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1 The elements of perspective

Look at Masaccio's Trinity (Figure 1-1), the oldest surviving
painting that uses perspective rigorously. Why it looks
compellingly three-dimensional will be explained in Chapter
3. In this chapter, we will discuss the geometry that
underlies perspective; in Chapter 2, we will look at the
origins of perspective.

We, in the late twentieth century, take photography for
granted as the prototypical physical embodiment of picture
taking, and perspective as its mathematical model. But for
the artists-scientists of the Renaissance the introduction of
perspective required a complex mesh of innovations: They
had to define the very concept of taking a picture, to understand
the optics implied by this definition, to abstract
the geometry underlying the optics, and finally to discover
ways of translating these abstractions into practical rules
of thumb that anyone could apply in order to draw scenes
in perspective. In the Introduction, we have already encountered
Alberti's description of his key concept of taking
a picture. Let me quote it again:

Before we analyze the relation between the Alberti window
and perspective, a distinction must be made between
the study of perspective as the theory of picture taking and
the practice of drawing in perspective. In the remainder of
this chapter, we will deal mostly with the theory of perspective,

[ILLUSTRATION]

Figure 1-2. Representation of Alberti's
window (perspective drawn using
a front picture plane).
Engraving (modified) from G. B.
Vignola, La due regole della
prospettiva practica, 1611.

that is, with the nature of the geometric transformation
that allows us to represent a three-dimensional
scene on a two-dimensional surface, and with certain features
that all perspective representations have in common.
After presenting these notions in some detail, we will dwell
briefly on the procedure that Alberti invented, but because
this is a very complicated topic to which numerous textbooks
have been devoted, the reader should not expect to
learn from this book how to draw a perspectivally correct
representation.[1]

Perspective is easiest to understand once we are familiar
with the camera obscura (“dark chamber” in Latin) illustrated
in Figures 1-3 and 1-4. Although the issue is shrouded
in uncertainty, there is some evidence that the device was
invented by Alberti (Pastore and Rosen, 1984). It is no
more than a box, or a room, with a relatively small hole
in it, called a pinhole. If the box is to serve its purpose as
a camera obscura, light should not enter it except through
the pinhole. The side of the box opposite the pinhole is
called the picture plane. If the picture plane is painted white
and all the other sides are lined with light-absorbing black
velvet, we can be sure that all the light that falls on the
picture plane has traveled in a straight line from an object
outside the box through the pinhole and that none of it
has been reflected from the walls. So, moving into geometry,
a camera obscura creates an image x of an object
point v by ensuring that one and only one ray of light,
called a projecting ray, coming from v hits the picture plane
at x after passing through the pinhole. Unless the camera

[ILLUSTRATION]

Figure 1-3. Earliest known illustration
of a camera obscura. Engraving
from R. Gemma Frisius, De
radio astronomico et geometrico
liber, 1545. The legend translates:
“Observing solar eclipse of 24 January
1544.”

obscura is a room in which a spectator can stand and look
at the picture plane (in which case the picture will be both
large and very faint), we must devise a way of showing
the picture it takes. There are two ways to do that: Either
replace the wall of the picture plane with a piece of ground
glass and view the image from outside, or replace it with
a photosensitive plate that can be developed into a photograph.
In the latter case, we will have a pinhole camera,
that is, a photographic camera with a pinhole for a lens.[2]

A camera obscura does not correspond exactly to Alberti's
window, for it inverts right for left and top for bottom
(as Figure 1-3 shows). To understand the basis of perspective
as discovered by Alberti, consider Figure 1-5: We
project a point M onto a point m on a front picture plane
(P), replacing the pinhole with a center of projection (O), and
replacing the optical process underlying the camera obscura
with a geometric transformation called central projection. It
is essential to keep in mind that the scheme of central
projection is a convenient geometric fiction: No optical

[ILLUSTRATION]

Figure 1-4. Geometry of the camera
obscura

[ILLUSTRATION]

Figure 1-5. Main features of central
projection

system can be devised to perform such a transformation
in this way.[3] Because central projection is a geometric abstraction,
it can be misleading to link it too closely with
vision — which is based on an optical process — as textbooks
almost invariably do. It is better, I believe, to first comprehend
perspective as a purely geometric procedure for
the representation of a three-dimensional world on a two-dimensional
surface before one tries to understand its perceptual
effects on viewers. One should avoid thinking of
it as a method that mimics, in some sense, what we see.
We will see later (Chapter 3) that there is a sense in which
a perspective representation mimics what could be seen
from a certain point of view, but I have found it useful to
postpone that interpretation of perspective and to consider
it in the course of an exploration of the reasons why a
perspective picture looks compellingly three-dimensional
(see Chapter 3). The traditional representation of Figure
1-2, which places an eye at the center of projection and
calls the center of projection the “vantage point” or the
“station point,” is a convenient pedagogical device — except
that it takes the step of relating vision to central projection
before the purely geometric nature of the transformation
is made clear.

