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1. 16TH
INTERNATIONAL CONFERENCE ON GEOMETRY AND GRAPHICS © 2014
ISGG
4–8 AUGUST, 2014, INNSBRUCK, AUSTRIA
Paper 67
APPLICATION OF THE CROSS-RATIO
TO THE ANALYSIS OF ARCHITECTURE
Riccardo MIGLIARI, Leonardo BAGLIONI
Sapienza University of Rome, Italy
ABSTRACT: Rudolf Arnheim, in his essay dedicated to the dynamics of architectural form, de-
scribes in a simple and clear way the two possible perceptions of an architectural space: as it is and
as it looks. The architecture, in fact, appears to our eyes seconding the laws of perspective and ap-
pears to our mind in its solid formal regularity. Actually, between these two opposite poles exists a
means, which allows to pass with continuity from one to another and which, as we will see, moves
towards one another in a metamorphosis without continuity solution. When we observe a perspec-
tive that simulates the depth of an architectural space or that simply alludes to that depth, inevitably
is activated the visual perception and the perception of mental space and the dialogue between them
develops the continuous transformation and engages, therefore, at the same time, the idea of move-
ment. The critique of architecture has recently emphasized some of the facades of Roman buildings,
and not only, which, using the above relations, induce, in the onlooker, illusory impression of
over-hangings and depths. Sandro Benedetti called these “pulsating facades”, referring to the sensa-
tion of motion that they evoke. The historical collocation of these buildings is the same as the dis-
covery of the perspective, of its first codification and its refinement, and it therefore seems right to
employ analysis of projective type to verify whether the perspective could have given rise to these
compositional solutions. The study that we are now presenting began with the measurement of the
intervals (“as they are”), apparently casual, that line these facades, in order to find the perspective
rule that generated them and to define the depths that said intervals are capable to evoke (“as they
seem”), depending on the observer's position. Relations between real and virtual distances, can be
implemented in a simple instrument (using a graphical algorithm editor) which can be used to ana-
lyze the “virtual” façade (created by the perception of an observer who changes his position) of
Roman building. A more detailed analysis of the cross-ratio of the above-said intervals enables in-
stead the measurement of the acceleration of the perspective view, or better, their variation, which is
in fact the characteristic that produces the sensation of a “pulsation”. This method is based on the
fundamental characteristic of cross-ratio to be an invariant for projective transformation. Since the
cross-ratio for four points which separate three segments with equal intervals, measure 4/3, we can
analyze how much the cross-ratio of adjacent segments diverge from this value to quantify the pul-
sation of the façade. The comparison between obtained results with the two analysis methods, the
pictures of the façades taken from different points of view and measured elevations demonstrate the
existence of the perspective matrix in the façade composition.
Keywords: Perspective, Cross-ratio applications, Pulsating façade, Visual perception.
1. PERCEPTION OF ARCHITECTURE
Rudolf Arnheim, in his essay dedicated to
the dynamics of architectural form [1], de-
scribes in simple and clear way two possible
perceptions of an architectural space: as it looks
and as it is. The architecture, in fact, appears to
our eyes seconding the laws of perspective but
appears to our mind in its solid formal regular-
ity. Who enters a Renaissance and Baroque

