Figure 1I shall limit this discussion to single lens photography. Various forms of stereography and holography, since they are not in common use at this time, shall not be considered, herein.
I shall also limit these notes to regarding objects of a rectilinear aspect, that is, of straight edges and right angles, since curved objects present sufficient ambiguity in the image as to preclude the possibility of simple analysis yielding useful results.
Rectilinear objects contain redundancies that can be used to advantage. For example, a rectangular box that is three inches high at one corner is also three inches high at another corner. Since these two corners are not in the same place they present the film with the same information taken from two different angles. When you have two photos of the same object taken from two angles, the differences between them are caused by parallax. Parallax is the phenomenon that allows you to see things in three dimensions using two eyes that see things from different angles. The synthetic parallax present in the single lens photos of rectilinear objects behaves the same way in that it can be used to extract information about the third dimension.
Think of the lens of the camera as a single point in space, called the point of view (PoV). In front of this point and in infinite planes lies the world; behind it lies the single plane of the photographic film. There exists an imaginary line running through this point and perpendicular to the plane of the film. It is along this perpendicular, or normal, axis of observation that the camera is "pointed" and it defines the center point of the photograph. (See figure 1)
Figure 1
Unfortunately, some lenses will distort the projection, rendering it difficult to analyze. The two common types of lens distortion are pillow distortion, where the corners of the photo seem stretched out, and pincushion distortion, where the corners are compressed towards the centre. In both of these, lines that are straight in real life end up with a curve on the photograph. You can check for lens distortion by finding a line in the photograph that corresponds with a straight line in the object and checking it with a straight edge or ruler. If it does not line up with the straight edge, then lens distortion must be accounted for, which is very difficult to do. Since museum photographers know this, museum photographs are usually taken with cameras and lenses that minimize these distortions. If you are taking your own photographs, you may minimize distortion in your photographs by using a long focal length lens. Failing that, take a photograph of rectangular graph paper using the lens you intend at the distance you intend. The resulting photograph will help you determine and correct for distortion.
The scale is the ratio of the size of the object as measured on the photograph to the size of the object that was photographed. This is not a constant from one photo to another and it not a constant from one point to another within the same photo, if the object that was photographed was not flat and parallel to the plane of the film.
Figure 2
By way of example, please consider figure 2. A and B are of equal size and, for convenience, are flat and parallel to film plane f. When the image of A is projected to A' on the film, and the image of B is projected to B' on the film, each has a different scale, due to the fact that B is further from the PoV than is A. Subject to distortion by the lens at PoV, the differing scale is strictly linear; a fixed function of the ratio determined by calculating (distance from PoV to film) / (distance from PoV to object).
Enlargements made from the negative follow the same rules, so although the scale will be different, the linearity is preserved. This means that you can work from an enlargement of any size using the same methods.
The apparent distortions of close things appearing larger than far away things is called perspective.
Figure 3
You can take advantage of the linearity of perspective to interpolate measurements. Consider figure 3. The object being photographed is a rectangular box and is at an angle with respect to the plane of the film in the horizontal axis (as seen from above), although the box and the film are parallel along the vertical axis (as seen from the side). Because of this, the farther end of the box will have a smaller scale than the closer edge, resulting in a photograph that would look somewhat like figure 4. Although the box is perfectly rectangular, the photo is full of different angles and one end of the face is smaller than the other.
Figure 4
Exercise:
There is an object about halfway along the face of the box that you wish to measure (see figure 4, again). You know from previous calculations or from a direct measurement that the height H of the box is 145mm.
From this, h1 and h2 both represent 145mm. Since h1 is 12mm and h2 is 7mm, when measured on the photograph, the scale at A is h2/H or 7/145 or about 1:21 and the scale at B is h1/H or 12/145 or about 1:12. Line A-C is 30mm, thereby indicating a change of scale (called dS) of
dS = (H/h2 - H/h1) / AC
= (21-12)/30
= 9/30
= 0.3 scale points per mm
We wish to measure the height of the object. Determine point B between A
and C such that a line B-B' drawn parallel to the corner C-C' will fall
across the object that we wish to measure and measure that distance (labeled
x on the figure). The distance CB measures as 14mm. The distance x is
3.5mm.
