Rotation,angle of vector from axes ?

[atrus_ovis]

How to find that? In R3.
I want to rotate everything around a vector, at an angle A. (making a n openGL game at my free time)

I tried , for normalized vector V = <x,y,z>:
Displace V to start of axes.

angleToYZ = acos(y);
Rotate all around Z with that angle. (1)

angleToZ = acos(z);
Rotate all around X with that angle. (2)

Now V is on the Z axis.
Rotate all around Z with angle A.

perform (2) , (1) with opposite respective angles.


Problem is, when i implement that for rotation around another vector, individually the rotations work, but when i.e. i do a rotation around V1, then the rotation around V2 gets messed up.

Is there a mathematical error in the above?

 

[Stephen Tashi]

Problem is, when i implement that for rotation around another vector, individually the rotations work, but when i.e. i do a rotation around V1, then the rotation around V2 gets messed up.

Your question isn't clear. Are you trying to rotate an object simultaneously about two different axes, V1 and V2?

 

[D H]

Is there a mathematical error in the above?

Yes. That's just not how rotations in R³ (three dimensional space) work.

You appear to be confusing an Euler rotation sequence with a single axis rotation (or eigenrotation, or angle/axis representation). The latter are very closely aligned with quaternions. openGL provides various quaternionic representations of a rotation. The easiest thing is to convert that single that single axis rotation to a quaternion and let openGL do the work.

This wikipedia page provides a good start if you want to understand the math: http://en.wikipedia.org/wiki/Axis-angle_representation.

 

[atrus_ovis]

Your question isn't clear. Are you trying to rotate an object simultaneously about two different axes, V1 and V2?

I have two vectors, v1, v2 ( the "up" and "left" vector of an object) and i want to rotate with respect to them, to implement left/right rotation and up/down rotation.

Yes. That's just not how rotations in R3 (three dimensional space) work.

But since i am aligning the reference vector with an axis, rotating via that axis, and performing the inverse alignment operations, shouldn't the desired rotation be the result?

The easiest thing is to convert that single that single axis rotation to a quaternion and let openGL do the work.

I had tried an "intro to quaternions" via a linear algebra intro pdf, but failed.I'll try wikipedia.

 

[Stephen Tashi]

I have two vectors, v1, v2 ( the "up" and "left" vector of an object) and i want to rotate with respect to them, to implement left/right rotation and up/down rotation.

If you rotate about the "up" axis, it moves the "left" axis to a new position. Is your intent to do the next rotation about the "new left" axis? Or are you trying to rotate about the original left axis?

 

[atrus_ovis]

If you rotate about the "up" axis, it moves the "left" axis to a new position. Is your intent to do the next rotation about the "new left" axis? Or are you trying to rotate about the original left axis?

Yes, of course. The left axis changes when rotation around the up is performed, and vice versa.

 

[Stephen Tashi]

Yes, of course..

Which alternative are you agreeing to?

 

[atrus_ovis]

Which alternative are you agreeing to?

To this:

Is your intent to do the next rotation about the "new left" axis?

Yes , like a regular 3rd person camera, viewing left/right and up/down.

 

[Stephen Tashi]

It isn't clear what algorithm you are using to perform a rotation. Let's assume you are multiplying a column vector v on the left by a matrix M to produce the new column vector u. So u = Mv. You're idea is to use a matrix T (which is the product of two other matrices) to transform 3 axes of object back to the xyz axes, use a matrix R (whichis is the product of two othe matrices) to perform rotations about the xyz axes and use the matrix T-inverse to transform the the xyz axes back to the original axes of the object. I think this idea works if properly implemented, but I can't tell from what you wrote exactly how you implemented it.

 

[atrus_ovis]

I use the matrices from here

http://en.wikipedia.org/wiki/Rotation_matrix#Basic_rotations

to perform the rotations of each vector.

Instead of using a "combined" matrix i do the rotations one after another,which us equivalent.
Instead of using the inverse, i just use the negative angles, which is also equivalent for these rotation matrices (negative angle = inverse)

Problem is,as i said : individually doing left/.right or up/down rotations works fine.
But when doing a left-right and then up-down (and vice versa) , the 2nd rotation gets messed up.

The cross product of the 2 rotating vectors doesn't equal the third (the reference for the rotation) after the above.

 

[Stephen Tashi]

That's not a sufficiently specific description of your method for anyone to check. You need to explicity write out the product of matrices that you used.

 

[D H]

Instead of using a "combined" matrix i do the rotations one after another,which us equivalent.

Equivalent to what?

Rotations in 3D space are a bit counterintuitive. They do not obey nice laws. Unlike rotations in 2D space, they are neither commutative nor additive. Rotation A followed by rotation B is not the same as rotation B followed by rotation A. Example: Rotate by 90 degrees about the x axis, then by 90 degrees about the y' axis (the orientation of the y-axis after being rotated by 90 degrees). This is quite different from rotating by 90 degrees about the y-axis and then rotating by 90 degrees about the x' axis. The former is equivalent to a single 120 degree rotation about an axis pointing along \hat x + \hat y + \hat z, while the latter is equivalent to a single 120 degree rotation about an axis pointing along \hat x + \hat y - \hat z.