Tomás García-SalgadoFaculty of Architecture, Autonomous National University of México (UNAM)
The ability to measure without physical contact has been a constant pursuit throughout the history of science. We now know, for example, that the cosmic horizon -- the limit to the observable universe -- is at a distance of 1-2 x 10 ^{26} meters. Recently,
in the search for planets beyond our solar system, the transit
method of measurement has yielded astonishing results. Charbonneau
and Henry asked themselves whether a planet passing before a
star along our line of sight would cause the star's light to
be diminished [Doyle, et al. 2000]. Doyle asserts that "[f]rom
our perspective, the star would then dim in a distinctive way"
[Doyle, et. al. 2000: 40]. The first logical question here concerns
the probability of our view being aligned with the orbital plane
of that "invisible" planet, no matter how distant it
may be. His answer is that "[f]or planets that orbit very
close to their stars, such as the one around HD 209458, the chance
of the correct alignment is one in ten"[Doyle, et al. 2000:
40]. It may sound incredible, yet with the aid of the drawing
below we shall see that the foundation of the argument is in
fact quite simple (Figure 1).Figure 1. At cosmic distances, the lines of sight passing through a star are considered parallel. The drawing therefore represents ten possibilities for the orbit of a planet to fall into the observer's line of sight of the star. Naturally, the planet's distance from, and size relative to the star are determining factors in the observation. What I find interesting about all this is that perspective is part of this search both to appreciate the reduction in the light from the source, as well as to determine the inclination in the planet's orbit relative to our line of sight.
Veltman identifies a dual origin of the concept of "distance". On the one hand are the theories of vision and representation, and on the other are the techniques of observation in the field. Optics is the first science to indirectly address the problem of distance by attempting to explain the appearance an object may have. Euclid (c. 300 BC) postulated that the magnitude of the visual angle determines the apparent size of the object viewed. Thus, in Euclidian geometry, distance represents an implicit, indeterminate value in angular measurement. Panofsky termed this principle the "angle axiom".[1] After Euclid, Alhazen, Witelo, and Pecham,
the subject of optics appeared to have been exhausted. Over time
the techniques of field observation were systematized, giving
way to a practical geometry, experimental to some extent, aimed
at the design and construction of novel measurement tools. The
commonest of these was the When quattrocento artists sought a sensitive representation of the object seen -not its measurement as a visual or practical problem- they introduced the concept of distance as a measurable relationship between object, pictorial plane, and observer. This is the first geometric foundation upon which the theory of perspective is built. Its interpretation has diverse connotations in Medieval -particularly Alhazen's[2] - and Renaissance treatises due to the variety of methods and application procedures. Its geometric interpretation is still debated today, as no unified theory of perspective has yet come to fruition. We shall now turn to how some of these Renaissance authors, painters, and architects understood and applied the concept of distance. In his Figure 2a. Measurement on a rod showing the intersection with the lines of sight, from Francesco di Giorgio Martini's La Pratica di Geometria, according to García-Salgado.In another drawing, Francesco constructed a system of visual
rays, measured in braccia, originating from a point he termed
"the center". He used another point called "the
countercenter" to associate the observer's lateral projection
with the same frontal view ( Figure 2b. Center and countercenter of the model by Francesco di Giorgio Martini. These correspond to the vanishing point (vp) and the distance point (dv) in the Modular model. The intersection of dv with the sightlines directed toward vp determines depth. The center, he explained, is the point and termination of all rays and of the eye. The countercenter is the eye that views the point generated by all oblique lines that cross the center rays, which cause the impression of reduction. He pointed out that the space between these two points, center and countercenter, may be as distant as one pleases, although recommending that they be neither too close nor too far. Francesco clearly understood that the geometric relationship of separation between these two points is the controlling factor in diminution of the ground checkerboard, because he considered the center and countercenter to be the fundamental principles of perspective. These principles remain valid in modern theory, even if Francesco only stated them in a general sense. Manetti did not explain clearly how the panels ( This hypothesis of orthogonal drawings presupposes a site plan and elevation of the Battistero and surrounding buildings, but even conceding that these could have been produced, the procedure would seem an unlikely one at that time.