**BArch, MSArch, PhD, UNAM prize, SNI n**

*Faculty of Architecture- National
Autonomous University of México*

*E-mail* *tgsalgado@perspectivegeometry.com*

Apparently there are many methods for perspective but if we
categorize them there are just a few. Some criterions of classification relate
perspective to the so-called 1-point, 2-point and 3-point methods, others
—more formally— to projective geometry, descriptive geometry or
vectorial algebra. Of course we cannot forget to mention the early treatises on
perspective such as Alberti’s *Della Pittura* or Piero’s *De Prospectiva Pingendi*, which escapes any classification. Our aim on this
article is not precisely to solve the classification problem rather we propose
a new comprehensive method for perspective, capable of 3D representation
without using vanishing points.

The *modular perspective*
method allows us to work in true three-dimensionality on the perspective plane
(*PPl*).
We will explain how to measure directly on the *PPl* the triad coordinates (*x, y, p*) of a given point into the visual space, and how to
play with the symmetrical planes X and Y (*SPl X/Y*) in order to generate or recover
data. Finally we will explain how to employ *modular perspective* in *generative design* formal-process through an example of application.

Our first analysis is focused on the difference between the perspective of the observer’s visual space and the perspective of the objects. Alberti’s method for instance, describes the observer’s visual space by means of a reticulated grid —as a measuring system; opposed to traditional methods that pursuit solving the perspective of the objects alone. These methods lead us to understand perspective through two different models: The Albertian, as a geometrical system of human vision and the traditional as a repertory of geometric recipes to solve figures in perspective.

But what does spatial perspective versus object’s perspective mean? A scientific explanation can be given through the human vision itself, but our purpose here is not to study the neuropsychological processes of vision —a very complex process indeed—, rather we pursue to interpret human vision throughout geometrical concepts.

From Renaissance perspective we inherited the *visual
pyramid* model —with its cutting plane
or *window*. But this model has an
important feature not very well understood even nowadays, and that is that only
*one vanishing point *can be
located in it. In other words we can ask: How many vanishing points does the *visual
pyramid* has? A question that challenges us
to explain how perspective can be conceptualized through the so-called vanishing
points methods. As we know these methods are based in the cube’s
geometric properties instead of a model of the human vision [1].

The *modular perspective*
model represents the observer’s visual space in the same manner the
Albertian *visual pyramid* does, in
such a way that our sight line runs from the pyramid’s vertex toward
infinity. The symmetrical plane X (*SPl X*), the symmetrical plane Y (*SPl Y*) and the *PPl* are the basic components of the *modular
perspective* model, as it is show in **Figure
1**.

What the observer perceives through the *PPl* is the object’s apparent size because its
length, width and height dimensions change as we move forwards or backwards. In
any case the apparent size of what we perceive is ruled by visual triangles.
This geometrical feature means that all visual triangles are proportional and
thereby the *PPl* can be placed it
at any depth in the model.

Notice that the *observer*
(o) and the *SPl
X/Y* vertex share the same origin
—contrary to those vectorial models in which two origins are
required—, so we can easily measure the width, height and depth (*x,
y, p*) of any given point into the visual
space. As the three planes of our model correlate to each other, the
coordinates (*x, y, p*) can be read
directly on the *PPl*. The
remaining question here is: where the *vanishing point* (*pf*) should be placed? As the reader might suppose, the
intersection of the *SPl X/Y* determines
the *symmetrical sight line* (*vs*), where (o) and
(*pf*) are located at its ending points.
In other words, the point at the center of *PPl* represents the (*pf*). This geometrical feature allows sketching in
perspective in a true three-dimensional plane, that is, on the *PPl*.

Our method consists on determine the (*x, y, p*) coordinates of a point (P*n*) in the space under the *modular
perspective* geometric procedure
—since the numeric one is more suitable for computer applications. There
are five cases of punctual perspective projection into the *PPl*. We will present here in full description the first
case, since the other four cases will be posted on the Internet soon [2]. Our
description, rather practical than theoretical, starts with the problem
statement and then after with its solution.

