This page derives a geometrical and mathematical model for the
lens and camera combination used most often in the Apollo lunar
surface photography. This model then becomes useful in reconstructing
the three-dimensional scenes captured in the photos.
Aim your camera straight down at the floor. Look through the
viewfinder and place an object on the floor at each corner of the area
visible in your viewfinder. These objects represent the corners of a
base of a pyramid. The apex of the pyramid is your camera lens. This
pyramid is what can be called a "view volume". Everything your camera
sees in this arrangement is inside that pyramid.
When you aim your camera at real subjects, that pyramidal view
volume is still there, but it's lying on its side. And instead of
stopping at the floor, it continues on forever. The line coming
straight out of the camera is what we will call the "optical axis".
It's the line from the apex of the pyramid to the center of the base.
ANNOTATIONS BY CLAVIUS
If we discover something about this view volume and learn to
describe it mathematically, we can often use this information to help
us understand what we see in photographs.
Your view volume is probably a little squashed. That's because
video cameras and 35mm cameras use a rectangular frame format that's
longer horizontally than it is tall. The Apollo Hasselblad cameras
used a square frame which helps us out a lot.
The basic dimension of the view volume is the angle the pyramid
walls make with each other. A narrow pyramid contains only objects
near the optical axis while a wide pyramid includes all objects in the
vague direction of photography. If we relate this to camera lenses,
we find that telephoto lenses (and zoom lenses that are zoomed in
tight) have narrow view volumes while wide-angle lenses have wide view
volumes. This relates to the apparent magnification of the image.
Both types of lenses record images on a frame of the same size. So if
the contents of a narrow view volume are presented on a television
screen, the individual objects appear larger in the frame. Similarly
if the contents of a wide view volume are presented, each individual
object appears smaller or farther away.
The manufacturers of camera lenses provide measurements that help
the photographer determine the shape of the view volume. Usually the
lens maker lists the "focal length" of the lens. If you know the size
of the film frame, a formula coverts this to the "field of view"
(FOV) angle. Sometimes lens makers will supply the FOV angle directly
for certain film sizes.
The Zeiss Biogon lenses provided to NASA for the lunar surface
cameras had a focal length of 60mm. This provides a FOV of 71°
for the 70mm film frame supplied in the Kodak Ektachrome longroll.
A BIT OF
This 71° FOV angle is a diagonal FOV, measured from
diagonally opposite corners of the frame. Going back to the pyramid
example, draw your line from the lens to one corner of the viewfinder,
then another line from the lens to the diagonally opposite corner of
the viewfinder. Then measure the angle between these two lines. With
the Zeiss Biogon lens this angle would be 71°.
This isn't very practical for measuring angles in photos. We're
more interested in horizontal and vertical angles -- the angle between
opposite faces of the pyramid. Fortunately a little
trigonometry helps us.
The view volume pyramid's diagonal
angle is 71°. We arbitrarily set the height at 1, and we note that the angle from the
pyramid's axis to any of the edges is
35.5° -- half the diagonal angle. It also creates a right
triangle between the apex, the center of the base, and the corner of
This allows us to compute the distance from the center of the
postulated base to any of the base corners as the trigonometric
tangent of 35.5° -- 0.713. This also forms the hypotenuse of an
isosceles right triangle lying in the pyramid base.
The Pythagorean theorem allows us to compute the length of the sides of this right triangle, 0.504. The angle
from the optical axis to the center of the edge is the arctangent of
this length (since we arbitrarily defined the length of the axis as
1), or 26.75°.
Therefore the horizontal (and vertical, since the frame is square) FOV
is twice this angle, or 53.5°.
ANNOTATIONS BY CLAVIUS
The Apollo lunar surface cameras were fitted with reseau plates
that overlays a grid of small crosshairs, called "fiducials", on every
photo. The center fiducial represents the optical axis and it's a
little larger than the others. The reseau grid can help us locate
objects in the photo, and sometimes even to measure them. A moment's
thought tells us that a fiducial corresponds to a certain angle from
the optical axis. And if we can measure objects relative to the
fiducials, we can determine their angular size, or the difference in
angle between them.
We know from our work above that the horizontal and vertical FOV
for the Biogon lens is 53.5°. But in most of the photos the edge
of the image is not a reliable indicator of the full FOV. The film
can jiggle around in the magazine, or slip in the camera. But we
still need a way to relate the fiducial coordinate system with the
theoretical edges of the frame.
When confronted with a problem we can't solve analytically, we
have to resort to empirical methods. That is, we have to measure some
stuff and hope we measure enough so that any errors cancel each other
First we consider the reseau grid to be a coordinate system. The
origin is the optical axis, the center fiducial. One fiducial to the
right is 1.0 on the x-axis, while one fiducial down is -1.0 on the
y-axis. It then follows that the edge of the photo should have some
value in that coordinate system.
We turn to Michael Light's excellent book Full Moon. Light
claims to have preserved the entire transparency frame in his prints.
So we can measure where he thinks his idea of the frame edge lies. It
turns out that the edges of all his photos are 2.59 fiducials away
from the center.
Next we turn to the Apollo
Lunar Surface Journal (ALSJ), which maintains online a
repository of scanned Apollo transparencies. The JPEG images it makes
available contain the entire frame, plus a surrounding black border.
Picking 20 images at random and measuring where the image border falls
in the fiducial coordinate system, we find that Light's 2.59 fiducials
is consistent with ALSJ usage.
So if 2.59 fiducials corresponds to 26.8° (half the horizontal
or vertical FOV), then we have another right triangle we can use to
measure any angle in the photo whose distance is represented in the
fiducial coordinate system.
The other leg of the triangle is computed as 5.127 (fiducials),
meaning that for an object measure at F fiducials left, right,
up, or down from the center fiducial, the corresponding angular
difference, A, from the optical axis is given by the formula
A = arctan ( F / 5.127 )
This can be used to locate objects in the frame relative to the
optical axis, or to orient the optical axis relative to some known
object in the frame.