Fundamentals of Alignment Patterns
The magic of two mirrors
Useful things happen when two mirrors face each other. If one mirror has indicia (marks of some kind) on or near it, the indicia reflect back and forth between the two mirrors, making a pattern from the repeated images. It is easy to recognize when the pattern shows misalignment, whether you view it when mirrors are stationary, or while you raise and lower an enlarger head, reposition a copy camera, or adjust view camera standards. The pattern points in the direction where planes are closest.
These are examples of alignment patterns:
Figure 1. Patterns
Using two mirrors is the most sensitive optical method. A system using a single mirror is not nearly as sensitive because it makes only one repeat, which does not create a pattern. Click here for a basic introduction to alignment patterns.
- An alignment pattern created by two mirrors is so sensitive, precise, and easy to use that it motivates you to find the cause of alignment problems.
multiple repeats – pattern power = amplification
Lack of parallelism changes the shape of an alignment pattern in the same way as compound interest changes principal. Principal does not increase noticeably for the first few interest periods, but the increase is obvious after awhile. Interest periods are like visible repeats in the pattern; each subsequent repeat doubles the error shown by the previous one. The more repeats, the more the "tail" (the part of the pattern closest to the view hole) amplifies error, the more sensitive it becomes, and the more accurate you can be.
When looking at the view hole portion of a two-mirror alignment pattern, you see only the first repeat of the view hole if the mirrors are already close to parallel. Additional repeats of the view hole are smaller and therefore out of sight behind the first one. But repeats of the indicia are far enough away from the view hole that they continue to be visible until they reach the hole where repeats can no longer continue.
The first few images of view holes may come into sight if parallelism is poor enough. But when the pattern is symmetrical, all view hole images are concentric.
Click the thumbnail image below to view an animation of the power of multiple repeats.
Picture an enlarger with its negative stage two feet from the paper, one mirror on the negative plane, another mirror on the paper plane. Individual repeats that form an alignment pattern make the two-foot actual distance become an even larger effective distance that increases with every repeat.
Effective distance of a given repeat is simply twice the actual distance between mirrors multiplied by the repeat number of the indicia (x = 2d #), where x = effective distance, d = actual distance, and # = repeat number).
The formulas used in Advanced Topics assume that the planes are parallel. Formulas for non-parallel planes are complex, but the simpler formulas can apply when alignment patterns of repeated images are almost perfect.
virtual images – behind the scenes
A picture can be worth a thousand words.The diagram below shows, among other things, the unseen events that take place when images are repeated between parallel planes.
Figure 2. Parallel Planes |
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If you look carefully at the image of yourself in a mirror, you will see that your image is beyond the plane of the mirror. Such an image is called a virtual image. Apparent location of a repeat's virtual image is the intersection of two lines, one normal (90 degrees) to the indicia and the other an in-line continuation of the repeat's last ray leg.
If strings of Christmas tree lights were connected in series and one light went out, all the lights would go out. The visible components of a two-mirror alignment pattern are like strings of lights connected in parallel; it is possible to put "out" some repeats while others remain "on". No visible repeat comes from the previous one. Each one is independent of the other.
This independence can be illustrated by a simple experiment using zig-align's LED module. Place your fingertip on the plain mirror opposite an LED; the last repeats disappear while the first ones remain. Your fingertip blocks initial ray legs of the last repeats but not of the first repeats. Now cover up an LED. All images are eliminated, not just some as before.
There are actually many other rays coming through the view hole from the indicia. As you move your head while you watch the pattern, you continue to see an image, but it comes from different rays taking different paths. This introduces error that, for a small amount of head movement, is negligible. (see system sensitivity chart in Accuracy is not the Same as Precision and following calculations).
ray paths
Rays reflecting between two mirrors form a progression of repeated images. The entire ray path of each repeat has a set of incident and reflected legs. The total sets of ray-legs equal the repeat number of the indicia. (See Figure 3.)
When a repeat is visible at the view hole, inclination of the repeat's first leg is due to the original parallelism, or lack of parallelism. And because the angle of incidence equals the angle of reflection, the reflected leg contains the same information as the first leg.
In this way, each V (created from incident and reflected legs) doubles the error accumulated by the previous V, so each subsequent repeat exponentially amplifies error shown by the first repeat. The more repeats in the pattern, the more the "tail" moves, and the easier it is to notice lack of parallelism.
In the drawing below, the alignment pattern – formed by rays –
points to the left where planes are closest. There are always more Vs
on the side where planes are furthest apart, but depending on the degree
of misalignment, not necessarily in the 2:1 ratio shown.
Figure 3. Non parallel planes
The process begins with the first repeat:
Figure 4. Non parallel planes
Each additional repeat makes an additional V, doubling the previous repeat's error – amplification in progress.
viewing error
A plane can rotate about X, Y, or both.
Figure 5. Error
Indicia project light rays in many directions, but only one set of the projections has the required inclination to pass through the view hole to your eye. Rays and their inclinations are invisible, but they create a visible alignment pattern. As parallelism varies, different rays reach your eye at the view hole. Thus, different repeats become visible, causing the pattern to change shape. In this way, the pattern becomes a two-dimensional representation of what is happening in three dimensions.
If a million repeats were in a pattern and you could see them all, the pattern would exactly represent parallelism of planes. Find out why you need only a few repeats in the "System Sensitivity" section in "Accuracy is not the Same as Precision".
it's graphic
Now back to amplification. The equation y = 2x (y = amplification;
x = repeat #) gives the amount of amplification of the xth repeat. The
first graph is the usual plot of this equation.
Figure 6. Uniform Spacing
The second graph maintains the same amplification as the first graph but shows the actual position of repeats on the x-axis rather than uniform increments.
Figure 7. Actual spacing
The Parallel Planes drawing (Figure 2) shows the locations of reflections and illustrates their actual positions on the x-axis of the second graph. The first repeat is always halfway between the indicia and center of the view hole; the second repeat, 3/4 (halfway between the first repeat and view hole); the third repeat, 5/6 (halfway between the second repeat and view hole); etc. Thus, all repeats maintain their sequence as they appear closer to the view hole.
Two mirrors exaggerate the exponential nature of the amplification as the second graph illustrates. Images of the last repeats appear closer and closer to each other, creating a sharp bend in the curve. As seen from the view hole, repeats also diminish in size. Finally they seem to merge. The combination of ever-closer distances between repeats and their growing accumulation of error makes an alignment pattern easy to use.
The second graph also shows there is space between the last distinct repeat and the edge of the view hole. The merged repeats are in this space. But if you magnify the pattern, you can see additional distinct ones. That makes the pattern even more sensitive, which is especially important for large prints.
a universal language – math
The math of how repeats amplify error is like the progression of f/stops or the Zone System's exposure units.
Figure 8. Equivalents