This case occurs when the values of (*x*) as well as those of (*y*) are not greater than ±5 *m*, and when (*p*) has a positive value. As can be seen through the *SPl
X/Y*, in **Figures 2** and **3**, the coordinates of point P1 are plotted within the
shaded area of both planes.

This case is the easiest of all to solve. When (*x*) and (*y*)
are less than or equal to ±5 *m*, they are drawn directly using the *Salgado
Modular Scale* (RMS). We can see in **Figure 4**
how to draw P1 in perspective:

*x*= 4.00*m*, measure on lower border and take to (*pf*).*p*= 8.00*m*, measure on the left border and draw it the width of the visual field, where it intersects (*x*) draw as**^**to (*p*). The problem now consists on finding (*y*) on this**^**.*y*= 2.00*m*, measure in right border and take to (*pf*). At the intersection of (*p*) with the diagonal raise another**^**to (*p*) until it meets (*y*), from there draw a horizontal line to the**^**(*x, p*), and this intersection determines P1 in perspective.

I am aware that this could sounds very much mechanical
because we are not using simple words to tell what is going on the *PPl*, but this is the only way to avoid mistakes.
Nevertheless, lets try another manner of understanding what we did to obtain
P1:

- Imagining
the
*PPl*, as a sheet of glass in front of you and a*point*behind it, then try to follow by sight its three spatial coordinates into the real space while observing the*PPl*, and you will realize exactly what we did above to obtain P1 in perspective. Repeating this procedure for other*points*will confirm you that (*x*) and (*y*) always vanish to (*pf*) while (*p*) relates to them transversally in depth.

Lets see now the classical example of a cube in perspective
this time without using any vanishing points aid but the (*pf*) alone, that is, taking the cube as a referred
object into the visual field. **Figures 5** and **6** show us the cube’s projection on the *SPl
X/Y*, so the (*x, y, p*)
coordinates of each of its point can be read onto them.

Do not worry about the geometrical accuracy in reading
coordinates on these planes since our *visual-measuring* approximation is reliable enough to accomplish this
task. For instance, when reading P3 coordinates we can have:

*x* ≈ – 4.05 or – 4.10

*y* ≈ 4.40 or 4.50

*p* ≈ 0.50 or 0.60

When the *SPl X/Y* are
larger than the illustrations posted here the coordinates read would perform
the same way as if using the customary scale-meter, which in our case would be
done through the RMS complementary scales.

Remember that we are employing *modular-scale* values all the time, which means that the cube can
take any size depending on its pre-establish dimensional equivalence, so our
example can either be as short as a cube-toy or as large as *La Grand
Arche* building (Paris). You can image these
two extreme examples on the cube’s **Figure 7**.

The three scales on the RMS can be enlarged or shortened as
we choose beforehand. If you want to render a perspective on a mural surface
you can outline it right there on the wall, avoiding the inaccurate *quadratura* procedure or wondering where to pass through on the
contiguous walls to locate the vanishing points you might need. The RMS more
noticeable advantage is the freedom we get in controlling the drawing’s
size, as big or small we wanted to.

It is easy to render a perspective by using the *SPl X/Y* data. This technique is part of the students
training in *modular perspective*
in order to enhance their spatial abilities. They have to read the*
SPl X/Y* data in order to interpret it on
the *PPl*, that is, to learn how
the 2D-3D transformation works. In the outmost level of training, students
learn how to visualize objects in 3D only, by attempting to design
architectural forms directly on the *PPl*, the same way they use to do in planar projections. Thinking in
two-dimensions can be complicated some times but thinking in three-dimensions
is a real challenge, due to the fact that our brain’s right side must
operate highly complex spatial relations.