2. 2
church has no doubt that the pillars that support
the broad arches of the nave are all of equal
height, but he can not help observing the ap-
parent convergence of the frames in the altar, as
well as any other scenic effect, and he can not
help being emotionally involved .
The issue was already known and debated in
the Renaissance, when it was encoded the “le-
gitimate construction” and, simultaneously , the
first theoretical discussions about the represen-
tation of architecture. The circle of artists who
hides behind a letter to Leone X was well
aware of the ambiguity mentioned by Arnheim:
if they had to survey and draw the Roman sur-
vivor antiquities, they refused the way of picto-
rial perspective which shows things as they
appear, to the benefit of plan, section and ele-
vation, which show for what they are.
Apparently, the perception of architecture
moves between two extremes: “as it looks and
as it is”, to quote Arnheim. But, actually, be-
tween these two opposite poles exists a vehicle,
which allows to pass with continuity from one
to another and that, as we shall see, moves one
towards the other in a metamorphosis without
continuity solution. To realize this vehicle im-
mediately, just think of the apse of Santa Maria
in San Satiro: here the distinction between as it
looks and as it is, is no longer so simple and
clear. In fact, the “as it looks” alludes to an “as
it is” wholly illusory. The real object, in this
case, acts as intermediary between the visual
perception (perspective), and the mental per-
ception (orthographic), which, in turn, is re-
sponsible of the illusion of depth, in the con-
tinuation of the nave beyond the transept. Ac-
cording to the studies of the Transactionalist
School of psychology [2], in fact, is the com-
parison between the known models of a ste-
reo-metric space and the perspective image,
which induces in the mind the illusion of depth .
Who knows these features of perception is thus
able to control perfectly the illusion, as demon-
strated by Ames’s experiments.
Let’s get back to the idea of the movement.
The apse by Bramante is, in fact, a phase of the
movement that transforms the as it is in as it
looks, that is the church choir in the vision we
have of it from main nave. The vehicle, or solid
perspective, made in the slim thickness of 97
cm, it is just a phase of transformation, as only
one frame in the sequence of a film. But the
transformation is continuous and has as its ex-
tremes: on the one hand , the space of regular
shapes we call orthographic, which we recog-
nize in the plan and elevation drawings of the
letter to Leone X, and on the other the flat per-
spective which is obtained by canceling
three-dimensional perspective space in the two
dimensions of a drawing.
At a closer look, every flat perspective, con-
tains this potential. In fact, in one perspective
image, without its geometric code, correspond
infinite three-dimensional realities. And the
transformation that translates ones into others
infinite possibilities, it is still continuous and
can be seen as an action film. When we place
ourselves in a perspective that simulates the
depth of an architectural space or simply al-
ludes to that depth, so inevitably, it activates
the visual perception and the perception of
mental space and in the dialogue among them,
the continuous transformation which we talked
about, develops and engages, therefore at the
same time, the idea of movement.
This is what we are going to demonstrate
with a few examples.
It is not uncommon the case of historical
buildings where the façade is not composed
according to a regular scan of the space, but
according to a non uniform scan, which seems
to allude to perspective intervals and that, in
fact, is able to conjure up illusions we men-
tioned and, in particular, the idea of movement.
Quite appropriately, Sandro Benedetti has de-
cided to define these façades, pulsating façade
[3] [4]. Is it possible to analyze these simple
allusions to the perspective, in order to describe
geometrically the feelings that are able to
evoke?
To answer this question, we should start
from a consideration both simple and obvious:
the regular scan of a façade (Fig. 1) (with the
axes of the openings arranged in equal intervals)

3. 3
turns, when it is observed in perspective, in a
perspective scan, in which the same intervals
follow the law of reduction of apparent mag-
nitudes (Fig. 2).
Figure 1: regular scan of a façade
Figure 2: reduction of apparent magnitudes
Reason why, if you want to create the illu-
sion of movement of the front, in the sense of
spatial depth that moves away from the observ-
er, will be sufficient to have the axes of the
openings (or other equally strong compositional
element), at declined intervals, rather than reg-
ular.
But when, and how, it happens that a series
of irregular intervals can be said to be properly
declined in perspective illusion?
The question is not trivial, because if there is
a unintuitive matter in perspective, this is pre-
cisely the reduction of apparent magnitudes.
The convergence of the images of parallel lines
to a point is a phenomenon that can easily be
seen (just using a mirror as the ancients knew)
and you can easily play with (just using a nail
on the part to “break”). But if we try to do the
same with the gradual reduction of the images
of objectively equal intervals, known as reduc-
tion, then the common sense is not enough and
you need the geometric science. In fact, even if
you generate the reduction thanks to the mirror,
the law that governs it is not at all obvious, as it
is instead, a clear convergence of intersecting
lines at the same point. And any intuitively at-
tempt to reproduce that reduction in the draw-
ing fails, because it conflicts with the correc-
tions induced from physical experience and its
perception of mind, which tell us that the in-
tervals are equal, while the visual experience, at
the same time, tells us that these intervals are
reduced. The result of an intuitive reduction,
such as the one that tries an unconscious draw-
er, is, in fact, a succession of intervals that ex-
ceed or do not sufficiently contract.
So, if you want to look for a test of
knowledge of the ancient perspective, it would
be simplistic to look only in the presence of a
single vanishing point, as Panofsky did on the
frescoes of the second style [5]. Instead, it is
much more useful and convincing investigate
the reduction of apparent magnitudes to check
if it is correct or it is not according to the laws
of perspective [6].
That said, we want to provide some useful
tools for analysis of the reduction and therefore
also for an analysis of the movements induced
by tricks of perspective on the façades we dealt
with.
We will use two method: the first one based