[6] Also, the octagonal nature of the Battistero floor meant that four of its eight faces generated 45° diagonals, such that when viewed from the Cathedral door its diagonal faces would necessarily run into the left and right distance points. This coincidence may have brought Klein to conjecture on the use of the distance point, or even what is known as the bifocal perspective mode.[7] To explain this phenomenon, study Alberti presented the first theoretical conceptualization
of perspective in his treatise Alberti defined the concept of distance thus:
He revisits the term Figure 4. The Albertian perpendicular is better understood as a plane than a as line, because this perpendicular represents the picture plane viewed laterally. In his To execute the strokes precisely, Piero della Francesca recommended
using nails, silk threads or horsetail hair. The Atlas of Drawings
from Piero said that the observer may be placed anywhere for a
frontal view, but the most comfortable position is to place the
eye at the center of the quadrato. We find the description of
the first method in theorem XIII of his treatise [della Francesca
1984: 76] (Figure 6f), which we shall interpret step-by-step
with the series in The function played by distance is clearly understood in Figure 6b, from the observer's eye, indicated by point A, to the perpendicular BF, which also marks the border of the quadrato. Nonetheless, when Figure 6f is compared with the series 6a to 6e, several constructive features arise meriting further scrutiny. These, however, I will set aside for a later study dedicated exclusively to Piero's treatise. Perspectiva Pictorum et Architectorum (1693-1700) is a classic
text on perspective, with truly advanced craftsmanship [Pozzo
1989]. The Jesuit friar Andrea Pozzo, the author of this masterpiece
and other works, presented the monumental ceiling of Sant'Ignazio
in Rome, possibly the most spectacular illusory fresco of all
time. To execute this Pozzo presented the principles of perspective right from the
first paragraph of his treatise. He dealt with an imaginary apse
onto which an illusory fresco was to be painted. We have put
aside the architecture of the church to simplify description
of the procedure. In the series of Figure 7a (left) Explicatio Linearum Plani &
Horizontis; Figure 7b (upper right) Ac
Punctorum Oculi; c (lower right) Distantiæ.Pozzo was the first commentator to systematize use of the
'vanishing distance' point (
Obeying the Euclidian concept of angle size, Veltman made a drawing that exemplifies what happens when four objects of equal size are seen from point A. Figure 8 establishes that BC<EF because angle BAC<angle EAF. It also establishes that BC<EF because BC is further away than EF. The proposition is correct but not entirely convincing because it raises the question of whether plane BF is the best measurement of the angle or whether there may a better means of measurement. Let us compare this with Veltman's idea of "equiangular" measurement. Revisiting the concepts in Figure 8. Angular size of equal-sized objects, according to Veltman. In
Therefore, it is not a function of the perspective plane to interpret the retinal image, but rather to represent geometrically the perceivable image on its flat surface. So far so good. One could imagine there to be a multiplicity of geometric procedures in measuring distance, but fortunately they may be grouped into three types:
Figure 10. Measurement of the distance to the perspective plane (PP) seen laterally. The depth of the ground checkerboard transversal lines is
determined by where the visual rays from the baseline to the
observer cut the picture plane. This procedure has three basic
distinguishing elements: the observer, the picture plane (also
called the perspective plane PP), and the object (the marks on
the ground in this case). Some variations on this application
have been ingenious, (Leonardo's geometric riddle in
Figure 11 (left). Direct measurement of depth in the perspective plane PP. Figure 12 (right). Square grid of module m.If the reader analyzes this carefully, he will find not only
a grand synthesis easily committed to memory, but also one of
the best and briefest lessons in perspective. The whole dilemma
in this discipline, of determining perspective depth in frontal
modulation ( This is the golden rule of perspective. A visual ray is traced
from Concerning the distance vanishing point, I have previously
established in my book,
The concept of distance is universal, but variation in its application can lead to unreliable conclusions, as is the idea of bifocal perspective derived from Gauricus's method, an issue I will deal with below.