I do not know what could it be for other schools of
architecture, but at mine —and for many years until nowadays— the
perspective outlining of a project is the last thing to do. In other words, the
customary design process is born in 2D sketches, matures either way, and gets
old in 3D. So, our perspective’s *vangelio* suggests for it to be born in 3D at once.

Celestino Soddu quotes in *From Forming to Transforming*: *“In every project there is a first
step. The designer knows that his first act has a precise purpose: he has to
trace a system of relationships that must be adaptable to each possible
development.”* [3] Certainly the *idea* must be grasped as the first act of design, but what
matters more is the way we choose to approach it. As we know creativity
involves the utmost complex brain operations not susceptible for translation
into a method of any sort, so there can be many ways to approach the *idea*. The more we can say about it are generalities such
as choosing between sketching in 2D or in 3D, a choice that depends upon our
spatial abilities to perform the one or the other.

For sketching an *idea*,
either in 2D or in 3D, computers are not yet suitable for problems solving into
the vast field of creativity, that is to say, they seem not to enhance human
creativity, and even more, as Van Doren says about *companion
computers*: *“They will make
life very pleasant, but they will no much change, and certainly not improve,
human nature.”* [4] They are just
tremendous processing tools but incapable of really helping the act of
creativity that involves personal perceptions and emotions. Artificial
intelligence (AI) plays an important roll in computer systems, pursuing the way
to emulate human behavior. Burton points out that: *“Both AI
(particularly symbolic AI) and drawing theory came to model human behavior as
information processing.”* [5] Cleary
the new lead to follow seems to be the so-called *drawing theory* that becomes a seriously research topic for
neuropsychologists, meanwhile designers starts to be interested on the subject
too.

When a children is asked to draw that what he is looking at,
he draws what he knows about that thing instead of what he actually is seeing,
because he has not been training to interpret and play with spatial relations.
But when an architect is asked to draw that which does not even exist is a big
challenge, because he does not know from where to grasp an image? As we know he
need to crate it. Before to conceiving any image designers start working very
much like a child does, gathering first all the pieces that they already know
about the theme and then after reassembled them in a new way, by adding or
taking away elements, repeating this process several times until the *idea* is done.

For our 3D sketching example *idea* we choose to design a Christian Church, to be
settled in the valley of México. It is worldwide known that the *cross* is the most powerful symbol for Christians. There
are many churches around the world expressing the *cross* in many ways, mainly through its architectural plan
layout or at the top of the towers and vaults as a sculptural ornament.
Thinking about this symbol —during a sermon in Union Church
(México)— my very first thought was: why not consider the whole
building as a cross? Perhaps a church like this already exists somewhere. I am
not claiming here its originality but its origin, as an insight for an
architectural *idea*.

Our first attempt was to visualize the *idea* formally, as it is showed in **Figures 8**
and **9**. During its execution some
questions arouse immediately, such as: What dimensions are we dealing with?
What structural system and materials must we think over? What colors would be
suitable for the curtain glass windows? Under which criteria should we select
the landscape settlement? And so on, many other questions come up. Our second
attempt went into the volumetric configuration, discovering more problems to
solve. See **Figure 10**.

Having these questions in mind, we introduced in our third
attempt the *architectural layout of proportions* in order to rule the composition, as we can see in **Figures 11**
and **12**. Vignola and Palladio were the
masters in architectural design through *proportions* —as
well for historical building analysis—, from which we can learn how to
apply them even to modern buildings.

As a result of this 3D sketching process we grasp a *code* to explore, as Soddu says: “This first design
act is the occasion to use the code as a means of possible transformations, and
the traced form is a frozen moment of this transformation process. We are not
able, in fact, to transform a white sheet: we trace a form transcribing our
memory a spark of the idea.” [6] So we decided to explore the *x-cross* composition as another possible transformation of
the original *code*, being careful
to notice when its meaning changes radically, to step back immediately, as it
occurred in **Figure 13**. We realize at this point that a *code* cannot be broken without changing the *idea*. A *code*,
as a system of architectural elements, can be transformed over and over until
it reaches one of its possible arrangements.