4. 4
on a simple direct comparison between inter-
vals and the second one based on an examina-
tion of the cross-ratio values that generate the
same intervals.
2. FIRST METHOD: DEPTH ANALYSIS
Imagine dividing the façade into as much
segments as there are intervals between the
windows axis. These segments will have the
rectangular aspect of as many wings of a thea-
tre stage, all located in the same plane. If all
intervals are equal, the corresponding segments
of the façade are perceived in the position
where they are, and that is like wings belonging
to the same plane. If, however, one segment is
smaller than the largest, it is perceived as if the
segment façade was further away, while the
largest interval “stays” on the façade.
The variation of the (virtual) distance of a
shorter interval can be calculated as follows: set
the focal length f (or principal distance) equal
to the unit (which we can assume equal to the
distance of the observer from the front), it is
shown that one object distant d=f from observer
appears in true form, one object at d = 2f is re-
duced by half, one object at d=3f is reduced by
one third, and so on, according to the relation:
f
d
a
a 







' (1)
where a’ is the width of reduced interval, a
is the width of the reference integer interval
(the largest among those present), d is the vir-
tual distance (distance of the illusory plane
which hosts a’ from the parallel plane passing
through the observer) and f is the distance of
the observer from the façade, which serves as
module or unit of measurement. In fact (Fig. 3),
if O is the observer and AB is a segment locat-
ed on the projection plane at distance f from O,
a segment PQ (equal to AB but far from O at
d=2f) is projected on the plane in a segment
P’Q’ which is half of the size of PQ, as can be
seen by comparing the two similar triangles
OQP and OQ’P’ in which OQ is double of
OQ’ for construction and, consequently, also
the sides PQ and P’Q’ are in the same ratio.
A similar reasoning allows to establish that
R’S’ is one third of RS which is distant d=3f
from O, and so on. Therefore, the (virtual) dis-
tance d of any illusory segment a’, painted or
drawn on projection plane (the façade plan) can
be calculated with the expression:
Figure 3: variations of virtual distances

5. 5





 

'
a
f
a
d (2)
while the distance d’ of the virtual wing a’
from the façade, measures evidently d-f and
therefore it is obtained by:
f
a
f
a
d 





 

'
' (3)
Now imagine a designer who, well aware of
the laws of perspective, has desire to give the
façade of the building which is planning an ef-
fect of concavity .
Divided the front in wings of equal width a,
b, c, d, e (in grey), he moves back b, c and d as
appropriate (Fig. 4).
Figure 4: creation process of concave façade
At this point, choosing a central location of
the observer O, he projects backward on the
plane façade the ends of the wings, getting the
segments b’, c’, and d’, with perspective reduc-
tion. With these segments and with the ex-
tremes a and e (which have remained in true
measure) our designer can build a scale (that is
a scan of the façade with new modules a’ b’ c’
d’ e’) that will serve as a model for the compo-
sition of the façades. Applying this scale to the
note construction that divides a segment into
equal or proportional parts, the designer will
get a scheme of a façade divided into perspec-
tive parts. The process, however, remains em-
bedded in the composition result so that we
(viewers of finished result) can easily rebuild it
using the formula (3) to the intervals in which
the façade is divided.
The formula returns the illusory distance of
the central segments to the façade and, with
them, a table that illustrates in a symbolic but
expressive way, the concave trend of virtual
façade (Fig. 5). Relations between real and vir-
tual distances, can be implemented in a simple
instrument (using a graphical algorithm editor)
which can be used to analyze the “virtual” fa-
çade (created by the perception of an observer
who changes his position) of a building.
Taking advantage of this device, we can also
verify a peculiar feature of this optical correc-
tion, as well as perspective in
Figure 5: virtual façade of a building
general: the intensity of the effect depends
on the observer’s position. In fact, the effect of
concavity is stronger the more the observer is
far from the front and decreases with distance
to zero when the observer crosses the threshold
and enters, so to speak, the illusion (Fig. 6).
It’s time to move from theory to practice,
examining a building that presents the charac-
teristics of the pulsating façade and precisely
the Lateran Palace in the elevation overlooking
Piazza di Porta San Giovanni in Rome, de-
signed by Domenico Fontana. In this pulsating
façade, the largest of the intervals f measures
8.10m and therefore the above formula gives

