• Concave mirrors (“mirror lenses”) can be used to
create images;
• Mirror lenses leave the symmetry of the scene
unchanged;
• The image formed by any simple lens—mirror or
refractive—will exhibit significant aberration that can
be reduced only by decreasing the diameter of the
lens;
• Spherical aberration, astigmatism, and coma limit
the useful area of an image, even if the diameter of
the lens is reduced.
Husband and Wife
, Lorenzo Lotto, circa 1523-4
Figure 5 has turned out to be a remarkably rich painting
in the context of our work, so we will discuss our analy-
sis of it in some detail. Figure 6 shows the central geo-
metric pattern on the tablecloth in this painting. What is
immediately striking is that Lotto painted a pattern that
appears to go out of focus as it recedes into the scene,
just as happens in a photograph when the DOF of the
lens is exceeded. Since the human eye automatically refo-
cuses as it traverses different depths of a scene, such an
out of focus feature would not have been visualized by
Lotto’s unaided eye alone. Although this painting pro-
vides striking evidence of having been based on a pro-
jected image, and even though eyeglass lenses were pro-
duced at least as early as circa 1300, documentation of
the first optical instruments to employ refractive optics
does not appear until the mid-16th century (Janssen’s
compound microscope of 1590 and Galileo’s telescope of
1609 are commonly cited, but the earliest known exam-
ple is Girolamo Cardamo’s 1550 description of a camera
obscura incorporating a lens). Demonstration that an
optical instrument was in fact in use prior to circa 1550
is thus an object of interest in the context of the history
of science as well as the history of art.
Interestingly, as can be seen in Figure 6, in the same
region of the painting where the image loses focus, the
vanishing point also changes. Had Lotto laid out the
pattern geometrically, following, for example, the prin-
ciples articulated in the 15th century by Leon Battista
Alberti,
5
the chance of such a change taking place would
have been minimal. However, had he traced the pattern
from a projected image, and had he moved the lens in
an attempt to refocus after the DOF had been exceeded,
such a change would be completely natural.
If we examine the triangular pattern toward the right
edge of the table, we discover that the vanishing point
changes here as well, at the same depth in the scene that
the central feature goes out of focus and the vanishing
point changes. However, unlike the central feature, the
triangular pattern on the right remains in focus all the
way to the back of the table. As discussed below, this can
also be easily explained by an optical analysis of the
painting. In fact, not only does this work provide con-
vincing scientific evidence that the artist used a lens,
there is actually enough information to allow us to cal-
culate its physical properties.
Assuming the width across the shoulders of a typical
woman is 50 cm, and measuring the corresponding
width in the original painting to be 28 cm, the magnifi-
cation M 0.56. For reasons explained in detail below,
we believe that intrinsic aberrations of the lens did not
allow Lotto to project the entire image onto the canvas
at one time. Instead, he was forced to piece it together
from several projected “frames,” each of height and
width 30-50 cm. (It should be noted, however, that
neither this assumption, nor the precise dimensions, are
critical to our analysis of this painting.) Since the visible
portion of the tabletop occupies a width of approxi-
mately 52 cm on the original painting, this portion cor-
responds to one projected frame.
To determine the precise focal length of the lens Lot-
to used would require accurate measurements of his
camera obscura. However, we can make a reasonable
estimate if we assume Lotto’s studio was roughly 3 m
deep. Allowing 1 m of that for the table and subjects
leaves a 2 m working distance. As Figure 7 shows, if the
lens were located 1.5 m from the subjects, the magnifi-
cation of M = 0.56 would result in an ample 84 cm
working distance between it and the canvas. With these
values for the lens-subject and lens-canvas distances,
and using Equation 5, we find the focal length of the
lens was f 54 cm. Interestingly, the diagonal of our
assumed range of projected “frames” is in the range 42-
70 cm, so such a lens would have provided a “normal”
perspective for this frame. Also, whether Lotto used a
refractive or a mirror lens, the diopter strength was
K
diopters
= 100 cm / 54 cm = 1.86 diopters
so the curvature of its surface was equivalent to that of a
pair of reading glasses.
The triangular pattern in the tablecloth of Figure 5
serves as a built-in fiducial that allows us to determine
additional information about the lens. There are seven
repeats of the triangular pattern across a 14.48 cm span
of the painting at the front of the table, so there is a
spacing of 2.07 cm per triangle. Since the magnification
is 0.56, this means there was a spacing of 3.70 cm per
triangle on the original. The central pattern seems to go
“out of focus” at 5-9 triangles deep into the scene.
Since the pattern is at an angle of about 30˚ with respect
to the camera lens, this is a distance of (5-9) 3.70 cm
cos30˚ = 18.5-33.30 cm 0.866 22.56 cm from
the front edge of the table. If we assume that Lotto ini-
tially focused on the front edge of the table, we now
know he exceeded the DOF of his lens at a distance of
22.56 cm.
Equation 2 allows us to calculate from this informa-
tion the physical size of the lens Lotto used. If we assume
a circle of confusion on the painting of 2 mm from his
simple lens, we find f# 22, and thus the diameter of
the lens D 2.4 cm. As we have confirmed from our
own experiments, a concave mirror of this focal length
and diameter projects an image that is bright enough,
and sharp enough, for an artist to use when the subject is
56
Optics & Photonics News / July 2000