To overcome this ambiguity we shall delineate the difference between a space lattice reference system such as the Albertian checkerboard, and a grid to aid the drawing of objects in perspective. Both appear to possess the same properties, but the conception is different. A space lattice reference system represents the entirety of an observer's visual field in all its extent, modulated three dimensionally to its limits. On the other hand, the perspective life of an isolated object begins with its geometry, at times referred to a supporting grid, which cannot extend throughout an observer's visual field or is oriented in a different direction. I will now set two questions to demonstrate this constructive, but paradoxical difference. Observe Figure 13. To how many vanishing points does the cube in the perspective plane vanish? Figure 14. To how many vanishing points does the cube in perspective vanish? This pair of questions is my regular opening for the course on Modular Perspective, taught at the UNAM's faculty of architecture since 1992, normally eliciting the answers, "one, two, and three vanishing points" to each question. When the difference between the two questions is underlined, doubt sets in. The first question is recast without infringing on its content:
So the correct response is "ONE." Wherever sight is directed - center, up, side, or down -- the observer's vanishing point will always be directly ahead along the line of sight: struck against something or lost in the horizon of the landscape or the cosmic horizon. Similarly, recasting of the second question as, "How many vanishing points does the cube have?" leaves the correct answer as "THREE," in all three cases. Quite so. A cube or a rectangular prism has three parallel
systems, necessarily having three asymmetric vanishing points,
a In conclusion, then, and strictly speaking, there is no correct taxonomy of perspective defined by whether it has one, two, and three vanishing points. One issue is the spatial structure of the observer's visual field, represented as the perspective plane, which has a single vanishing point. And a separate issue is the geometric structure of the object(s) in the visual field that will have as many asymmetrical vanishing points as there are parallel systems conforming to its shape.[12] I now put to the reader the standard question to close that first class in the course: "To how many vanishing points does an icosahedron in perspective vanish?"
Alberti began by defining the observer's visual field as follows:
For Klein, the fundamental distinction between the methods
of Alberti and Gauricus is that the former was unaware of the
distance point,[14]
while Gauricus was unaware both of the central vanishing point
and the horizon line.[15]
Neither of these claims is geometrically correct. Firstly, the
term "distance point" is a way of constructively interpreting
distance ( The most controversial part of Alberti's method is the way of interpreting the correlation between frontal and lateral views. According to Klein, (and similarly, to Ivins and Panofsky):
Let us go through this paragraph step by step, taking the
author's interpretation of Alberti's method as the point of reference
[García-Salgado 1998b: 123-128]. Firstly, Alberti did
not begin by dividing the base line (linea bassa), but rather
began with the observer's visual field. Secondly, he did not
choose the central point arbitrarily, because its determination
is a logical consequence of the first step: Thirdly and most controversially, Klein asserted, "the profile line, on other paper and to scale." Let us take Alberti literally:
This passage must surely refer to a complementary drawing,
as at no time does it say, "on other paper, to scale, and
that the outline must be restituted to the original scale."
It would be risky to attribute to a text a particular geometrical
meaning which it lacks. As we know, Alberti did not include a
single illustration in his work; perhaps he considered a practical
situation in aiding his painter friends, in front of the wall
fresco to be painted (
Veltman began by explaining why Gauricus started his procedure
with the two semi-circles, claiming that the horizontal line
(parallel to the baseline) resulting from the intersection of
the semi-circles determines the upper limit of the picture's
diminished perspective. A couple of observations are in order
at this point. One is that Gauricus's original description makes
no mention of any central vanishing point, which leaves no geometric
evidence for Veltman's deduction of an simplified outline ( Figures 15a and 15b. Gauricus according to Veltman. The second observation concerns Veltman's claim that the countercenter must be situated at a precise height exactly vertically above the right edge of the square panel; but if it strays to left or right of this, the lower transversal will not coincide exactly with the line of intersection of the semi-circles.[23] Nonetheless, this lowest transversal must be calculated from baseline modulation (Ducatur itaque quot uolueris pedum linea hec). This is why the outline of the floor reticulation does not necessarily depend on fixing the lower transversal or in precision in placing the countercenter.[24] To my way of thinking, Decio Gioseffi gave the clearest interpretation