6. 6
Figure 6: variations of intensity of concavity
depending on the observer’s position.
the following results :
Table 1.
Interval Width (m)
Virtual Distance
from façade plane
(m)
a’ 6,57 2,328767
b’ 6,63 2,217195
c’ 6,93 1,688312
d’ 7,95 0,188679
e 8,1 0
f’ 7,88 0,279188
g’ 7,6 0,657895
h’ 6,86 1,80758
i’ 6,68 2,125749
l’ 6,83 1,859444
The table reads as follows:
the first column gives the name of progressive
intervals, from left to right; the interval f, being
the largest, was hired as belonging to the plane
of the façade;
the second column gives the width of the
intervals expressed in meters;
in the third column figures the result of the
formula (3), which expresses the distance from
the façade to which each segment is perceived
with respect to the reference, which has null
distance.
This means, for example, that an observer
placed in front of the façade at 10 meters, per-
ceives the segment a’ as if it were about 2,32
meters farther away. Of course, this value in-
creases if the observer moves away and de-
creases if the observer approaches, as has al-
ready been said. It should be noted, however,
that the greater distance also means, of course,
a “crushing” perspective, as it happens, for in-
stance, in photographs taken with a telephoto
lens. And so the two opposing effects balance
each other, making the choice of the distance of
the observer from the façade, at the end of the
calculations that we have illustrated, merely
conventional.
If we put the above values in a graph, in
which the X-axis is aligned with the façade and

7. 7
Y-axis indicates the illusory depth, we obtain a
drawing that describes significantly the succes-
sion of concavity and convexity produced by
the perspective correction (Fig. 7). The conti-
nuity of the masonry then merges together the-
se illusory depressions and recompose impres-
sion of motion that has been called “pulse”.
You can read even more about this effect by
interpolating the data with a curve, taken back
to the front of the building.
3. SECOND METHOD: CROSS-RATIO
INVARIANCE
The eastern façade of the Lateran Palace,
demonstrates how the application of perspec-
tive corrections we have illustrated is more
complex than one might expect. In fact, the
continuous variation of the intervals is used not
just to create an effect of concavity or convexi-
ty, but to switch to one another, with greater or
lesser intensity.
It is opportune to think of a more appropri-
ate method of analysis that provides a measure
of the variation of the effect of perspective.
This method is based on the cross-ratio invari-
ance. To understand how it works, we must
remember that if a series of contiguous seg-
ments are aligned and subjected to an operation
of central projection (as it happens, in fact, in
perspective and in human vision), their
measures and their mutual relations are as a re-
sult of the projection. For example, if you build
the perspective of three segments AB, BC and
CD, aligned and contiguous and of equal length
(Fig. 8), their lengths result altered and, gener-
ally, also the relationships you can identify by
comparing a segment with each other. However,
there is a special relationship, which does not
suffer any alteration: it is called cross-ratio and
Figure 7: pulse analysis of eastern façade of Lateran Palace

8. 8
it is an invariant in projective transformations.
Figure 8: cross-ratio in equal segments
We see, then, how to build and measure the
cross-ratio. Consider extremes A, B, C and D
of the three segments aligned and contiguous
that we have described above, without exclud-
ing, however, that may be different from each
other and of any length.
We can build the following simple ratios:
AC/BC and AD/BD
The cross ratio (briefly indicated with
(ABCD)) is the ratio of simple ratios con-
structed as above, that is:
BD
AD
BC
AC
Suppose now that the three intervals AB,
BC and CD are all equals to the unit and ask
what will be, in this case, the value of the cross
ratio. In the event that we hypothesized the
segment AB measures 2·(AB + BC), the seg-
ment BC is worth the unit, and the segment AD
is worth 2·(AB + BC + CD), the segment BD is
still worth 2· (BC + BD). Therefore, the
cross-ratio is:
3
4
3
2
2
2
3
1
2










