Matrix of a Rotation Tensor

In summary, the conversation discusses finding the matrix of a rotation tensor for a rotation by angle θ about an axis aligned with e1+e2. The attempted solution involves using the rotation tensor for e3 and modifying it to fit the given axis, with some discussion about the determinant of the resulting matrix.
  • #1
samee
60
0

Homework Statement



Write the matrix of a rotation tensor corresponding to the rotation by angle θ about an axis aligned with e1+e2

Homework Equations



I know that the matrix for a rotation tensor about e3 is;

cosθ -sinθ 0
sinθ cosθ 0
0 0 0

The Attempt at a Solution



I assume that the rotation would be changing only on the e3 axis because the axis are aligned with e1+e2, right? So the matrix will be all zero with the R33 component being some complicated rotated value?
 
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  • #2
hi samee! :smile:

(try using the X2 button just above the Reply box :wink:)

how about rotating e1+e2 onto e1, then rotating through θ, then rotating e1 back again onto e1 +e2 ? :wink:
 
  • #3
Okay! I had some new revelations.

I know that (R-I)u=0 where R is the rotation tensor, I is Identity and u is some vector, here it's e1+e3.

So, this equation will be true if the R matrix is;

0 1 0
1 0 0
0 0 1

BUT! I think it needs to be in sinθ and cosθ instead of 1... right?

Also- thanks for the tip on the X2 ^_^
 
  • #4
samee said:
0 1 0
1 0 0
0 0 1

erm :redface:

the determinant of that is -1 :biggrin:
 
  • #5


Correct, since the axis is aligned with e1+e2, the rotation will only affect the e3 component. Therefore, the matrix for the rotation tensor will be:

1 0 0
0 1 0
0 0 R33(θ)

Where R33(θ) represents the rotation about the e3 axis by angle θ. This can be calculated using the same formula as for the rotation about e3, but with the angle θ instead of the angle φ.
 

1. What is a rotation tensor?

A rotation tensor is a mathematical representation of a rotation in three-dimensional space. It is a 3x3 matrix that describes the transformation of coordinates from one frame of reference to another.

2. How is a rotation tensor different from a rotation matrix?

A rotation tensor is a more general representation of a rotation than a rotation matrix. While a rotation matrix is always orthogonal (with a determinant of 1), a rotation tensor can also be non-orthogonal (with a determinant other than 1). Additionally, a rotation tensor can represent rotations in higher dimensions, while a rotation matrix is limited to 3-dimensional space.

3. What is the relationship between a rotation tensor and quaternions?

A rotation tensor can be converted to a quaternion and vice versa. Both representations describe a rotation in three-dimensional space, but quaternions have certain advantages, such as compactness and ease of interpolation.

4. How do you calculate the determinant of a rotation tensor?

The determinant of a rotation tensor can be calculated by taking the dot product of any two rows or columns of the matrix. Alternatively, it is equal to the product of the eigenvalues of the tensor.

5. How are rotation tensors used in robotics and computer graphics?

Rotation tensors are used in robotics and computer graphics to represent the orientation of objects in 3D space. They are essential for performing transformations and animations, such as rotating a 3D model or controlling the movement of a robotic arm.

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