspective. His equations are analogous to those I have employed in the Modular Model. See Gino Loria, Storia della Geometria Descrittiva (Milan, Italy: U. Hoepli, 1921) pp. 68-69. See also Salgado, "A Modular Network Perspective Model" [5] p. 278.

23. "Dicono i mathematici diamitro d'uno quadrangolo quella retta linea da uno angolo ad un altro angolo, quale divida in due il quadrangolo per modo che d'uno quadrangolo solo sia due triangoli. " See Alberti [3] pp. 73-74.

24. When the diagonal is drawn to the distance point at an angle other than 45?, the coherence principle holds in the grid, but the modulation cor- responds to rectangles instead of squares. In this case, the distance point becomes the diagonal point.

25. See Salgado, "A Modular Network Perspective Model" [5] p. 280: "The application of Salgado's Modular Scale (SMS) can be demonstrated in an example consisting in obtaining the perspective of a cube." See also Salgado, Perspectiva Modular Aplicada al Diseno Arquitectonico [5] pp. 61-67.

Glossary base line-the border of the floor at zero depth in the projective plane (in the Moduar model this is the inferior limit of the visual field).

central vanishing point (punto centrico)-the limit to which any coordinate (P) tends in the symmetrical line of sight when its value approaches infinity. When projected on the projective plane, this point is determined on the visual horizon.

diagonal line (diamitro)-the line that intersects the corners of the floor in order to legitimize its reticulated outline (theoretically, this is a square grid in 1:1 proportion).

distance (distantia)-the interval between the observer and the projective plane, established by Alberti in the lateral view of his model. When the distance is defined in the frontal view, then it is called the distance point (see Fig. 5').

limit of the visual field-the geometrical place where every absolute value of the projective coordinates (Xo/Yo) given in the projective plane is equal to and only equal to five modules.

Modular model-the geometric representation of the Modular Network method [5] (a new method of perspective that has the following basic concepts: focal distance in function with the aperture of the visual field; only one central vanishing point to solve any perspective projection case; and a projective plane defined by the limit of the visual field).

projective plane (quadrangolo, finestra, superficie)- any virtual surface perpendicular to the symmetrical line of sight given by the limit of the visual field.

symmetrical line of sight (razzo centrico)-the line from the observer to the central vanishing point; also the intersection of the symmetrical planes (X/Y).

vanishing point-any point at infinity, within or without the projective plane, where the system of parallel lines in space converges, regardless of the orientation and direction of such lines.

visual horizon (linea centrica)-a line in the projective plane that marks the height of the observer's eyes and contains the central vanishing point.

Manuscript received 12July 1996.

Salgado, Geometric Interpretation of the Albertian Model 123



geometric interpretation of the albertian model

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