BD
AD
BC
AC
Figure 9: painting in House of Augustus
Thus, the cross ratio of four points separat-
ing equal intervals worth 4/3, that is 1,33 ... But
since the cross-ratio is an invariant in projective
operations, even any prospective of these three
segments aligned and contiguous, will provide
the same result.
Consequently, the knowledge of this value
allows an immediate verification of the cor-
rectness of a perspective: if any of the three in-
tervals that in real world represent three equal
intervals between them, have a cross-ratio
equal to 4/3, the construction of the reduction
of apparent magnitudes is legitimate. It respects
exactly the theoretical principles of central
projection. Note that this is always, whatever is
the point of view, and how many times those
three segments aligned and contiguous have
been subjected to projection operations. If, for
example, we measure the cross-ratio of four
points on one of the paintings in the House of
Augustus on the Palatine, we can verify that
this value is very close to the theoretical value

9. 9
of 1.33… (Fig. 9). This observation could per-
haps lead us to revise the reading given by
Panofsky about perspective of the ancients. If
we repeat the measurement on any photograph
of the fresco, also taken from a lateral and
oblique point of view, the value of the
cross-ratio does not change, precisely because
it is invariant in projection operations.
Figure 10: Vignola’s drawing of four Tuscan
columns
The same check conducted on the drawing
of four Tuscan columns, which is located on p.
142 of the edition of the Two Rules of Vignola
in 1583[7], provides similar results (Fig. 10).
But these are just simple examples of the use
of the cross-ratio as a tool for qualitative analy-
sis of perspective.
The case that we are examining is more
complex, because here the searched effect is
not that of a constancy in succession in the
depth of the intervals, but of a variation such as
to bring forward what was just before backward
and vice versa. We are therefore in a situation
that sees varying the widths of the illusory in-
tervals created by perspective and, therefore,
sees increasing and decreasing the value of the
cross-ratio compared to the value of 4/3 that
describes the constancy of the intervals.
This change introduces a new concept,
which we call perspective acceleration.
Consider, therefore, some wings of equal
width that, instead of being arranged all on a
plane, are arranged in space so as to describe a
sinuous shape, as seen in the figure (Fig. 11),
which for simplicity is in top view.
Figure 11: design process of a pulsating façade
We may use this model as polygonal chain of
the movement that we want to impose to a fa-
çade. Imagine, then, that the façade is repre-
sented by the straight line AQ and project, from
a center of projection S, the vertices of the po-
lygonal chain on the straight line AQ at points
A, B, C, ... O, P, Q. We will obtain, in this way,
the widths of eleven intervals that punctuate the
façade according to a perspective rhythm. Ap-
plying to this configuration the method that we
have proposed for the analysis of illusory depth,
we get the graph shown in Figure 12 (Fig. 12).
Figure 12: illusory depth analysis

10. 10
As you can see this chart proposes the sinuous
course of illusion, tracing the starting model,
except in the extreme. These appear almost re-
treated to accentuate the effect that we wanted.
This happens because the projections of related
segments are shorter than the projections of the
segments that follow (on the left side) and pre-
cede (to the right side) and, therefore, appear as
retreated respect to the plane of the façade. This
effect tends to disappear, increasing the ob-
server’s distance.
We now want to consider the acceleration of
the perspective illusion. For this purpose, con-
sider the fourteen segments of the polygonal
chain, and its fifteen vertices A, B, C, D, E, F,
G, H, I, L, M, N, O, P, Q.
With these points is possible to build twelve
quadruples:
ABCD, BCDE, CDEF, DEFG, EFGH,
FGHI, GHIL, HILM, ILMN, LMNO,
MNOP, NOPQ. Each of this quadruple in-
cludes three intervals of unequal lengths, which
give rise to the cross-ratios: (ABCD),
(BCDE), ... , (MNOP), (NOPQ).
We would like to mention that these
cross-ratios are equal to those of the angles
subtended by the points that generate them in
the viewer’s eye. Reporting on the X-axis the
natural succession of intervals and on the
Y-axis the values of the relative cross-ratios,
you get, based on the principles we have set out,
a graph that describes the acceleration of per-
spective illusion applied to the façade (Fig. 13).
Figure 13: perspective acceleration analysis
This chart does not describe the shape of the
virtual façade, but the intensity of its “pulse”.
Quick changes in the cross-ratio signify a sud-
den accentuation of depth, as is done in ex-
treme ranges, while delicate variations describe
a regular pattern, as is the center of the façade.
To apply these theoretical considerations to
a particular case, we will examine again the
eastern façade of the Lateran Palace. Here,
however, the number of intervals is odd, be-
cause, as almost always happens, the central
one is dedicated to the portal of entry. In cases
like this, you can not build many quadruples of
points to cover the entire length of the façade,
because the range will always be excluded. It is
possible, however, to build two series of quad-
ruples proceeding from left to right and from
right to left. The central quadruple, while hav-
ing the opposite trend, for example (DEFG)
and (GFED), will have the same value, because
this is a known property of cross-ratio [8].
Figure 14: depth analysis and perspective
acceleration of eastern façade of Lateran Palace
This method, applied to the façade of the
Lateran Palace, gives this result (Fig. 14) which
here is compared with the previous depth anal-
ysis. In the figure the lower graph describes the
deformation produced by the illusory perspec-
tive of the façade, while the upper describes,
interval after interval, the intensity of this
changes, that is the vibration of the façade or its
“pulse”, to use Sandro Benedetti’s expression.
4. CONCLUSIONS AND FUTURE WORK
The presented investigations, show the evi-
dence of perspective matrix in the composition
of the façade, they can measure the intensity,
they demonstrate the trend, but nothing they
say yet about the theoretical tools, reasoning
and methods used to obtain these qualities.
The legitimate questions that stem from this

