Of The Way Life Used To Be

Didja ever notice, as Andy Rooney might have said, how well you can see parked cars at night as you drive up to them? Specifically, how their taillights seem to glow in your headlights? This isn't all that easy to arrange, if you stop to think about it: the shiny chrome isn't all that visible, and it would seem to be a much better mirror than the transparent red plastic.

The obvious idea of a simple mirror to reflect your headlights has an obvious simple flaw: mirrors reflect at the angle the light comes in, just like a ball bounces off a surface. The curved chrome surfaces reflect the incoming light in all directions. Only a tiny fraction from one point comes back to your eyes. If the parked car had a flat mirror pointed at your car it would be great, but how do you manage to have the mirror pointed in the "right" direction all the time? I know, a microprocessor-controlled mirror motor and light tracker! Only $600 at Sharper Image!

The answer!

Luckily for safety, a clever and entirely non-electronic idea is even better, because it manages to reflect multiple light sources at once back at themselves! (No, it's not the same principle as those portraits in old mystery novels where the eyes follow you around the room.) It's called a corner reflector and many small ones are molded into the taillight. To see how this can possibly work, let's start with a simple case, where all the light is horizontal, which is close to the headlight situation. You can make a reflector for this situation by just putting two vertical flat mirrors at right angles to each other! You can easily try this in the so-called real world, but since this is the Web, you expect some computer version, so here it is (if your browser supports Java) Your're looking down from above on the two mirrors on the right side, so you see their top edges making a right angle. The red light beam comes from the laser pointer, reflects off each mirror, and exits in the same direction it entered the corner. Grab the laser and swing it around (by clicking and dragging with your mouse) and see how the in and out beams stay parallel. If you click above or below the laser, you move the laser up or down. If you click and drag in the middle of the picture, you swing the laser so that the beam passes thru your mouse cursor.

To see that the in and out beams aren't automatically parallel for any old set of mirrors, try clicking and dragging over the mirrors to change their angle from 90 degrees. You can make some interesting patterns with multiple reflections, but the right angle setting gives parallel beams for all directions. Try 45 degrees and see if you can tell why it isn't as practical as 90. Are there any other interesting angles? (If you'd had some plane geometry, and you know that the angle of incidence is equal to the angle of reflection, you should be able to figure out a proof of this parallel reflection property of right angles. If you try but find you need a hint, try here.)

Images

For most purposes of corner reflectors, that's all you need to understand, but if you actually haul out two mirrors in the real world and look at them in that configuration, the striking thing you'll see is that the image is reversed in the corner. You can understand this, too, from the above simulation. The idea is to realize that the path of the light beam is the same regardless of which way it's moving: the laser at one location pointing in different directions could be replaced by an eye looking in different directions. The other end of the light beam could then be coming from an object at that location, and the eye would see it in the direction the light beam enters the eye. That's a bit much to follow in words, so click and drag on the beam with the mirrors set at 90 degrees: if you swing the beam from top to bottom, you'll see the reflected beam move in the opposite direction, bottom to top. If you imagine the different directions as coming from parts of an object, like the letters, you'll see that the whole image is reversed. No, huh? Try our mirror image page now or later, but come back here and keep reading! Incidentally, images in a mirror have been called virtual by many generations of physicists, and that's the source of the term "virtual reality"!

What about three dimensions?

It turns out that the same corner idea works for full three-dimensional light direction! Just put three flat mirrors together, each at right angles to the others, like the corner of a room with walls and floor. Incoming light in any direction bounces off all three mirrors and exits in the same direction it entered. What's going on in 3-D is not as obvious, as shown by a physical model, and a clickable virtual model to the right. If you found it a little magic that 2-D corner reflectors work, it's a lot magic in 3-D. If you physically put three mirrors together in a corner and look in it, the image is reversed and inverted.

The trick to understanding that 3-D works the same as 2-D is projection in the mirror planes. Imagine yourself viewing the light beam (visible like the physical model or the simulation) from various angles. In particular, look from the three directions directly into each of the three mirrors. (Try this in the cube at the right, by click and dragging to rotate.) In each of these views, the beam path is exactly the 2-D case! You don't "see" the third reflection, the one in the mirror you're looking straight into, because the bounce is always in the plane perpendicular to the mirror. (This is like an idealized bouncing ball: it'll go straight forward and not turn on a bounce... in that situation, the impact force is perpendicular to the surface, so that only the perpendicular component of velocity changes)

Since all three perpendicular 2-D views of the 3-D case show parallel beams, all vector components of the incoming and outgoing beams are equal, so the beams are in the same direction in 3-Space, and the corner reflects light from any direction back at its source.

How useful are they?

Corner reflectors are in those reflective bumps in the highway centerline, on bicycles, and whatnot. Corner relectors are useful for precision measurement too: they are used to measure tiny ground movements around active volcanoes. Corner reflectors were even placed on the moon by the Apollo astronauts and are still used today to measure the distance to the moon by timing laser light pulses reflected from earth, to an accuracy of inches! (this isn't overkill: there's important science like tests of Einstein's General Relativity here) Of course, if you want precision, you'll pay for it.

The same idea works for anything that travels in straight lines and reflects like light, such as microwaves from satellites (with samples from Alaska and the Poles). Most unlikely topic: what's the connection with alien landings?

So why do mirrors reverse left and right but not up and down? try our walk thru the looking glass.

-- Alice, to her kitten
Through the Looking-Glass

David Griffiths did that waving effect in the title, and Sun Microsystems did the cube

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