The Arctangential Error Meter Written December 11, 2006   Alongside the road, there's a fence.  When you look at the nearest part, the fenceposts are a reasonable distance from each other.  But as you turn to see the part that is farther away, the posts seem closer together.  Towards the horizon, the distant posts are so close to each other that you can't see anything between them. When I was a young man riding down the highway, I pondered this effect.  Having paid attention during high school trigonometry, I realized that the apparent distance between the posts was related to the tangent of the angle at which I was viewing them. What's a tangent?  A right triangle includes a hypotenuse (5 in this example) and two shorter sides (3 and 4) that meet at a 90° right angle.  One of the non-right angles is marked "theta" in this example.  The tangent of theta is the ratio of the length of the side opposite theta to the length of the side adjacent to theta.  In this example, that would be the ratio of 4 to 3, or 1.333. Then there's another function called the arctangent, which is simply the inverse of the tangent function.  The arctangent of 1.333 is the angle (theta in our example) whose tangent is 1.333.  What's the numeric value of theta?  By using a calculator or a set of tables, we find that it's 53.13º.  Or, in symbolic terms, ARCTAN (1.333) = 53.13º. The arctangent function became relevant in the 1970s, when I was thinking about an ideal display for a rally computer. In the type of rally in which my friend Terry Rockhold and I competed as driver and navigator, our car toured country roads.  We tried to drive at the assigned speeds to remain exactly on schedule, neither early nor late. Bouncing along in the right-hand navigator's seat, I read the route instructions to Terry while trying to follow our course on a map, recalibrating a clock from time to time, and preparing charts of time versus distance. Terry, the driver, had to look at four readouts to stay on time:  his speedometer, his odometer (to determine when he reached half-mile reference points), the clock, and my chart showing what the clock should read at each reference point.  In addition, he had to stay on the road! His job would have been considerably easier, and both of us would have been safer, if there had been a computer on board to handle the calculations.  Some rallyists did have these computers, many of them homemade.  Those rallyists competed in an "equipped" class, while Terry and I were "unequipped." Here is some commercially available rally equipment, circa 1970, including a Swedish-made Halda Twinmaster and Halda Speedpilot (which apparently was a mechanical analog computer). I dreamed about rallying with the help of such equipment.  What would an ideal computer be like?  In particular, what was the clearest method by which the computer could tell Terry whether he should be going faster or slower, and by how much? A digital readout would be a straightforward solution.  The computer, noting the car's location, would calculate that it should be at that location at 2:45:16 pm.  Subtracting the actual time of 2:45:36 pm, the computer would obtain a difference of -20 seconds.  A display of  -20  in front of Terry (or L20 for "late") would show his error, or how much he was off schedule.  He would know to drive faster until the error reached zero. Were I designing a rally computer today, a digital readout would be a no-brainer. But there's another option.  An analog display, though perhaps not as precise as a digital one, can be read and understood more quickly.  You can glance very briefly at the hands of a clock and get a feeling for the time, and then you can return your eyes to the road. The Halda Speedpilot had two analog dials, in addition to an odometer.  The dial on the left in this picture is set to a specified average speed of 40 mph.  The dial on the right is actually a clock with only two hands, both of them minute hands. The black hand moved like the minute hand on an ordinary clock.  The red hand was driven by the computer to show time T = D / R, where D was distance (obtained from the odometer) and R was specified speed (obtained from the left-hand dial).  If the car was moving at the correct speed, the red hand would circle the dial once per hour, just like the black hand.  The driver's job was to keep the two hands superimposed, as shown in the photo above. But this display wasn't accurate enough.  For our purposes, the gearing needed to be changed to make the red and black hands act as second hands instead of minute hands. The Autonav, a more advanced computer selling for $745 in 1972 (installation not included), was “constructed using the latest Motorola microcircuits assembled on the most durable plug-in printed circuit cards.” For an extra $95, you could augment the Autonav with this single-hand Driver Guide display and mount it just below the driver’s line of vision.  Its range was plus or minus 30 seconds (shown as 50 hundredths of a minute). Even then, it would be hard to tell the difference between being right on time and being a second early.  That difference would be only six degrees on the display — too small to be perceived with a quick glance. Conversely, we didn't care whether we were 20 seconds late or 30 seconds late; in either case, Terry had to drive as fast as safely possible until caught up.  But the difference between 20 and 30 seconds (equivalent on the Autonav to the difference between 33 and 50 hundredths of a minute) is a whopping 60 degrees on the dial.  That's a waste. Therefore the seconds on my ideal display would not be evenly spaced.  I needed a display like a fence, with the posts comfortably apart in the center but compressed when they're far away.  And that led me to invent the Arctangential Error Meter. An AEM looks like this.  The single hand is reading -2 seconds.  In the top half of the dial, each mark represents one second, but as we move away from zero the marks gradually get closer together.  In the bottom half, each mark represents ten seconds.  As you can see, the meter gives high precision around the critical zero point, with less precision when the error exceeds 20 seconds or so. The angle of the pointer, measured clockwise from straight up, is controlled by a digital computer.  It is equal to twice the arctangent of one-tenth of the error in seconds.  In symbols, that's 2 * ARCTAN (0.1 * E).  For example, if the error were +10 seconds, the angle would be twice the arctangent of +1, which is twice +45º, which is +90º, and the pointer would be horizontal pointing to the right. For even greater precision around the zero point, we could adjust the formula.  Here are examples of how the dial would display an error of -2 seconds by using twice the arctangent of two-tenths and four-tenths of the error, respectively. In other competitions, we might only care about negative errors, not positive ones.  For example, maybe the "error" is the time remaining.  By using four times the arctangent, the dial could be arranged asymmetrically like the one below. This example counts down from 200 minutes.  We don't much care about the difference between 90 minutes and 80 minutes.  The difference between 30 and 20 minutes is more important, so it gets a larger display area.  In the last ten minutes, every minute counts.  But once the time reaches zero, the game is over. All of this is theoretical.  But in the 1980s, I wrote a rally simulator program on my Radio Shack TRS-80 Model 1 computer, and as part of it I actually simulated an Arctangential Error Meter on the computer's monitor. One subroutine calculated the error and its arctangential equivalent, converted that angle from polar to Cartesian coordinates, and lit up the corresponding pixel in this little box.  After waiting a second, it erased the pixel, recalculated, and lit up a pixel again (often the same one as before).  So the little blinking dot corresponded to the point of the arrow in my drawings above. Did it work?  Of course. As the simulated rallyist, parked on the side of the road, waited for his assigned time to leave the checkpoint, a pixel in the lower right quadrant kept blinking.  But then, like the closely watched ball in Times Square at one minute before midnight, the pixel slowly began moving.  It crept counterclockwise along its predetermined arc, gradually moving faster. By pressing buttons on the TRS-80 keyboard, the simulated rallyist stepped on the simulated gas and accelerated to rally speed, slowing down the pixel's movement and hopefully stopping it at top dead center — meaning exactly on schedule.