Up to this point, we have been talking as if central
projection and perspective were the same. The truth is that
central projection is a somewhat more general geometric
transformation than perspective. In central projection, one
defines (see Figure 1-5) a picture plane P and a center of
projection O (which is not contained in the plane P). The
central projection of any point M (distinct from O) is the
point m, which is the intersection of the projecting ray OM
with the plane P. It is important to keep in mind that this
definition does not restrict the location of M with respect
to P: It could be anywhere in space relative to the picture
plane — in front of it or behind it. Now because perspective
is a model of the picture-taking process, it restricts itself
to a 180-degree field of view from the vantage point of O,
pointing toward the picture plane. That is why perspective
applies only to the projection of points M contained in the
half-space (an infinite region of space bounded by a surface)
that contains the picture plane P and that is bounded by a
plane P' that contains the center of projection O and is
parallel to the picture plane P. This much constraint upon
the geometry is imposed by the optical nature of the phenomenon
we are trying to model: The maximum field of
a pinhole camera is 180 degrees. This formulation of perspective
is still somewhat more general than Alberti's window,
which implies that only objects behind the window
can be projected onto the picture plane. Indeed, in the large

majority of the cases, perspective is restricted to the region
beyond the window.

There are two ways of formulating this point, geometric
and perceptual. Geometrically speaking, the points M are
limited to the closed half-space which does not contain O,
and is bounded by the picture plane, P (see Figure 1-5).
The geometric evidence for this point can be found in the
size of the depictions of known objects. The geometry of
perspective implies that the painting of an object which is
in front of the picture plane will be larger-than-life; since
Renaissance painters very rarely painted larger-than-life
figures, most figures must be behind the picture plane.[4]

Psychologically speaking, most pictures look as if they
were seen through a window — none of the objects seems
to pop out into the space of the room; the scenes they
depict appear to be entirely behind the surface of the painting.
The evidence for this psychological observation comes
from the exception to this rule. In these rare exceptions,
we see the lengths to which an artist must go to coax the
spectator into relinquishing the assumption that the entire
scene is behind a window. Take for instance an anonymous
illumination of the twelfth century (Figure 1-6). Because
the fish occludes part of the frame, we assume that it is
flying in front of the page. But to attribute this perceptual
effect to the occlusion of a section of the frame implies
that, in the absence of this device, the scene would be
perceived to be entirely beyond the page. In other words,
even in preperspectival pictorial representations, we tend
to perceive a picture frame as the frame of a window through
which we can look into the virtual space depicted by the
picture. One of the most brilliant applications of this method
is in Jan van Eyck's Annunciation (Figure 1-7). In this diptych,[5]
we see the angel and Mary represented in a grey
simulation of figures sculpted in the round, standing on
octagonal pedestals. Behind them, just touching the far

[ILLUSTRATION]

Figure 1-6. The Flying Fish of
Tyre (ca. 1170). Ms. 81, the Pierpont
Morgan Library, New York.

surface of the pedestals, we see a black mirrorlike surface
in which the statues appear to be reflected, framed by dark
moldings, part of which are occluded by the angel's left
wing and by Mary's cloak.

It is possible to achieve the same effect by propagation,
that is, to have object A occlude the frame, and to suggest
(whether by occlusion or other means) that object B is in
front of object A: The result is that object B seems to be
in front of the picture plane. Such is one of the interesting
perspective effects used by Mantegna in his frescoes for
the Ovetari Chapel (Figure 1-8). In the Martyrdom of Saint
James, the railing appears to be attached to the front of the
picture frame; that is why the torso of the soldier leaning
over it appears to emerge into the space of the chapel,
above the floor onto which Saint James's head will roll
when it is severed.

[ILLUSTRATION]

Figure 1-7. Jan van Eyck, Annunciation
(after 1432). Thyssen-Bornemisza
Collection, Lugano,
Switzerland.

[ILLUSTRATION]

Figure 1-8. Mantegna, Martyrdom
of Saint James (1451–5). Fresco.
Ovetari Chapel, Eremitani Church,
Padua.

Returning now to our exposition of the elements of
perspective, there are three geometric properties of central
projection that we must understand in order to proceed
with our analysis of the psychology of perspective.

Proposition 1. The perspectival image of a straight line that
does not pass through the center of projection is always a
straight line (the image of a straight line that does pass
through the center of projection, i.e., of a projecting ray,
is a point).