of Gauricus's text, because it follows most closely the text's
geometrical description. Gioseffi conceived the procedure in
twelve steps, of which steps 10 to 12 may be simplified ( Figure 16. Gauricus according to Gioseffi. Gioseffi's step 10 relies on the furthest transversal to generate the system of orthogonal lines, giving greater certainty in line orientation (Figure 16a). Nonetheless, as is evident in Figure 19f, the rest of the transversals serve as intermediate points in orienting the orthogonal lines -an advantage no geometrician or draughtsman would miss. Panofsky introduced the central vanishing point and test diagonal
in his interpretation, elements that may be clearly appreciated
in his figure 23 which accompanied note 60 to his renowned essay
"Die Perspective als 'Symbolische Form'" ( Figure 17. Gauricus according to Panofsky. Klein's interpretation is practically a carbon copy of Gioseffi's
step 12 ( Figure 18. Gauricus according to Klein. To recap on these comments I now present my interpretation
of Gauricus's method, respecting the order and meaning of the
description as it appears in the original text (which may be
consulted in Veltman [1975] and Panofsky [1991]. The intention
is not to produce a literal translation, but to achieve a comprehension
of the passages in which he described the procedure. These I
synthesize in six steps in In sum, as may be seen in Figures 19a to 19f, Gauricus described a system for modular reticulation of the ground. Yet unlike Alberti, restricting his drawing to the furthest transversal and not making use of the vanishing point (Figure 19f) left him without a geometrical solution to continue depth degradation to infinity. Yet curiously Gauricus did not discover the significance of the furthest transversal, that is, that it intersects all lines reaching the vanishing point at the same depth (Figure 19e); and, moreover, that these lines all run to infinity at the distance vanishing point (the countercenter, in Veltman's parlance). Gauricus was able to imagine the observer simultaneously before the lateral and frontal vistas, thanks to the geometric ambivalence of the perpendicular. But what he was unable to intuit or discover in the overlapped construction of these projections is that the perpendicular is not a line but a plane; viewed laterally it projects as a line but frontally it represents the perspective plane. The fact that these views are superimposed, makes it inevitable to interpret the furthest transversal as a flat drawing instead of as the projection of a three-dimensional construction, because there is no way to deduce its depth directly on the perspective plane.
What is known as bifocal perspective also arises from linear perspective. It consists of beginning the drawing of the grid from the distance points (or "distance vanishing points") which, due to symmetry with the picture plane, are situated at each side of the central vanishing point. Klein saw in Gauricus's system a moment of transition between the central pyramid and the oblique pyramid, attributing a wider angle to the bifocal construction than Alberti.[28] This idea sounds convincing in principle, but in geometric rigor the oblique (visual) pyramid may only be defined in anamorphic perspective, which is constructed precisely when the perspective plane is cut obliquely. This explains why marginal distortions are not anamorphic but rather the result of an excessively wide field of vision. Another consideration is that the perspective called bifocal derives from the picture's diagonals, that is, from a two-dimensional geometric property. This property is retained in the perspective plane by virtue of symmetry with three-dimensional space. Therefore, the idea of an oblique pyramid, in Klein's terms, simply translates into reticulation of the ground based on the lines that generate the diagonals. Viator ( Now, whatever position the grid is transposed into, frontal
or oblique, it must satisfy the properties of the square: the
square's diagonals in both cases must generate their own asymmetric
vanishing points. Note how the distance vanishing points or asymmetric
vanishing points (a This geometric overlapping which at times occurs, has prevented the concepts of 'distance vanishing point' and 'diagonal vanishing point' from maturing, leading to the conclusion that Gauricus came to, that they are a single geometric element. Finally, the following premise must be met to validate the procedure for drawing the grid when rotated on an axis:
I will use In this fashion we have rotated the grid from position A to
position B while conserving the [ Figure 21. Bifocal method according to Klein's figure 8: Ground-plane construction in the bifocal method. When the rotation is other than 45°, the diagonal vanishing
points will not coincide with the distance vanishing points,
to which purpose one should follow the rotation procedures I
set out in
[2]
Alhazen (c. 965- c. 1039 A.D.) was a Muslim scientist and philosopher.
