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Going in circles

Badly Shaved Monkey

Anti-homeopathy illuminati member
Badly Shaved Monkey
Joined
Feb 5, 2004
Messages
5,363
No, not a homeopathy thread.

Here's a maths puzzle that defeats my common sense, but I'm not sure how I'd go about analysing it mathematically.

Picture a sphere. Spin the sphere about a random axis.
The path followed by a point on the surface is a circle.

Now choose an extra axis orthgonal to the first and spin the sphere about that as well.
What is the path of the point?

If I add extra axes of spin at various angles so that asymptotically the sphere is spinning about all possible axes what is the path of a point? (Assume you can only spin one way round each axis, i.e. once you have a N-S axis and spin clockwise about it viewed from above N, you don't merely cancel that out by coming across the same axis again and labelling it S-N and applying a clockwise spin viewed from S, so I think I'm constrained to stipulate that spin is in the same direction, say clockwise, about all the axes when each is viewed from N).



(p.s. I can't link to a page with the correct answer, because I came up with this myself and it's been like an unscratchable itch ever since I thought of it)
 
I'm tempted to add the rotations like vectors, and therefore for the first part (with perpendicular rotations) if you have two sets of the same velocity rotation, it would be equivalent to having one rotation with axis 45deg off each of the originals, with magnitude root two times angular velocity.

That's if you can add it up that way. I'm working on a horribly convolouted way in MATLAB now to test it. (Damn you BSM, you've piqued my curiousity.)
 
wittgenst3in said:
(Damn you BSM, you've piqued my curiousity.)
Click to expand...


Thanks. I had a gut feeling that the answer to the first part would just be another circle. My guess would be that the extrapolation to infinity might turn out to be just a circle as well, but as I said I can't do the sums myself.

Ah, ha, see my pretty how easily these mathmos are lured in.
 
Originally posted by Badly Shaved Monkey
Picture a sphere. Spin the sphere about a random axis.
The path followed by a point on the surface is a circle.

Now choose an extra axis orthgonal to the first and spin the sphere about that as well.
Click to expand...
I'm not sure what this means. Do the axes (one or both) move with the sphere or not? And how do you rotate a sphere about more than one axis at a time, anyway?

Suppose I want to actually build one of these things and move it the way you have in mind. How do I do it, exactly?
 
Okay, here's what I did.

Created a coordinate frame and placed a point at (x1,y1,z1).

loop for n=1:1000
Rotate that coordinate frame about the X axis by n/1000 * 360 degrees.

Rotate this again around the new Y axis by n/1000 * 360.
next n

Save all the points and graph.

Note that this means (for simplicitys sake) both the angular rates and angles are the same.

Result is not what I expected. Some form of spirography thingy. I may have made a mistake in the code but I doubt it, I have my robotics texts in my lab and the code is small.

I'll throw the MATLAB code up if anyone wants to see:

clear;
p=[1;1;1;1];
for n=1:1000
ang1=n/1000*2*pi;
ang2=ang1;
t1=[1 0 0 0 ; 0, cos(ang1), -sin(ang1), 0; 0, sin(ang1), cos(ang1), 0 ; 0 0 0 1];
t2=[cos(ang2),0,sin(ang2), 0;0,1,0,0; -sin(ang2), 0,cos(ang2), 0; 0 0 0 1];
tmp=t2*t1*p;
r(:,n)=tmp(1:3);
end
plot3(r(1,:),r(2,:),r(3,:)); grid on;

69Dodge
My reading of the problem is like so: Say I have an object suspended in a frame that is free to rotate. That frame is suspended in another frame which allows the first one to rotate (Like a gyroscope).

If you're after a practical application of these, NASA used them to send man to the moon. There were 3-axis gyros, which were a problem because in certain angular configurations they would lock up, and the craft would literally forget which way it was pointing. (Movie Apollo 13 had an excellent rendition of them using the '8-ball' to see when they were close to gimball lock).
 
To give a clearer idea of what is going on I've added a second locus (in red), of a point located at x0.5 y0.5 z0.5
They are both rotating about the origin at 0,0,0.
 
wittgenst3in said:
My reading of the problem is like so: Say I have an object suspended in a frame that is free to rotate. That frame is suspended in another frame which allows the first one to rotate (Like a gyroscope).
Click to expand...


Thanks again. I'm glad it's not just working out a as circle, makes th ehowle exercise more interesting.

"Like a gyroscope" is exactly what I had in mind. I was looking at something inside a set of gimbals (right word?) and could see what happened to a point on the surface of the sphere inside if you just spun one of the rings, but it was not immediately obvious what the path would be if I spun the second and the third, which obviously (well, obvious because I'm me) led me to wonder about an infinite set of nested gimbal rings.

Strictly, by looking at somthing inside gimbal rings, is the inside out version of the way I have asked the question. My original thoughts were very casual and hazy and I suppose really what I was considering was the path of a point in the 'sky' over the sphere at the centre when one gimbal was spun and what would happen to that path if the ring to which that first ring is joined, was also given a spin. Only later did I start wondering about the abstracted version in which it's just a sphere, itself spinning on multiple axes, which is easier to conceptualise, but a bugger to construct, instead of something suspended inside gimbal rings, which would be easier to fabricate for a few layers, but makes the question less intuitive.

Edited to add: Gimbal. Right word, and mine is the correct spelling.

http://machaut.uchicago.edu/cgi-bin/WEBSTER.sh?WORD=gimbal
 
Next part of the problem that I envisaged, is, i think inherent in the way you have answered it.