The second property of perspective concerns the representation
of sets of parallel lines (such as you would have
to contend with in drawing a box). First, consider a set of
vertical lines. If the picture plane is not vertical, the images
of all vertical lines converge onto the one and only vertical
vanishing point. Figure 1-9 illustrates this property as it
occurs in a picture of a box projected onto a picture plane
tilted sharply downward. Next, consider a set of horizontal
lines. They too converge onto a vanishing point. To specify
the location of this vanishing point, we must first define

[ILLUSTRATION]

Figure 1-10. Definition of the horizon
line

the horizon line of the picture (see Figure 1-10), which is
the line defined by the intersection of the picture plane
(which need not be vertical) and a horizontal plane that
contains the center of projection. The perspectives of any
two horizontal lines that are parallel to each other intersect
the horizon line at the same point, which is their horizontal
vanishing point.

Proposition 2. The perspectival images of parallel lines that
are also parallel to the picture plane are parallel to each
other; the perspectival images of parallel lines that are not
parallel to the picture plane converge onto a vanishing
point (which is not necessarily within the confines of the
picture).

It can be confusing at first to realize that when one looks
at a picture like Figure 1-9, one cannot tell whether the
box was tilted and the picture plane was vertical, or the
box was upright and the picture plane tilted (as we described
it to be). Had we described it as the picture of a
tilted box projected onto a vertical picture plane, then we
would have had to relabel the vanishing points, for there
would be neither horizontal nor vertical vanishing points
and it would therefore be incorrect to label a line connecting
two of the vanishing points “horizon.”

This point can be further clarified by noting that in Figure
1-2 one projecting ray is singled out and given a name
of its own: It is the principal ray, the projecting ray that is

perpendicular to the picture plane. (It is along this line that
one measures the distance between the center of projection
and the picture plane.)[6] In Figure 1-4, the principal ray
intersects the picture plane at the horizon. This is true only
in the special case when the picture plane is vertical. Nevertheless,
although it is inappropriate to call a horizontal line
drawn through the foot of the principal ray the “horizon,”
this line is meaningful and important. It is the locus of the
vanishing points of lines orthogonal to the picture plane
and parallel to each other.

We are now ready for the third property of perspective:

Proposition 3. The location of an object point cannot be
determined uniquely by its image. However, it is possible,
by making assumptions about properties of the scene, to
solve the inverse problem of perspective, namely, given
the central projection of a scene, to reconstruct its plan and
elevation.[7]

For instance, Sanpaolesi (1962)[8] proposed the reconstruction
of Masaccio's Trinity (Figure 1-1), shown in Figure
1-11.

To gain some insight into what Alberti taught his contemporaries,
we should examine what Renaissance artists
came to call the construzione legittima (legitimate construction)
of the perspective drawing of a pavement consisting
of square tiles. Figure 1-12 summarizes the geometry underlying
the construction. To carry it out, one needs a plan
(view from above) and an elevation (side view) of the
pavement, on which are indicated the picture plane and
the center of projection. In Figure 1-12, I have numbered

[ILLUSTRATION]

Figure 1-11. Plan and elevation of
Masaccio's Trinity according to
Sanpaolesi (1962, figure C, opp.
p. 52)

[ILLUSTRATION]

Figure 1-12. Construction of perspective
representation of a pavement
consisting of square tiles

[ILLUSTRATION]

Figure 1-13. Leonardo da Vinci,
Alberti's construzione legittima
(ca. 1492). Manuscript A, Fol.
42r. Bibliothèque de l'Institut,
Paris.

the steps involved in constructing the perspective: (1) Use
the elevation to draw the horizon. (2) Use the plan to
determine the vanishing point on the horizon. (3) Use the
plan to mark off the front of the pavement on the bottom
of the picture. (4) Connect these points to the vanishing
point. (5) Transfer the locations of the tiles from the plan
to the elevation. (6) Connect these points to the center of
projection in the elevation. (7) Transfer the intersections
of these lines with the picture plane in the elevation. Figure
1-13 shows how Leonardo represented the procedure. Because
the tiles in the pavement are square, Alberti (and
Leonardo after him) was able to combine steps (3) and (5).

To verify the correctness of the construction, Alberti
recommended that the artist draw the two sets of diagonals
of the square tiles. Because each set consists only of parallel
lines, each should converge to a vanishing point on the
horizon; these are the distance points D1 and D2 in Figure
1-12. These distance points are important for a reason suggested
by their name: In the construzione legittima, the distance
between the vanishing point and each distance point
is equal to the distance d between the center of projection
and the picture plane.[9]