We now have enough information to measure the quantity on the original box that x represents. First we determine the scale at B-B' by multiplying CB by dS and adding to the C-C' ratio thus:
scale = 1 : (C-C' + (CB X dS))
= 1 : (12 + (14 X 0.3))
= 1 : (12 + 4.2)
= 1 : 16.2
The value 3.5 mm can then be multiplied by this scale
x(orig) = x X scale
= 3.5mm X 16.2
= 56.7mm
yielding a height for the object of 56.7mm.
The width of the object is a much simpler measurement. Since the box was parallel to the film in the vertical axis, there is no changing of horizontal scale when referring to a straight edge, such as the box. Simply measure AC on the photograph and divide it into the actual length AC of the box. Multiply this result by the measured width of the object along line A-C to yield the actual width of the object.
Please note: In these examples I have rounded the numbers so as not to clutter the text with excessive decimal numbers. Left this way, these cumulative effect of these round-offs will introduce error into the final result. In the real world, I would carry the measurements to the limit of their significance. That means that I would keep decimals to the first place after the resolving limit of my measuring device (ruler). Upon division or multiplication, I would keep two places past, but round off the final result.
The distance of the PoV from the object being photographed will affect the amount of perspective distortion you will see. Some measurements cannot be made without determining the actual geometry of where the PoV lies in relation to the object or objects being photographed. Because of this, I shall consider this topic in depth.
Figure 5a
Consider figure 5a. A cubical box is photographed off axis but with the front face parallel to the plane of the film. Because the box is off-axis, the top is visible but the back edge, being further away, appears smaller than the front edge, as shown in figure 5b. (Incidentally, because the front face of the box is parallel to the plane of the film, that is to say, the angle is known, if you continue the lines of the sides until they converge at the vanishing point, you will have also defined the point of the axis of observation; the place where the camera is pointing.)
Figure 5b
To assist in my measurements and in the geometric construction that is to follow, I trace the photograph onto a work sheet (see the right hand side of figure 5c).
Figure 5c
Exercise:
Determine the PoV for figure 5b, given that you know the measurements of the box.
Since we wish to know the elevation, orientation and distance from the front face of the PoV, we first must rotate the scene to look at it from the side. In the middle of figure 5c, we have done this. Since we know the dimensions of the box, we can draw it accurately to the same scale as the front face of the box (since that is the reference from which we wish to determine the PoV). Extend the face plane to F-F'. Find where it intersects with the virtual image of the back edge E2 of the box by projecting the back edge (E2-E2'). Draw a line P-P' through this intersection and originating at the back edge of the side view of the box. The point of view will fall somewhere along this line.
Now we must determine where upon line P-P' the PoV lies. If we had a second object in the field, it could be used to generate an intersecting line. Since we don't, we must look elsewhere.
We know from the description of the circumstance that the front face of the box and the film were parallel planes. This means that the axis of observation is at right angles to the plane of the front of the box. Since this axis runs from the PoV to the vanishing point, it can be used to generate that needed second line.
Drawing line V-V' such that it is at right angles to F-F' yields a line that intersects P-P'. This intersection marks the location of the point of view!
Exercise:
Figure 6a
A rectangular box of known dimensions is photographed. The front and top of the box are open. The film is parallel to the front face of the box. The resulting photograph looks like figure 6a. The exercise is to determine the PoV and without using the axis of observation.
As in the previous example, the box is drawn from the side and the back edges E1 and E2 are projected forward to intersect the forward face F-F'. Lines P1-P1' and P2-P2' drawn from the back edges of the side view through these intersection points extend to intersect at the PoV (see figure 6b).
Figure 6b
A rectangular box of known outside dimensions, open at one end, is photographed. The end face of the box is parallel to the plane of the film. The result is shown in figure 7a.
Figure 7a
On first examination of this figure, it appears unremarkable. Closer examination shows a discrepancy. The parallel vertical lines of the back of the box, if extended, do not intersect with the points shown to be the back of the box as seen on its top. There is some sort of septum or false end.
