Camera Obscura (from latin: “Dark Room”) can be thought of as an an...
Jean-Auguste-Dominique Ingres was a 19th century French Neoclassica...
### Normal Lens
A scene viewed through a normal lens appears to ha...
### Focal Length
The main purpose of a convex lens is to take pa...
### Estimating focal length
First the authors estimated the angl...
Depth of Field decreases as you increase aperture:
):
...
A diopter is a unit of measurement of the “optical power” of a lens...
Pattern in question ([full res](https://i.imgur.com/e9oILh4.jpg)):
...
Hans Holbein Georg Gisze ([full res](https://i.imgur.com/CT7ov1C.jp...
1047-6938/00/07/0052/08-$0015.00 © Optical Society of America
In this feature, world-renowned artist David Hockney
and University of Arizona optical sciences professor
Charles Falco explain how Hockney’s observation that
certain Renaissance paintings seemed almost “photo-
graphic” in nature led them to launch an inquiry into the
possibility of finding scientific evidence that some of the
Old Masters relied on optical aids. Hockney’s visual
observations received scientific validation when applica-
tion of basic optics principles to a number of Renaissance
paintings began generating remarkably consistent results.
52
Optics & Photonics News / July 2000
The Hermitage, St. Petersburg
A
n article published earlier this year in OPN
stated “The camera obscura has enjoyed
two lives, one that has been fully docu-
mented by art historians, and a second,
comparatively unknown, as an object of
scientific speculation.”
1
The author of that
piece may have given a bit too much credit
to art historians since, as we describe below, very recent
work now shows that the use of projected images in art
goes back at least 150 years further than previously
thought. Since portraits painted by Renaissance mas-
ters now provide important scientific documentation
of the early use of optical instruments, this discovery
has significant implications for the history of science as
well as the history of art. In an era in which the divide
between artists and scientists was not as large as it is
today, and with a dearth of contemporary written
accounts, the paintings themselves have become pri-
mary source documents.
How did we decide to undertake this research? At a
recent exhibition, David Hockney observed a certain
quality in the 19th century portraits of Jean-Auguste-
Dominique Ingres that suggested that the artist had
used some sort of optical device as an aid.
2
This led to
a detailed examination of a large number of European
paintings. The outcome? The “photographic quality”
observed in the Ingres portrait was traced back to as
early a work as that of Robert Campin (circa 1430).
For a complete account of this extensive visual investi-
gation, including its significance within the context of
our current understanding of Renaissance as well as
modern art, please see the list of references.
3
This article describes the variety of scientific evi-
dence we discovered to support and extend our inves-
tigation. We begin with a brief review of the relevant
properties of imaging optics. The discussion employs
the language of photography since, as will be shown, a
number of Renaissance paintings share the same opti-
cal basis as these modern photographic instruments.
Focal length and geometry
The “normal” lens for a given film format is one with a
focal length roughly the diagonal of the negative, or
43 mm in the case of 35 mm film.
4
Lenses of focal
length significantly shorter or longer than the film
diagonal result in perspectives that are termed, respec-
tively, wide-angle or telephoto. Although the reader
will find this obvious, it’s worth restating in the context
of our work that if a larger piece of film is used, since
its diagonal will be longer (325 mm in the case of an
8" 10" piece of film), it will be necessary to use a lens
of equivalently longer focal length to obtain the same
perspective.
Some photographs contain enough information to
allow us to make an estimate of the focal length of the
lens used. In the case of Figure 1, starting from the
height of the people (1.8 m) we can compute the
horizontal distance across the scene at the location
where the most distant two are standing as approxi-
mately 2.2 m. If we estimate the length of the console
Optics & Photonics News / July 2000
53
Figure 1.
Photograph taken with a 35 mm camera (negative size 24 mm 36 mm).
As described in the text, this photograph contains enough information to allow us to
estimate the focal length of the lens used.
Figure 2.
Frontispiece to Opticae Thesaurus, with Vitellionis Thurinopoloni
Opticaue Libri Decem, edited by Federico Risner (Basle, 1572). A variety of
optical phenomena are illustrated in this engraving. To the right we see
Archimedes use of “burning mirrors” to set fire to the Roman fleet. In the
foreground we see an image projected by a mirror (albeit, not drawn invert-
ed, as would have to be the case for a concave mirror).
of a molecular beam epitaxy (MBE) machine to be
2 m, the horizontal included angle is given by:
tan(/2) = [1/2 horizontal distance across scene]
/ [length of MBE console]
or
= 2 tan
-1
[2.2 m / 2 2 m] (Eq. 1)
58˚
which corresponds to a 22-mm focal-length lens (the
lens used was actually 24 mm).
Effect of aperture on depth of field
Although a lens can be focused perfectly at only one
specific distance at a time, a field on either side of that
distance will be acceptably sharp. How far that depth of
field (DOF) extends depends on three factors: the per-
missible circle of confusion C, the relative aperture of
the lens f #, and the image magnification M relative to
the original scene, as given by:
4
Total DOF = 2 C f # (M + 1) / M
2
(Eq. 2)
The circle of confusion C is the smallest feature on an
image that a viewer can distinguish from a point. For
purposes of photography, it is commonly accepted that
on a high-quality print, an image area smaller in diame-
ter than V/1000, where V is the viewing distance, will
be indistinguishable from a point. As an example, for a
large photograph produced with modern camera and
enlarger lenses, and examined at a distance of 50 cm, fea-
tures smaller than about 0.5 mm would appear to be
points.