11. 11
gap will have to answer a further study which,
based on a careful metrology analysis, could
explore the relationship between integer
measures, as the graphic constructions that can
establish such relationships.
For example, the construction of reduction
of apparent magnitudes based on “point of dis-
tance”, described by Vignola in the Two Rules,
can lead to relationships between integers such
as those expressed by sets of three segments:
5, 3, 2, 10, 5, 3, 10, 6, 4, 20, 12, 8, 24, 18, 14,
30, 18, 12, 36, 27, 21 ... etc. and multiples,
where all are able of 4/3 cross-ratio. In fact
given two segments aligned and contiguous, it
is possible to build a third, a fourth and nth
segment so as to form a perspective sequence
with constant speed, that is to form the per-
spective of equal segments in real space. Given
three intervals aligned and contiguous, AB, BC
and CD, the value of the third interval is ob-
tained by solving, compared to CD, the equa-
tion:
3
4

BD
AD
BC
AC
Where:
CD
BC
BD
CD
BC
AB
AD
BC
AB
AC







for which:
3
4





CD
BC
CD
BC
AB
BC
BC
AB
that is:
 
 
BC
AB
BC
BC
AB
CD




3
2
So it can not be excluded that beginning with a
regular scan of “constant speed”, Domenico
Fontana has also introduced some simple cor-
rective to avoid fractional measures and dosing
the illusory movement of the façade according
to the sensitivity of his eye and the mastery of
his art.
Figure 15: pulsating façade perception in Ca-
pizucchi Palace (Giacomo della Porta, 1593)

12. 12
ACKNOWLEDGMENTS
Authors would like to acknowledge Michele
Calvano who contributed to the development of
parameterization algorithm used in this re-
search.
REFERENCES
[1.] R. Arnheim, The Dynamics of architectur-
al form, University of California Press,
Berkley - Los Angeles - London, 1977.
[2.] F. P. Kilpatrick, La Psicologia transazio-
nale, Milano, Bompiani 1967.
[3.] S. Benedetti, I palazzi romani di Giacomo
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ABOUT THE AUTHORS
1. Riccardo Migliari is Full Professor in
Fundamentals and Applications of Descriptive
Geometry since 1990 at the Faculty of Archi-
tecture at the ‘La Sapienza’ University in Rome.
He is assiduously engaged in research, particu-
larly in the areas of Descriptive Geometry and
of Representation and Instrumental Survey of
Architecture. He directed, as Scientific Manag-
er, the architectural survey of Coliseum in
Rome during the preliminary studies for the
restoration of the monument undertaken by the
Archaeological Superintendence of Rome.
From year 2003 he deals, in particular, with the
renewal of the studies on the scientific repre-
sentation of the space, within in the evolution-
ary picture of the descriptive geometry, from
the projective theory to the digital theory and
from the graphical applications to the digital
modeling.
2. Leonardo Baglioni, is Assistant Professor
in Fundamentals and Applications of Descrip-
tive Geometry at the Faculty of Architecture at
the ‘La Sapienza’ University in Rome. His re-
search focuses on issues related to descriptive
geometry and renewal through the introduction
of methods of digital representation. In this
field of research, he is interested in new appli-
cations of descriptive geometry with reference
to the development of the recent achievements
of discrete differential geometry which today
play a significant role in contemporary archi-
tectural projects.