Fig. 2.13
Plan and elevation of Masaccio's Trinity (according to Sanpaolesi,
1962, figure C, opp. p. 52).

Proposition
4:It is possible, by making assumptions about properties
of the scene, to solve the inverse problem of perspective: given the point
projection of a scene, we can reconstruct the scene including its plan and
elevation.^{14}

For instance, Sanpaolesi (1962)^{15} proposed the reconstruction of
Masaccio's Trinity (Figure 2.1), shown in Figure 2.13.

To gain some insight into what Alberti taught his contemporaries, we should
examine what Renaissance artists came to call the costruzione legittima
(legitimate construction) of the perspective drawing of a pavement consisting
of square tiles. Figure 2.14 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 2.14 have numbered 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 2.15 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).
Comment: I am not really sure how to revise this section.
We need to discuss the relation between the costruzione legittima,
which proceeds by plan and elevation, and the costruzione abbreviata,
which is more like Leonardo's method and is all that is needed for constructing
a pavimento. The plan and elevation approach is really needed for
more elaborate building structures.

LEFT
Fig. 2.14 Construction of perspective representation of a pavement consisting
of square tiles.

RIGHT
Fig. 2.15 Leonardo da Vinci, Alberti's construzione legittima
(ca. 1492). Manuscript A, Fol. 42r. Bibliothèque de l'Institut,
Paris.

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 pointsD_{1}
and D_{2} in Figure 2.14. These distance points are important
for a reason suggested by their name: In the costruzione 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 (see Box 2.2).^{16}

Box 2.2
In one-point perspective, the distance between the vanishing
point and a distance point equals the distance between the center
of projection and the picture plane To demonstrate this fact,
find the central projection of a horizontal line, passing through
the center of projection O and forming a 45° angle with
the picture plane (line 9 in the plan of Figure 14). Because,
as we have seen, all lines that pass through the center of projection
are represented as a point, the representation of this line
is the intersection of line 9 with the picture plane. This intersection
is D1 for line 9 is parallel to the diagonals that converge
at that point. Now consider the triangle O VD1. Because it is
a right triangle with one 45° angle, it is isosceles; and
because the length of O V is d, the length of D1V is also d.
QED.

Box 2.3
In two-point perspective, to determine the center of projection
of a picture, it must represent at least two distinct objects
each with two conjugate vanishing points so that we have four
distinct vanishing points QED.

^{14} This problem was analyzed in great depth by
Jules de la Gournerie (1814-83) in his monumental Traité de Perspective
Linéaire (1884). Methods such as his have been applied to a substantial
number of works of Renaissance art. For a recent bibliography, see Welliver
(1973) and http://www-2.cs.cmu.edu/~ph/869/papers/Criminisi99.pdf.

^{15} Although Janson (1967, p. 88, footnote 25) argues convincingly
that Sanpaolesi's reconstruction contains errors, I have chosen to reproduce
his rather than Janson's because its elevation shows the location of the figures,
whereas Janson's shows only the architecture. (See also Battisti, 1971.)

^{16} The distance points are known as a conjugate pair of vanishing
points. For future reference, we may define this term: The perspective images
of any two lines pass through their respective vanishing points. If the lines
to be represented intersect, and if the angle of their intersection is a right
angle, their respective vanishing points are said to form a conjugate pair
that project at a right angle at the eye. It is, in fact, this right angle
property that allows us to use pairs of conjugate vanishing points to define
the correct viewing distance for the perspective construction.