He explained the concept of distance (
Thus, in Alhazen's optics, distance was understood
as the relation between the size of the object and its subtended
visual angle, and in its turn, the visual angle was understood
as the quantity of the wall's remoteness. [3]
To set up the San Giovanni experiment, Brunelleschi used a mirror
both to capture the image of the Battistero and to observe it
-- through an orifice at the back of the [4]
"[…] for Masaccio's Trinity, whose artificial architectural
scaffolding was probably drawn, if not by Brunelleschi himself,
at least according to his method, supposes an exact awareness
of the role played by the point of distance, which is, moreover,
indicated by a nail in the wall on which it is painted"
[Klein 1981: 131]. [5]
The hypothesis of the execution is that Brunelleschi, with his
back to the Battistero, placed -over the easel- a half braccio
mirror and next to it a tavola (panel) of the same size onto
which he painted what he saw and measured with the mirror. See
[García-Salgado 1998a]. [6]
As we know, scale drawing is exactly what would facilitate application
of perspective methods or procedures based on orthogonal drawings.
[7]
Klein [9]: "We do not know the precise moment at which the
two lateral points (…) received their theoretical explanation
as the 'point of distance.' Brunelleschi probably extended their
use from the plane to space; but did he know that their distance
from the central vanishing point represented, according to the
scale of the picture, the distance between the vantage point
of an ideal spectator and the plane of the image?" [Klein
1981: 134]. [8]
"E se bene Filippo non aveva lettere, gli rendeva sí
ragione delle cose, con il naturale della pratica e sperienza,
che molte volte lo confondeva" [Vasari 1986: 280]. [9]
"Questa linea perpendiculare dunque mi darà ne i
tagli suoi termini d'ogni distantia, lequali deuono essere fra
le linee trauerse del pauimento egualmente lontane: nelqual modo
io descritti tutti i paralelli dello spazzo" [Alberti 1435:
16]. [10]
"E con uno paio di seste farai quattro punti equidistanti,
e con linee diritte le agiugni insieme, e fa' uno cuadro, o vuoi
fare con la squadra; e fallo di quella grandezza che ti piace"
[Filarete 1972, II: 651]. [11]
Note I intentionally avoid saying, "… where both visuals
intersect determines the perspective depth of the transversal
( [12]
The vanishing point of the perspective plane is defined by Modular
Perspective as the vanishing point ( [13]
Cennino Cennini describes a similar procedure in his work, [14]
"Alberti, contrariwise, seems to have been ignorant of the
real significance (almost of the existence) of the distance point;
but he knew and well understood the central vanishing point and
the horizon" [Klein 1981: 112]. [15]
"It is astonishing that this construction was completed
without mentioning the central vanishing point or the horizon;
Gauricus seems to describe a thoroughly mechanized procedure"
[Klein 1981: 112]. [16]
Alberti [14]: "Dapoi ordino quanta distanza uoglio, che
sia tra l'occhio di chi guarda, e la pittura: e quiui ordinato
il loco del taglio, con una linea perpendiculare, come dicono
i Mathematici, faccio il taglio di tutte le linee, ch'ella ha
ritrouato" [Alberti 1435: 16]. [18]
In Perspective Geometry, a constructive principle denotes --
at least -- two fundamental characteristics: order and structure.
[19]
In Modular Perspective, the perspective plane is considered as
a true planar limit of 3D space. According to the geometrical
model of Modular Perspective, the distance between the observer
and the perspective plane is variable, and that between the observer
and the object is constant (such concepts lack clarity in traditional
methods, including that of Gauricus). Consequently, the distance
between the observer and the perspective plane is ruled by the
observer's visual angle and in its turn, interpreted as the distance
vanishing point of the perspective plane. In traditional methods,
on the other hand, the distance point will always remain in a
two-dimensional plane either on a simple lateral projection or
in a superimposed projection onto the picture plane. [20]
With regard to the practice of Gauricus method, Gioseffi suggests
the employ of both the vanishing point and the distance point:
"Gli artisti che practicarano il metodo del Gaurico avranno
indubbiamente prolungato per controllo le linee di fuga fino
al concorso nel punto principale: analogamente a chi per controllo
tirasse le diagonali, secondo l'Alberti. Al quale la concorrenza
in un punto delle diagonali stesse non è poi detto devesse
necessariamente essere sfuggita, solo perchè non fu messa
a profitto" [1957: 93]. [21]
Kitao closely analyzes the differences between Gioseffi and Panofsky
interpretation on Gauricus method; see [1962: 193]. [22]
"An elementary demonstration in geometry can prove that
the orthogonals obtained by Gauricus's method should converge
in a point situated on the median axis at the height of O, the
classical principal vanishing point" [Klein 1981: 112].
