Reversion to the Extraordinary

As I expected, the craft and trade of surveying is filled with all manner of neat geometric tricks: in the face of imperfect equipment and inevitable errors in measurement, one does some mighty clever things to eliminate imprecision.

Take, for example, the problem of levelling. One of the most fundamental techniques of surveying involves looking through a telescope which is "absolutely" level. Once you have a telescope known to be level, you can look through it at two different points where your assistants are holding up vertical rods with heights marked along them. The difference in the measurements tells you how far the one point is above the other. Of course, your computed elevations will only be as correct as your telescope is actually level. It's thus of paramount importance to have a way of making your telescope very close to horizontal.

The most common technique of doing this is to fit the telescope with a "level tube:" a fluid-filled tube, parallel to the telescope, with an air bubble inside. When the bubble is exactly centered in the tube, the tube is horizontal. And hence the telescope is horizontal.

Well, more or less horizontal. What if the tube is mounted out of alignment, so that there's a slight angle between tube and telescope? Well, then, and this is the brilliant part, you use a technique called reversion to align the telescope and tube.

So first, you level the tube telescope as normal. When the bubble is nice and centered, you look through the telescope, which will be some place close to -- but not exactly -- horizontal. You have your assistant pick up a measuring rod, walk a ways away, and hold it vertically in the line of sight of the telescope, with the bottom end touching the ground. You take note of where along the rod the telescope is aimed. Say, for sake of example, it's at 1.137 meters off the ground.

Now, you take the telescope and flip it upside down. In your standard surveying device (called a "transit"), the telescope is mounted on a pair of wheels, one vertical and one horizontal, so it can point in any direction and do complete 360-degree turns. Upside-down is no problem; you just give the vertical wheel a half turn. Since the telescope is now pointing in the exact opposite of the way it used to be pointing, you need to give the horizontal wheel a half turn, too. Now you take the scope and look through it and make small adjustments until you're looking at the rod your assistant is holding, and then you make tiny adjustments until you're staring right at the 1.137-meter mark again.

See what that's gotten you? The telescope is aligned exactly the same way it used to be, because you're still sighting along the same straight line. But the level tube has been flipped upside-down. Instead of hanging down from the telescope, it's now projecting up. If there was an angle a between the tube and telescope, the tube is now 2a away from being horizontal.

Think of it this way. You levelled the tube. It was horizontal to start with. The telescope was a away from being level (let's think of it as being slightly too high at the front end). When it was that far away from being level, you were looking at the 1.137-meter mark on a fixed rod. Then you fiddled with the wheels and did some mojo that ended with the telescope looking at the exact same spot. So it's still a above the horizontal The tube, however, has been mirror-reflected, as it were, through the telescope. If the front end were some smidgen further away from the scope than the back end, it's still that smidgen away, just on the other side. So the tube is at an angle of 2a to its original position.

From here, the game is easy. You adjust the screws holding the tube to the scope. You make sufficient adjustment to bring the the bubble halfway from where it is towards the center. (That's because a, the actual bad angle between scope and tube, is only half of 2a, which is how far the tube is currently away from level.) Now, you go back to the start. Put the telescope back in its normal position and level the tube. Take a sight on the rod; flip the scope over and align it with that sight again. If you've done things well, the bubble in the tube will still be precisely centered.

This is a great trick. It works anywhere you have two different systems of measurement, each of which can be checked for internal consistency, and whose relative divergence, while not itself directly measureable, can be reversed. By means of the reversal, you turn the error term into something you can add to itself and then measure on one of your two scales.

Its partner-in-crime involves the subtraction of systematic errors: you rig things so that the same error shows up on both sides of an equation and cancels with itself. For example, rather than measure difference in heights by starting at one point and looking at the other; you measure both points by looking at them from some arbitrary third point. That way, you don't need to worry about how far the telescope is off the ground; you just establish that height, whatever it happens to be, as the zero of your coordinate system.

Provided this neutral third point is roughly equidistant from the two endpoints, you'll also get rid of a couple of other systematic errors this way. You'll cancel out any residual levelling error too small to detect with the tube. It doesn't matter if you're dropping 2cm in every 100m; if both points are 200m away from your scope, you'll be 4cm too low on both measurements, but that 4cm will be subtracted from itself when you take the difference of your measurements. On long-haul high-precision surveys, you'll even cancel out the effects of the earth's curviture and of atmospheric refraction.

Or, even more cleverly, you can use this trick to get rid of horizontal deviations in your telescope when you're constructing a straight line. Typical surveying straight lines are set up by starting at A, marking a spot B along the desired line, then taking the transit out to B, flipping the scope over vertically to line up with A, then flipping it back to point in the correct direction, where you set up a new point C and repeat. But this technique is vulnerable to unevenness in the vertical wheel; if you skew a little left when you flip, your line will skew progressively more to the left as you go along.

Let's think about the geometry for a moment. Say that when I spin the wheel vertically, I'm off by one degree to the right. If I now turn around, face the way I was coming from, and spin the wheel back to point the way I came from, it'll be back where it started. This means that the reverse 180-degree vertical flip causes a deviation of one degree to the left. Forward gives me a one-degree deviation right; backwards gives me a one-degree deviation left. Left, right. Those sound like opposites to me. I smell a reversion, don't you?

When I get to B, then, I start off as above. I take a "backsight" (a sighting with the scope upside-down) to A, then flip the scope over vertically and have my assistant mark out the appropriate point C. (That is, along the line I'm sighting, we lay out a known length (50 meters might be typical) using the tape, the other standard surveying tool.) Now, I know that C is likely to be slightly out of alignment. So I turn my scope around horizontally and take a regular sighting of A again. Now I flip it vertically and take a backsight, using that line to mark out some other point C' (at the same known distance along the line I'm now sighting).

If there's, say a leftwards angular error a due to skewing when I use the vertical wheel to go from backwards to forwards, then C is 50 cos a meters too far to the left. On the other hand, C' is 50 cos a meters too far to the right. The true point I'm looking for should be exactly halfway between C and C'. But that point is amazingly easy to find; we just tromp on over to C and C', lay down the tape between them, and find the halfway point. This new point, C*, becomes the point we use when we extend the line forwards from B, and so on. Any error introduced by eccentricity of the vertical wheel's mounting point in the transit has been cancelled out.

It is left as an exercise for the reader to work out other, more metaphorical, applications of the principle of reversion.