There are two ways of measuring the depth of this false end. For the purposes of example, I shall ignore the method of intersecting the extended back vertical edge with the left or right top edge and working out the proportion. I shall concentrate on using PoV to determine this measurement.
I perform the same operations as in figure 5c. See the results in figure 7b (see next page), wherein I use the same nomenclature for the construction lines. In addition, I project the back bottom edge E3 by line E3-E3' onto F-F'. Running line P2-P2' through PoV and the intersection of E3-E3' will drop an intersect onto the floor of the box. This is the physical location of edge E3 and can be measured directly off the drawing.
Figure 7b
In figure 8 is a case similar to that of figure 5, but without the benefit of a parallel orientation of the box end to the film plane. The resulting photograph looks like 8a.
Figure 8
Here I will introduce to you the concept of the projection line, which must be parallel to the vertical edges of the box. With the outline of the photo to the right (see figure 8b), I project the top and bottom edges of the near end of the box to the projection line. This will define the front face of the box to be drawn in profile.
Projecting it thus has caused a change in scale. I then draw the rest of the box to that scale.
I then project the back edges to the projection line and solve for the PoV the same was as I did in fig 6. One thing to note: if you only have one edge to work from, you can synthesize another projection line by drawing a line from the vanishing point at right angles to the projection line. This only works because the scale and position are a result of projecting the lines and thus synthesizing the correct aspect for treating it as though the nearest end of the box was parallel to the film plane.
I hope that I have made it abundantly clear that keeping at least one edge of the artifact being photographed parallel to the film plane makes later analysis so much simpler. Keeping a plane of the object parallel to the plane of the film makes it even simpler, still.
Lens distortion can turn a simple analysis into an impossible dream. If you are not sure about your lens, take it to a professional camera repair depot where they can put it on a test bed and check its alignment and distortion specs.
Some museums may allow you access to those portions of their collections that are not on display. Write to the curator and ask what the procedure for this is.
If you have to settle for just those items that are on exhibit, bear in mind that some museums restrict access to their displays and collections by photographers. Check beforehand what you need to do to gain access to the museum with your camera. Sometimes it is simply a matter of signing a contract that puts limits on what you may use your photographs for. Occasionally there is a fee.
If possible, use a tripod. The exhibits in most museums are kept in subdued lighting to limit the damage that is caused by light. Using a flash is almost universally prohibited, thus obliging you to use time exposures. The temptation is to use a fast film, but these are usually too grainy to gather much information about an item. I have had many experiences where time exposures of over a minute were needed. You can cut down on reflections from display glass by wearing a dark shirt and carrying a dark cape. Stand behind your camera and spread it out, dracula style, during the exposure; this limits the amount of light reaching your side of the glass. If possible, use a black camera or cover your camera with a black cloth so you don't photograph its reflection. If your camera has any little lights on its front, cover them, also.
Many times you will find that your pictures allow you a much greater depth of detail than you could see with your naked eye. The time exposures fill in the shadows and make visible that which was lost to normal eyes during your visit. Viewing the photographs for the first time is like making another trip to the museum. "I didn't see that when I was there..."
Take notes! If possible, note the focal length of the lens you are using as well as the lens to object distance (read it off the focus ring). These two bits of info can help you measure an object! Remember the scale ratio of (distance from PoV to film) / (distance from PoV to object)? In this case, the distance from PoV to film is the focal length of the lens. Measure the object on the negative or slide (not from the enlargement; that would be a different scale) and scale it to get the desired measurement.
Finally, photograph the damaged sections of artifacts with especial care. It is often in the damage that you can find cues and clues that simply do not show up in the intact portions. Harpsichord jacks sitting askew provide angles to measure and intersects to calculate; missing pieces reveal the structure behind them; chipped paint reveals the underlying material; missing pieces of inlay reveal the method of excavating the channel it was put in; splits in wood reveal the direction of grain and, sometimes, method and orientation of sawing; and wear patterns on coverings reveal much about what lies beneath now or what lay beneath in the past. You could wish you would be allowed to take the thing apart to study it... but nature and time has already done that job in spots. Take advantage of it!