I'm assuming all the axes have equal priority. Everything spins at the same time. Your method has to take one axis first then add the second rotation. Does the order of addition matter? If it does then how does one get away from adding the axes sequntially?
 
One more file just for you.

If the angular rotation rates are not the same, then the paths are not closed loop (or at least in the single rotation I simulated).

I've just noticed you've posted again, so I'll upload this image and then post again.

Edit: The left picture is a top view of the system, and the right is the straight on view.
 
Originally posted by Badly Shaved Monkey My bolds
I'm assuming all the axes have equal priority. Everything spins at the same time. Your method has to take one axis first then add the second rotation. Does the order of addition matter? If it does then how does one get away from adding the axes sequntially?
Click to expand...

I'll just clarify to ensure we're on the correct page. The rotations are performed simultaneously, but they are relative to one another in a specific order.

Say you have the object (sphere, whatever), in order to be unambiguously located, it must be related to another coordinate system, or the 'world' system. So consider this with motors on gymbals. Each motor spins the frame next to it.

So the outside motor is bolted to the world coordinate system, and that motor spins a gimball, which has another motor mounted on it, etc. Since by spinning one motor, the direction the other motor is pointing is affected, the order is indeed important. Indeed I don't think it is physically meaningful to say that something has two rotations relative to one thing.

Problems crop up like this all the time in robotics. I've used some heavy assembly robots which have sophisticated joint control. Say you're doing an assembly and you want the robot to twist a cap on something, or tighten a screw. You can select 'tool coordinate system' which means you specify locations based on the current location of the robots gripper. So no matter which way the robot is facing it will rotate relative to it's gripper the same way.

Alternatively you can select the 'world' coordinate system in which the rotations and translations are relative to the fixed base of the 'bot.

Interestingly enough these robots aren't too inteligent. If I got in the way the robot would go straight through me, or knock me flying without even triggering an overtorque alarm. Fun times.
 
There must be an algebraic way to work this out by creating an infinite series of terms that needs to be summed.

We know the orbit about each axis is a circle, so can't we just start with the expression for position in a circular orbit at time, t, then come up with an expresssion for all the other orbits that vary in the plane of rotation by infinitessimal amounts until we have summed over all the possible orbital planes for axes that intersect with the surface of one hemisphere? Or does it have to be done by numerical approximation?
 
wittgenst3in said:
I'll just clarify to ensure we're on the correct page. The rotations are performed simultaneously, but they are relative to one another in a specific order.
Click to expand...

This is why my mind began to get boggled by it. Yes, I'd like an answer if you worked through the axes in turn, but I wonder what is the effect of choosing different sequences of selection. Then i kind of vaguely assume that if you averaged over all possible sequences then you'd end up with something in which all axes were chosen simultaneously. I had the same problem as you, although the question started out with a physically realistic model I'm wondering about the abstracted form, in which case I may not need to imagine actually constructing a set of gimbals and wondering what happens when I now spin the millionth ring.

So, my question was meant to be abstract, but part of answering it may include whether one can really consider all axes simultaneously even in Plato's world rather than the factory floor.
 
You can certainly express that kind of line mathematically, if you got the sheer masochism to combine that much trig and calculus.

I'm trying to get my head around what that line is in 3D, even with those funky charts I can't seem to visualise it. Are we talking like conical helix? Could you create the same line by laying a piece of thread around a cone?
 
p.s. I must say, I'm really pleased that this problem bugs more numerate people than me! I do like a good puzzle. This has that feeling of an intuitive answer just within grasp, but every time you try to guess at it you realise another complication and you know that it needs a proper mathematical solution.
 
Benguin said:
You can certainly express that kind of line mathematically, if you got the sheer masochism to combine that much trig and calculus.

I'm trying to get my head around what that line is in 3D, even with those funky charts I can't seem to visualise it. Are we talking like conical helix? Could you create the same line by laying a piece of thread around a cone?
Click to expand...

In the window I generated I can view it and turn it in 3D space. It's not like any simply expressable shape, or topology I can imagine. The best example I can give is the 3d locus of the points a poorly folded taco, with a section that crosses over itself.

Image a pin with 10 segments. Stick the pin such that the center is in the origin of the globe. My latest program will now plot the locus of the 10 points (from 0 0 0 to 1 1 1 in 0.1 increments in each dimension) as both axes make a 360 degree revoloution simultaneously.

Last post for tonight as it's after 1.

Edit: The colours in the image represent height in the z-dimension. Blue is lowest, and it ranges through to red. This might make it slightly easier to see.
 
Who believed me when I said I was going to bed?

Here's two more images that should clear up the whole thing. (I promise no more pictures! this has to be hell on the board)

The reason that the previous shape was so unexpected was due to the pin location. The 1 1 1 means that any rotation is going to show up as a change in space.

These ones represent what would happen if you stuck a pin in (from memory, I can't remember the exact order). at 1 0 0, 0 1 0 , and 0 0 1.

To start off check out what happens when 2 pins are stuck in. Note how one pin merely rotates in a flat circle, whilst the other one rotates in a folded figure 8 shape.

The next photo shows all 3 pins stuck in. The shapes traced by the last two pins are identical and they are bisected by a flat circular plane in the middle.

Hopefully this should give a good idea of what the shape looks like.
 
BSM made a christmas decoration.

This reminds me of the Topographic sculptures on the 2nd floor (I think) of the Science museum in London, just opposite that Babbage machine construction.

They have their mathematical descriptions on the tickets, so if anyone is really bored tomorrow ...
 
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