Next is the relative aperture f #, which is the ratio
of the focal length of the lens f to its effective diame-
ter D :
f # = f / D (Eq. 3)
From Equations 2 and 3 it can be seen that for a given
focal length lens, the DOF depends inversely on the
effective diameter D, and thus can be increased by mask-
ing off (“stopping down”) the outer area of the lens.
Finally, there is the magnification M. This is the
ratio of the image size to the subject size, which is also
the ratio of the lens-image distance to the lens-subject
distance:
54
Optics & Photonics News / July 2000
Figure 3.
The Doge Leonardo Loredan, Giovanni Bellini, 1501–05. This figure
shows how a real image would be projected onto film, or a canvas, by a
refractive lens [left] and by a concave mirror (“mirror lens”) [right]. The fact
that, after subsequently inverting the image, a mirror lens has left the origi-
nal symmetry of the scene unchanged, has significant advantages for an
artist using such a lens as an aid.
Figure 4.
The Marriage of Giovanni Arnolfini, Jan van Eyck, 1434.
59.7 cm 81.8 cm. The distorted image of the window reflected by
the mirror on the back wall shows that it was convex. If the back side
of this convex mirror had been silvered, the resulting concave mirror
could have been used to form images.
The National Gallery, London
The National Gallery, London
M = image size/subject size
= lens-image distance/lens-subject distance (Eq. 4)
Returning to Equation 2, if we are able to estimate
the DOF in a given photograph, and also to extract the
geometrical factors necessary to estimate the magnifica-
tion and focal length of the lens used by the photogra-
pher, we can then use Equation 3 to calculate the effec-
tive diameter of the lens.
Effect of focus on magnification and on vanishing points
The Gaussian lens formula relates the focal length of a
lens f to the distance between it, the subject and the
image:
1/f = 1/d
lens-subject
+ 1/d
lens-image
(Eq. 5)
Since as Equation 4 shows, the magnification depends
on the relative distances, a consequence of Equation 5 is
that if a lens is moved for purposes of refocusing, the
relative distances change, as does the magnification.
However, this effect is seldom noticed since the “typical”
photograph involves a small magnification. In a com-
mon snapshot, for instance, the image of a 180 cm tall
person ends up less than 24 mm high on the film,
resulting in a magnification of only ~10.
-3
As M increas-
es, however, changes in M upon alteration of the focus
can become noticeable.
Another principle of photography important to the
research at hand is that parallel lines converge to a van-
ishing point related to the focal length and position of
the lens. Thus, refocusing a lens, or indeed, moving it
for any reason, will change not only the magnification
of the image but also the vanishing point.
Optical aberrations
Various aberrations (astigmatic, coma, chromatic, etc.),
all of which degrade the quality of the image, are exhib-
ited by lenses. For example, in the case of spherical
aberration, rays from the outer edges of a lens are
brought to focus closer to the lens than are the central
rays. The only way to reduce this effect in a single-ele-
ment lens is to decrease its effective diameter. In the
range of lens sizes that concern us, other aberrations
are also reduced as the lens diameter is decreased.
Unfortunately, stopping a lens down reduces the
brightness of the image, resulting in a tradeoff between
sharpness and brightness.
The concave mirror as lens
The optical properties of concave mirrors have been
studied since the time of Euclid. Figure 2, for instance, is
an engraving from a 1572 book showing what appears
to be Archimedes’ use of focused light from several con-
cave mirrors to defend Syracuse from a Roman fleet in
212 BC. We discovered in the course of our research,
however, that outside the scientific community, there is
scant awareness of the fact that an image can be formed
with a concave mirror. For this reason, we will use the
term “mirror lens” in this article to make explicit the
imaging properties of the concave mirror. Although
both refractive lenses and mirror lenses can form
images, images formed by mirror lenses have a particu-
larly significant characteristic. In both cases the image is
inverted but, since a mirror reverses left to right, the
result is that the symmetry of the final image created by
a mirror lens is identical to that of the subject. This is
illustrated in Figure 3. The importance of this from the
point of view of an artist fashioning a painting is dis-
cussed elsewhere.
3
We will emphasize mirror lenses in our discussion
since we have uncovered a variety of circumstantial evi-
dence pointing to their possible use. However, it should
be noted that we have as yet found no scientific evi-
dence that might distinguish between portraits made
with the aid of concave mirrors rather than refractive
lenses. What evidence is there that the fabrication tech-
nology to produce such an optical element existed in the
early Renaissance? Jan van Eyck gives us one answer in
the 1434 portrait shown in Figure 4. Until an opaque
protective coating had been applied to the back side of
the convex mirror on the wall, its obverse side would
have been a mirror lens. In addition, concave mirrors of
polished bronze and speculum metal did exist in
Medieval times and in antiquity.
Summary of relevant lens effects
To summarize the specific aspects of lenses we have
drawn upon for our analysis of Renaissance paintings:
• If we know the geometry of the original scene, and
the size of the canvas (film), we can calculate the
focal length of the lens used;
• From the focal length and depth of field we can cal-
culate the diameter of the lens;
• If a lens is moved to alter the focus, the magnifica-
tion of the image will change;
• If a lens is moved between two exposures, a second
vanishing point will be created;
Optics & Photonics News / July 2000
55
Although both refractive
lenses and mirror lenses
can form images, images
formed by mirror lenses
have a particularly signifi-
cant characteristic.
• 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