I thought this week about a problem I originally considered about ten years ago. I imagined a source of a laser beam, mounted high up above the ground, shining straight down, and allowed to rotate upwards at a constant rate until it was shining horizontally. The point at which the laser beam touched the ground would travel from directly below the source to the horizon.
The question is, if I know where the source of the laser is, and how quickly it is rotating, can I know exactly where the beam strikes the ground over time?
The problem came about, I’m sure, as I was listening to Jeff Wayne’s Musical Version of the War of the Worlds, imagining beams of Martian heat rays mounted to towering tripods sweeping across the hull of the Thunder Child.
Trigonometry: What’s the length of a side of a right triangle with a constant height?
This is not so bad of a problem when only the geometry is considered. For now let’s call the angle that the laser makes with the “straight down” direction (the vertical) “omega-t”: . With a length of time since the laser started shining, we can see that is a sort of speed — when I multiply it by a length of time , it gives a total angle, which is like a distance. The product works the same way that 30 miles per hour times 2 hours is 60 miles. In physics we’d call this speed of rotation the angular speed, or angular velocity if you’re considering rotations in all three dimensions. For now, it doesn’t matter what the value of this speed of rotation is.
Setting up the laser at an angle from the vertical and a height from the ground, we find it shines at a point away, and a horizontal distance to the right.
If you’re not so used to working with trig functions, you could get to the image above by first setting up the “classic” trig diagram, with the point a distance (hypotenuse) away:
The above image has all the right ratios, but keeps the hypotenuse constant, not the height. Divide all the lengths by , and remember that secant = 1/cosine (by definition).
We’re already done. The point at which the laser beam hits the ground is
Tangent “blows up” to infinity at $\pi/2$, which corresponds to the laser shining parallel to the ground. It intersects the flat ground an infinite distance away, at the horizon. Hopefully this agrees with your expectations; tangent is defined to act this way.
The Tripod Problem: Incorporating the speed of light
So that’s not super interesting. The real “tripod problem” was this: Where is the point of intersection if the speed of light isn’t infinite? If it takes some time for the laser beam to travel from the source to the ground, and the laser continues to rotate, then the location where the beam strikes will lag behind the orientation of the laser emitter.
This results in a “floppy” trajectory of the laser beam, drooping down to the ground
The behavior of the point where the laser strikes the ground is very different with the speed of light restriction. It never reaches the horizon in finite time — the behavior for long lengths of time, and far distances, is totally dominated by the speed of light. It should travel along the ground at a speed approaching c.
The way the pointer location moves depends little on the speed of light directly under the source, where the distance to the ground doesn’t depend strongly on the angle, and a wider angle of the laser covers a small length. As the laser approaches the horizontal, the length covered by each small change in angle increases. The light that will eventually strike very far distances looks more like a point source, since the small angles will be covered in a very short length of time. The trajectory of the beam itself, while it’s still in the air, will look more and more like an expanding circle with time.
When I first mentioned the tripod problem to a friend recently, he had the insight of saying that a laser’s point could definitely travel faster than the speed of light. He could shine a laser at one side of the moon, and then the other. A quick enough rotation on a far enough canvas could result in a pointer appearing to travel faster than the speed of light. Remember, this doesn’t violate anything in relativity. No object is traveling faster than light, rather, a series of events in which different photons strike the distant moon are occurring. This situation is very much like that when the tripod laser is pointed nearly downwards. The speed of the pointer is dominated not by the speed of light, but by the rate of the laser rotation because the distance the light has to travel doesn’t change much when the laser is pointed directly downwards (or from one side of the moon to the other).
This suggests that the speed of the pointer, at very far distances from the tripod, would approach c from slower speeds if the laser were rotating slowly. But, if the laser were rotating quickly enough, could counterintuitively approach c from faster speeds.
Anyway, let’s try to deal with the problem. Take a look at our trig diagram again.
When a certain portion of the laser beam is emitted at a time (and an angle ), it has to travel a distance . Traveling this distance at speed c takes a length of time equal to
Any portion of the heat ray travels in a straight line. Although the beam as a whole is curved, we’re still assuming it’s always traveling directly away from the source (no diffraction, etc.). The laser travels in the same direction and therefore strikes the ground at the point
at a later time
This seems great. We have the basis for a complete understanding of the position of the laser pointer (or toasted Edwardian human) given some time. A portion of the laser, emitted at time , will strike the ground a horizontal distance away not at but at above. This allows us to find the location corresponding to any time of emission in the interval
If we were satisfied with this, the game plan would be to pick a time of emission , determine how long that portion of the beam traveled, and then pair up the resulting with the distance .
I’m not satisfied, though. I’d like to get a trajectory of the laser pointer: a location as a function of the actual time rather than the time that portion of the laser was emitted, . In order to do this, we’d need to replace the in the tangent function with an equivalent function of . In order to do that, we’d need to solve
for . Good luck.
What we’ve got above is a transcendental equation. This means it is not composed of a finite number of additions, subtractions, multiplications and divisions of our variable and the constants, as well as rational powers of these. In most cases, and I’m pretty sure in this one, we can’t solve a transcendental equation exactly for the input variable. We cannot write in terms of .
It seems like the best we could do, if we wanted to create an animation with a step by step progression of the position of the pointer, is to prepare ahead of time. Pick an emission time , find the value of the tangent function to find the distance, find the value of the strike time , and record that pair. Then do this many, many times to create a table with more values than we expect someone to ask us for. We could find the position as a function of time with as much precision as we wanted, supposing we were willing to put the effort in.
I wanted a closed form solution to the problem, a trajectory , and it seems more than out of reach. This annoyed me, until a friend (hey, there, buddo!) reminded me that “closed-form” is just a matter of what I’m allowing as a definition. In fact, like I mentioned in the last post, all of the trig functions are themselves transcendental: They can be written as Taylor polynomials, but these are polynomials of infinite length. The secant in the equation above can be estimated using
The problem with this, though, is that this isn’t much better. It would still take an infinite amount of time to achieve the exact value of secant given most x’s. The only reason I’m more comfortable using this and the other trig functions is because I’ve been trained to use this name for them, and rely on calculators or tables to give me the values whenever I need them. Anyone using a trigonometric function table is benefiting from someone else’s hard work to overprepare. When we use a calculator, we are relying on an estimation that is as precise as the manufacturer (or sometimes the user) dictates. One could make this estimation with a Taylor series, or with a more efficient method, but the calculator still wont give an exact decimal value.
Any single irrational number, whether it is the solution to an algebraic or a transcendental equation, is another instance of this. I’ve gotten used writing things like or as representations of numbers with clear definitions. These numbers have exact values, but it’s hopeless for me to try writing them down. In a very definite sense, these numbers elude us. I could write or use them to any finite precision I wanted, with millions and millions of digits, so long as I were willing to come up with and use an efficient algorithm to find them, or if I were were willing to wait or work a very long time, or both. But, I still wouldn’t have the “exact” value, just one that was plenty good enough for whatever application I had in mind.
These examples remind me: it’s convenient to have named functions like “Cosine” to cover a mathematical idea, but we can’t let this name cover up the meaning of that idea. There are an infinite number of angles whose cosine is a transcendental value. We’re able to use cosine because we can always (right?) reach a higher precision than is necessary in a physical application. I’ve gotten used to working with cosine, and mentally separated it from the solution to the tripod problem, because someone gave it a name that I’ve adopted.
So, I guess I should name the solution. Let’s call the composed function
where and are related by
the heat ray function. We could create a huge table for x(t), someone could come up with an efficient algorithm for calculating values of x, and in the future we could use these to invade infinite planes with laser pointers more effectively.