[23]
"Another obvious problem with having the countercenter positioned
beyond the frame itself is that it invites doubts whether it
should be the central line or the boundary of the plane that
serves to define the intersections" [Veltman 1975: 293,
Chap. III, n. 55]. [24]
In any case, Gauricus's method remains valid even though it does
not meet these conditions; as I postulated in my study on Perspective
Geometry, the position -- on the visual horizon -- of the distance
vanishing point ( [25] Panofsky reproduced this passage:
[27] "By natural perspective I mean that the plane
on which this perspective is represented is a flat surface, and
this plane, although it is parallel both in length and height,
is forced to diminish in its remoter parts more than in its nearer
ones. … But artificial perspective, that is that which is
devised by art, does the contrary; for objects equal in size
increase on the plane where it is foreshortened in proportion
as the eye is more natural and nearer to the plane, and as the
part of the plane on which it is figured is farther from the
eye" [Richter 1970, I: 63]. [28] "practically and historically, it was a simplified
bifocal construction; […] -more generally, of the oblique
pyramid- […] in order to obtain a more supple and open perspective
than Alberti's, thus infringing on one or the other of his postulates"
[Klein 1981: 134]. [29] "Le point principal en perspective doit estre
constitue assis au nyueau de lueil: lequel point est appelle
fix ou subiect. En apres/ une ligne produite et tiree des deux
pars dudit point: et en icelle ligne doiuêt estre signez
deux autres poits/ equedistans du subiect: plus prochains en
presente/ et plus esloignez en distante beue: lesquelz sont appellez
tiers poits" [Ivins 1973: Second Edition, A.ii]. Viator's
La Pittura. Tradotta
per M. Lodovico Domenichi, Con Gratia e Privilegio, In Vinegia
Apreßo Gabriel Giolito de Ferrari, MDXLVII (Milano, Biblioteca
Ambrosiana SQLX 63).Alhazen. 1972. della Francesca, Piero. 1984. Doyle, Laurence R., Hans-Jörg Deeg and
Timothy M. Brown. 2000. Searching for Shadows of Other Earths.
Filarete (Antonio Averlino). 1972. García-Salgado, Tomás. 1998a.
Brunelleschi, Il Doumo y el Punto de Fuga. García-Salgado, Tomás. 1998b.
Geometric Interpretation of the Albertian Model. García-Salgado,Tomás. 1998c.
La Última Cena de Leonardo da Vinci. García-Salgado, Tomás. 2000.
Gauricus, Pomponius (Pomponio Gaurico). 1989.
Gioseffi, Decio. 1957. Ivins, William M., Jr. 1973: Kitao, Timothy K. 1962. Prejudice in Perspective:
A Study of Vignola's Perspective Treatise. Klein, Robert. 1981. Martini, Francesco di Giorgio. 1969. Panofsky, Erwin. 1991. Pozzo, Andrea. 1989. Richter, Jean Paul. 1970. Serlio, Sebastiano. 1982. Vasari, Giorgio. 1986. Veltman, Kim H. 1975.
Leonardo (1988) or
in his book A
Modular Perspective Handbook. Some recent articles are:
"Geometric Interpretation of the Albertian Model",
Leonardo (1998); "The
Geometry of the Pantheon's Vault"; "Anamorphic
Perspective and Illusory Architecture" for the Generative
Art 2001 Conference (Milano, Dec. 2001); and "Rational
Design in Stained Glass Art vs. Artistic Intuition", to
be published in The Visual Mind II: Art and Mathematics,
Michele Emmer, ed. (MIT Press, in preparation).
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