how to rotate coordiante system about Z axis first then rotate about about the new Y axis.

Assume two coordinate systems, one is the global coordinate system(XYZ) and another is a local coordinate system which is obtained by rotating about the Z axis of the global coordinate system and then rotating about the new Y axis. how can I get the translated matrix of this process?

[KSSV]

Have a look on Affine transformations. You can frame your Rotation matrices and get what you want.

Answer 1

[David Goodmanson]

Hi Yang,

Suppose you have rotation matrices

Rx = [1 0 0      Ry = [c 0 -s      Rz = [c -s 0
      0 c -s           0 1 0             s c  0
      0 s c]           s 0 c]            0 0 1]

If p is the column vector [x;y;z], then for example pnew = Rx*p.

Here s is sin(theta), c = cos(theta), these are active rotations of the object, and looking down the axis of rotation toward the origin, the rotations are counterclockwise.

A rotation in z and then in the original y would be

vnew = Ry*Rz*v

A rotation in z and then in the new y is

vnew = Rz*Ry*v

which has the order reversed. That's because a rotation about the new y axis is equivalent to: rotate the new y axis back to the old y axis, then rotate about the old y axis, then rotate forward so that the old y axis is the new y axis.

Ry_new = Rz*Ry*Rz^(-1)
so 
Ry_new*Rz = Rz*Ry*Rz^(-1)*Rz = Rz*Ry

[Yang Lu]

Hi David, Thank you for your reply. I think the matrices are used to coordinate transformation, not for the coordinate system transformation. am I right? according to the following picture, i think 'vnew = Ry*Rz*v'is the rotation about the new Y-axis. how do you think.

[David Goodmanson]

Hi,

The rotation matrices depend on [1] whether you have a right hand or left hand coordinate system (I forgot to say that I used right hand as usual) [2] whether your rotations are clockwise or counterclockwise [3] whether the rotations are defined as active or passive.

I was using active transformations as I mentioned. If you are using passive transformations I believe what you say is correct (but I am not willing to bet my next meal on it).

[Yang Lu]

Hi, thank you very much. it is my first time to know the concepts of active transformation and passive transformation which are really useful for me.

[David Goodmanson]

You're welcome, I was glad to be able to provide some information.

[James Tursa]

@David: You have a mixed set of rotation matrices because the sine signs are not consistent. The sine signs of the Ry rotation should be reversed when compared to the others to make the set consistent. E.g.,

Rx = [1 0 0      Ry = [c 0 s       Rz = [c -s 0
      0 c -s           0 1 0             s c  0
      0 s c]          -s 0 c]            0 0 1]

See this link "In three dimensions":

https://en.wikipedia.org/wiki/Rotation_matrix

[David Goodmanson]

Hi James, thanks for pointing this out. I blithely borrowed the first ones from a set I made having to do with the earth's rotation, but for the problem here the consistent set would have been better.

Answer 2

[Yang Lu]

Ry_new = Rz*Ry*Rz^(-1)
so 
Ry_new*Rz = Rz*Ry*Rz^(-1)*Rz = Rz*Ry

Hi David,

Regarding to the Ry_new, I do not understand why do you multiply Rz^(-1). In my opinion, the new Ry can be obtained by rotating the Ry around the Z-axis.

Thank you!

Yang.

[David Goodmanson]

HI YL,

It sounds like your proposed solution for Rnew is Rnew = Rz*Ry, is that correct?

[Yang Lu]

Hi David,

We can think like this. Firstly, we have global coordinate system XYZ and the unit vector is (i; j; k). we want to get local coordinate system(u ;v ;w) by rotation the (i; j; k).

1. we rotate (i; j; k)around Z-axis by Rz*(i; j; k) then get the (u ;v ;w)
2. if rotate (u ;v ;w) around the golbal Y, it should be Ry*Rz*(i; j; k) and if rotate around the new y-axis which is v, then it should be Rz*Ry*(i; j; k).

Is that correct/

Thank you very much.

Yang.

[David Goodmanson]

could you write down the formula you are proposing for Rnew just in terms of Rx,Ry,Rz? It's Rnew = Ry*Rz, is that correct?

[Yang Lu]

if rotate around the global Y, the Rnew = Ry*Rz and if rotate around the local Y, the Rnew is Rnew = Rz*Ry.

[David Goodmanson]

OK, to, start with, a vector has one index so it is going to transform like

vnew(i)  = Sum{j} A(i,j)v(j)

for some A.

A matrix has two indices so under the usual assumptions involving transformations it is going to transform like

Mnew(i,j) = Sum{k,q} A(i,k)B(j,q)M(k,q)

for some A and B. WIth matrix multiplication convention you arrive at, for example

Rnew = Rz*R*Rz^(-1).

That's one argument.

Now suppose there is an arbitrary rotation about the x axis and a 90 degree rotation about the z axis:

Rx = [1 0  0;
      0 c -s;
      0 s  c]

 Rz = [0 -1 0;
       1  0 0;
       0  0 1]

Without getting entrapped in all the details, since the 90 deg z rotation swaps x and y (with a sign involved), using these two rotations one should be able to convert an x rotation into a y rotation.

However, doing what you propose,

Rz*Rx = [0 -c  s;
         1  0  0;
         0  s  c]

which is an orthogonal matrix as it must be, but not a useful rotation matrix. On the other hand if you work out Rz*Rrx*Rz^(-1) the result is

Rz*Rx*Rz^(-1) = [ c 0 s;    = Ry
                  0 1 0 
                 -s 0 c]

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