Working on the *code*
our *visual thinking* starts
playing its role in the process, as Arnheim says: “Our thoughts influence
what we see, and vice versa.” [7] And that indeed happens. Until we
turned the view’s perspective, as it can be appreciated in **Figures 14**
and **15**, we realize that the large
lateral walls forming the cross could be capable of holding all the
complementary activities of the church, as small independent buildings linked
by the sanctuary, in other words, *a church between living-walls*.

Of course we tried to find out a meaning for the *living-walls* from the historical point of view. Observing a
perspective section of San Peter (Rome, ca. 320-329), the four lateral aisles
accomplish an important structural function as well as housing complementary
activities. So we thought that the basilical spatial configuration —of
two lateral aisles, in this case— could match pretty well for the *living-walls* without destroying its vertical expression, that of
the cross.

Once the *idea-code*
was reached a feedback process begins reformulating our original inquiries, but
this time giving them architectural answers. That is the case of **Figure 16**
in which we feedback Figure 12, improving its architectural elements. After
this final attempt we find ourselves ready to develop the architectural
drawings; plans, elevations, sections and details mainly. The quality of this
process depends upon our professional training and capacity as well as the
teams we work with —for structural calculus, contractors and so on. At
this point there is no more *idea* to
grasp or a *code* to transform but
the building process alone.

I would like to close
this writing going back to the concept of *idea* in *generative design*. At
least in architecture, an *idea* is
always three-dimensional, or better said, fragmentary-three-dimensional. Soddu
gives us a great example recreating the possible requests made to Borromini for
Sant’Agnese: “I imagine the request made to Borromini for the
church of Sant’Agnese in Piazza Navona: I want the church be present in
the whole square, inside the square but, at the same time at the limit of it;
the dome has to be present in all the square, it has to move itself amplifying
the character of the square that is a lengthened elliptic Roman passage.”
[8] As we can notice all verbal-requests evoke 3D images otherwise it would be
as tedious and useless to translate it into 2D “ideas.”

When Renzo Piano
decided to build in stone the new cathedral for Padre Pio, in San Giovanni
Rotondo (Italy), he took the *arc* as its *code* through which a 360º view from the interior was
generated in response to the infinite landscape. His *code* rested upon the *stone* and the technology to carve stones in different
sizes using a computer program —because the arcs, all of them, have
different spans too— as if they were standardized production. This is one
of the many ways in which computers become a designer’s friend.
Borromini’s *idea* was
probably to connect in harmony its church with the surrounding space.
Renzo’s *idea* matches in his
own words: *“Architecture is always the construction of emotions
through technical means.”* Our
church’s *idea* suggests that
architectural forms can be conceived and transformed within its natural
dimensions.

[1] Tomás García-Salgado, “Distance to the Perspective Plane.” Nexus Network Journal (out coming issue on perspective and optics), vol. 5, 2003, at nexusjournal.com

[2] perspectivegeometry.com (This Web Site is directed by Tomás García-Salgado)

[3] Celestino Soddu, "From Forming to Transforming" (paper at GA 2000 Conference, Milan 2000) generativeart.com

[4] Charles Van Doren, *A History of Knowledge* (USA: Ballantine Books, 1992) p. 381.

[5] Edward Burton, “Artificial Innocence: Interactions between the Study of Children’s Drawing and Artificial Intelligence”, Leonardo, Vol. 30 No. 4, pp. 301-309, 1997.

[6] Soddu [3]

[7] Rudolf Arnheim, *Visual Thinking* (USA: Univ. of California Press, 1969), p. 15.

[8] Soddu [3]

[9] All perspective drawings were rendered using the *Modular
Salgado Scale* (RMS).

More information at perspectivegeometry.com (comments).