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In an off-topic discussion, I tried to explain to a student why a "ray-gun" that (somehow!) shoots a ray that followed a spiral path would be much more difficult to aim at a particular target (point in $\mathbb{R}^3$) than a gun that shoots a ray along a straight ray. (Added: The spiral has a uniform radius and uniform spacing along its axis.) My attempted explanation failed to get at the essence, and consequently failed to convince. Perhaps the ambiguity is that the notion of what it might mean to "aim" such a gun is not clear. But even with precise knowledge of the spiral's geometry, I feel that under any interpretation of "aim," it is still difficult.


SpiralGun


Q. Assuming I am not wrong that it is more difficult, can you think of a convincing way to explain this?

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    $\begingroup$ Last comment from me for now: Is the 2D version clear? Something like: It is difficult to predict where a sinusoidal "ray" (in the "ray gun" sense) with unknown amplitude will hit an object at an unknown distance? $\endgroup$ – Benjamin Dickman Oct 9 '14 at 0:23
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    $\begingroup$ Here's the most succinct way I can express my intuition that it is harder to aim the spiral gun. It "appears" that the distance doesn't matter when I'm aiming the linear gun. I just figure out the direction and I can hit anything on that line. But for the spiral gun I have to vary the direction I point the spiral based on how far away the point is. This can be seen in the 2D version that Ben suggested. The "angle" I aim the sinusoidal wave at depends on the distance to the point, not just the slope of the line that passes through it. Intuition-wise. $\endgroup$ – JPBurke Oct 9 '14 at 1:00
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    $\begingroup$ If they have ever played a game like Angry Birds they would know how hard it is to aim something that follows a curved path! $\endgroup$ – DavidButlerUofA Oct 9 '14 at 7:53
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    $\begingroup$ I think one major issue is what are the relative sizes of the objects. Say if you wanted to hit a target that is quite thick (or generally big) relative to the spiral then the aiming with the spiral does not seem harder, possibly even easier. (Say you want to hit a house with a spiral of radius a meter.) It is hard to say not having been there but it seems possible the student had some other scales in mind than you, and it was this discrepancy that causes issues. $\endgroup$ – quid Oct 9 '14 at 15:16
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    $\begingroup$ You might be able to actually build such a ray gun and play around with it! Connect two spheres with a small rod. Then you should be able to fire this thing and have it move in a parabolic arc along the center of the rod. The two balls will trace out a path identical to the ray gun you propose. Then just try to hit things with the spheres. $\endgroup$ – Steven Gubkin Oct 9 '14 at 19:16
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As a sharpshooter in the military (but nothing special, never used it) I would say the big difference is that you can just align things linearly: your eye, the first sight (usually a post), the second one, usually a notch and the target.

I'm assuming a conventional shot here, but even shots requiring windage or Coriolis or drop, can be still be sighted in the normal manner. You just use a vernier to adjust for them, using the sights, but the picture is still linear.

The loopedy-loop thing, you don't have sights to align. Instead, you need to use some equation. Plus, really, you need to know the distance (to know the part of the loop and then to compensate by off angle shot and/or adjustment of rotation phase). But for a straight conventional shot, you just need angle, not distance.

[There is an interesting segue here to target motion analysis and passive sonar (bearing only) versus radar (bearing and distance) and how much harder sonar TMA is. See. e.g. Ekelund ranging.]

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To riff off of the ideas I develop in the comments: A unit disk starts off in the $xy$ plane centered at the origin. It is spinning at $\omega$ revolutions per minute, and traveling straight up along the $z$ axis at a rate of $S$ units per minute. A target is located at the point $(1,0,L)$.

You obtain a different "ray gun path" by placing a mark at a different place on the boundary of the disk, and letting it trace out a path. Only one of these paths will pass through the target. To a human eye, all of these paths look qualitatively very similar: If your ray gun fired one beam versus another I doubt you would notice. It stands to reason that it would be very hard to hit your target, since only one out of an uncountable number of choices which are identical looking to you hits the target. Even if you give some width to the beam, say $\delta$, your chance of hitting the target should be somewhere around $\frac{\delta}{2\pi}$, which is not too great. You could, of course, calculate where your beam should start in this case at $(\cos(2\omega L\pi/S),-\sin(2\omega L\pi/S))$, but that is not the sort of thing which would be easy to "eyeball". On the other hand, humans are quite good at aiming things along straight lines, or parabolic arcs.

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    $\begingroup$ Very interesting comment re parabolic arcs. Of course, gentle parabolic arcs. Perhaps if gravity were $\times 10$, it would not be so straightforward (HaHa) to aim... $\endgroup$ – Joseph O'Rourke Oct 11 '14 at 0:07
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First we need a definition of "aim" that is interesting. If the gun had a laser sight, you would just wiggle until the red spot falls on the target (assuming, of course, that the laser also travels in a spiral). But that is just going in circles :)

So let's say that we are given 3D coordinates for gun and target, and our goal is to compute a direction vector such that when the gun is 'aimed' in that direction and fired, the target is hit. Thus, we have to determine the shot's axis; the line $L$ through the center of the spiral. But we also have to account for the initial direction of the ray: the muzzle can be placed on any point around a circle of radius $r$ orthogonal to $L$. It seems that "aiming" is more difficult due to our having one more parameter to determine. One would have to find the relative direction $\theta$ from $L$ to the target as seen from the gun, and then determine the initial position of the muzzle, given the spiral parameters and the distance.

However, the line $L$ is not unique. In fact, the is a circle's worth of lines around the target all of which give hits, provided the initial direction is chosen correctly. Once you work out the formula (which I won't do since this is just a conceptual discussion) you can hit the target in infinitely many different skew directions!

In any case, I would say that it is indeed more difficult to aim a spiral ray gun because the formula is more complicated than a straight line equation. This is an anticlimactic answer, but after trying to set up a reasonable model, it is the answer that matches my intuition.

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  • $\begingroup$ "you can hit the target in infinitely many different skew directions!"---Nice point! $\endgroup$ – Joseph O'Rourke Oct 11 '14 at 13:17

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