Coordinate System Rotation Matrix (global to local)

In summary, the conversation discusses the possibility of obtaining a rotation matrix from two matrices composed of orthogonal vectors. The conversation also includes a discussion on the use of MATLAB and the correctness of the computation of R(alpha) * R(beta) * R(gamma). It is noted that the determinant and orthogonality of rows can be used as good sanity checks for the computation.
  • #1
dhume878
6
1
I feel I may have improperly posted this thread https://www.physicsforums.com/showthread.php?t=469331" but am just not as knowledgeable in my matrix math as I need to be. One (me) would think that somehow you should be able to get a rotation matrix from these two systems.

Homework Statement


So I have two matrices composed of 3 orthogonal vectors

G = [1 0 0;
0 1 0;
0 0 1]
and
L = [0.96247 -0.03259 -0.266524;
0.02676 0.99932 -0.025578;
0.26718 0.018486 -0.962682]

I have a point in the global system, which i would like to rotate in the same manner one matrix is rotated from the other. (I think .. R(alpha) * R(beta) * R(gama))

It can be assumed the two systems have the same origin.

Homework Equations


Possibly relevant

x'_vector = R_matrix * x_vector

The Attempt at a Solution



Here is my MATLAB attempt. I feel I'm way off (no laughing!)

Code:
x = [1 0 0];
x_p = [0.96247 -0.0325928 -0.266524];

A = x_p \ x

y = [0 1 0];
y_p = [0.0267575 0.999315 -0.0255778];

B = y_p \ y

z = [0 0 1];
z_p = [0.267175 0.0174863 0.962682];

C = z_p \ z

Again I hope I'm not being too hasty with this repost, but i didn't want to be scorned for my incorrect post location.

Thanks for any help,

Cheers!
 
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  • #2
The matrix L is not a rotation of the matrix G. |det(L)| = 0.85, while |det(G)| = 1. Furthermore the vectors composing L are not orthogonal, e.g., {0.96247, -0.03259, -0.266524}.{ 0.26718, 0.018486, -0.962682} = 0.51. There is a linear transformation between the bases, but it is not an orthogonal transformation (rotation).
 
  • #3
:uhh:

Actually it was my mistake I typed in the numbers incorrectly.

[PLAIN]http://j.drhu.me/2011-02-03_1426.png [Broken]

dot = 0 as well.

Annnnddd, I believe the matrix I was referring to as L .. is just my rotation matrix and I should be able to multiply it by a point, and get that point in the local system.

Does this make sense at all?
 
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  • #4
dhume878 said:
:uhh:

Actually it was my mistake I typed in the numbers incorrectly.


dot = 0 as well.

Annnnddd, I believe the matrix I was referring to as L .. is just my rotation matrix and I should be able to multiply it by a point, and get that point in the local system.

Does this make sense at all?

I think it makes sense. If two coordinate systems are linearly related and have the same origin, then they will be related by a rotation matrix. So if you indeed computed R(alpha) * R(beta) * R(gamma) correctly, you should be ok. The determinant and orthogonality of rows are good sanity checks.
 

1. What is a Coordinate System Rotation Matrix?

A Coordinate System Rotation Matrix is a mathematical tool used to transform coordinates from one coordinate system to another. It is a square matrix that captures the relationship between the two coordinate systems and allows for the conversion of coordinates from one system to the other.

2. How is a Coordinate System Rotation Matrix calculated?

A Coordinate System Rotation Matrix is calculated using a combination of trigonometric functions and basic matrix operations. The exact calculation will depend on the rotation angles and axes used to transform from one coordinate system to another.

3. What are the applications of Coordinate System Rotation Matrix?

Coordinate System Rotation Matrix has many applications in various fields such as computer graphics, robotics, and geodesy. It is used to transform coordinates in 3D space, which is essential for accurate positioning and orientation in these fields.

4. How does Coordinate System Rotation Matrix differ from other coordinate transformation methods?

Coordinate System Rotation Matrix is a specific type of coordinate transformation method that uses a matrix to represent the transformation between two coordinate systems. Other methods, such as Euler angles or quaternions, use different mathematical representations to achieve the same goal.

5. Can a Coordinate System Rotation Matrix be used for all types of coordinate systems?

Yes, a Coordinate System Rotation Matrix can be used for any type of coordinate system, as long as the transformation between the two systems can be represented using a matrix. This includes Cartesian, cylindrical, and spherical coordinate systems.

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