angles between objects

If I have two objects--for example, two small spheres--and I want to connect them with a cylinder, calculating the angle between them is easy as pie if they share at least one of the XY, XZ, or YZ planes (i.e., one of their X, Y, or Z coordinates is the same). But what if they share no planes directly addressable by the viewer's edit system?

Here's an example of the first problem, which is easily solvable with simple trigonometry:

Sphere 1: <10.0, 20.0, 100.0>
Sphere 2: <10.0, 25.0, 105.0>

In this example, the two spheres share the YZ plane, and calculating the angle on that plane relative to the Y axis is simple:

tan theta = (z2 - z1) / (y2 - y1) = (10 / 10) = 1
theta = 45 degrees

I can then create a cylinder of the correct length, rotate it such that it is on the Y axis, place one of its ends on sphere 1, link the cylinder with sphere 1 as the root prim, and simply rotate 45 degrees on the Y axis. Easy.

Here's an example of the second problem, which may seem trivial in trigonometry, but actually isn't (and if you know how to solve this, you know this already):

Sphere 1: <10.0, 20.0, 100.0>
Sphere 2: <12.0, 25.0, 105.0>

In this example, the two spheres do not share any of the three addressable planes. Doing a similar calculation as above for the angle alpha (the angle relative to the Y axis on the XY plane, discounting the Z axis) and beta (the angle relative to the Y axis on the YZ plane, discounting the X axis), only beta will be correct. Alpha will be slightly too large, and when extreme accuracy is needed, "slightly too large" means useless. While I can get the distance very easily with the Euclidean distance formula and then rotate the remaining alpha by sight, I can never be sure to get the exact angle.

This may seem trivial, but consider this: Imagine these two spheres are 50 meters apart. While I could pivot a long, linked series of cylinders in other cases, the viewer will not let me link 50 meters of cylinder, so I absolutely cannot "eyeball" it in this case. There's no way to place this cylinder series without being able to calculate all angles precisely.

Does anyone know a simple way to do this using Euclidean rules? If not, using linear algebra or some other method?

You could script the prim to llLookAt the other position.

Do it in two steps?

Calculate the data for Object A and a point on the x-y plane that is at the same x-y coordinates as Object B, but shares the z-value for Object A. That gives you a different starting length and angle, that is on a plane with Object B.

Now calculate using that baseline in the x-y plane and the z-axis difference, and you can get the point to point angle and length between the two.

I'm not the greatest at rotations (see comment under my name...), but unless I misunderstand, I think you're making this more difficult for yourself than you need to?

sphere 1 position at (vector) pos1
sphere 2 position at (vector) pos2

(vector) dir = llVecNorm(pos2-pos1) will give a unit vector pointing from sphere 1 to 2

then use (rotation) rezRot = llRotBetween( <appropriate axis vector>, dir ), where the axis vector might be <0,0,1> or <0,1,0> or <1,0,0> to give the likely rotation you'll have to apply to your rezzed connection cylinder.

Depending how many cylinder links you need, use a loop to get positions between pos1 and pos2, rez a cylinder link at the appropriate place with rezRot rotation.

Does that do what you want? The details of the above might be wrong, but I think the concept should be sound.

I don't know if it does what the OP wanted, but it sure as heck helped me, thank you.

I had been looking for a better description of how to use llRotBetween(), and for whatever reason your description clicked for me where the others had not.

LOL, glad to be of help. And when you understand rotations, please explain them to me, because I still just hack at it, trying every which axis I can, until something vaguely resembling what I was trying for happens.

But more often, I end up with this:

Thanks for the explanation. I don't know how I missed your reply, but I ended up doing exactly (I think) what you're doing here.

Rotations confound me as well, BTW--or at least, what happens when applying various operations to them.