Determinant of rotation matrices

It's only purpose is to enable transformation of vectors. And since transformation of vectors can't be done with less than three independent parameters, that's how many the rotational matrix must have as well. Is this right?MoluIn summary, the conversation discusses the concept of rotation matrices in 2D and 3D space. It is mentioned that the 2D rotation matrix can be determined by a single adjustable parameter, while the 3D rotation matrix requires 3 parameters. This is due to the fact that a rotation in 3D space requires the rotation of an entire space, not just one vector. The conversation also touches on the determinant of 3D rotation matrices and the concept of Euler angles
  • #1
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Hi,

After obtaining the 2D rotation matrix (as a function of rotation angle) once by geometry and once by complex algebra, I tried to obtain it by invariance of the Euclidean metric. By this approach, the four elements of the 2D rotation matrix can be determined in terms of a single adjustable parameter, which is all right as rotation in 2D is characterised by a single parameter.

However when I try this with the 3D rotation matrices, I get 3 adjustable parameters. But 3D rotation is characterised by only 2 independent parameters! What's happening?

Further, I can show that the 2D rotation matrices must have determinant equal to 1/-1. Is the same true for 3 or higher dimensions?

Thanks.

Molu
 
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  • #2
No, 3D rotation is determined by 3 parameters: the angle of rotation and two angles to determine the direction of the axis. Imagining the axis of rotation going through the origin, those angles might be the angle the axis makes with the z-axis and with the x-axis. The angle it makes with the y-axis could then be calculated.

Equivalently, you can think of the "rotation" as a vector pointing in the direction of the axis of rotation with length equal to the angle of rotation- 3 parameters.

Finally, the rotation is equivalent to an anti-symmetric matrix and so has 3 independent values: all 0s along the diagonal and the lower triangle is just the negative of the three values in the upper triangle.
 
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  • #3
But aren't you talking about rotation of rigid bodies? A vector is characterised by three coordinates. If we take, for example, the spherical polar coordinate system, there is one scale coordinate and two angle coordinates. That means a pure rotation can be described completely by giving the changes in the two angle coordinates. I can't find anything wrong with this line of reasoning.

Molu
 
  • #4
loom91 said:
If we take, for example, the spherical polar coordinate system, there is one scale coordinate and two angle coordinates. That means a pure rotation can be described completely by giving the changes in the two angle coordinates. I can't find anything wrong with this line of reasoning.

Molu
The problem is that line of reasoning is a non sequitur. Here's a better way of looking at it: Those two angles are the equation of a line passing through the origin in spherical coordinates. Look at that line as one of the axes of another coordinate system. The other two axes form a plane, and as you know, rotation in a plane can be described in terms of one angle.

Here's another way to look at things. A 3x3 matrix has nine elements. In other words, the 3x3 matrices are nine dimensional entities. Nine numbers picked at random do not form a rotation matrix. Those nine numbers must obey some constraints. The rows (or columns) must be unit vectors (three constraints), and the rows (or columns) must be normal to each other (three more constraints). Nine degrees of freedom less six constraints => three degrees of freedom.
 
  • #5
I've thought long and hard but I really don't get this. How can the rotation of a vector *not* be described by 2 independent parameters? If I specify the change in angle with the z axis and the change in angle with the x-axis of the projection on the x-y plane, I've completely specified the new direction of the unit vector. What else is there to specify? If it's a rigid body, I completely understand why it has to be 3 parameters, but it really doesn't make sense for vectors.

Also, what will be the determinant of the 3D rotation matrix? Will it be 1/-1? Then how can I prove this?

Thank you.

Molu
 
  • #6
A little help please?

Molu
 
  • #7
loom91 said:
I've thought long and hard but I really don't get this. How can the rotation of a vector *not* be described by 2 independent parameters?

This is still a non sequitur. You are correct in the sense that it does indeed take two parameters to describe the rotation of a single vector in 3 space. However, a transformation matrix (or a rotation matix) describes the transformation (or rotation) of an entire space, not just one vector.

It does take just 2 independent parameters to describe the rotation of a single vector in three space. This means that it takes 2 independent parameters to describe the rotation of a plane (sans coordinates on the plane) in three space. Rotating a normal to the plane will do it. However, this does not give the requisite information for a coordinate transform in three space. The rotated plane needs to be rotated about its normal. Thus, three rotations are required: Two to properly orient a reference plane, and one more to properly align the reference plane.
 
  • #8
Think about an airplane. There are 3 different rotation parameters for that: yaw, pitch, and roll.
 
  • #9
Look at the description of Euler angles (especially the diagrams) in any mechanics book (Marion, Goldstein, Landau), or look here

You can see clearly why 3 angles are required to rotate a 3D coordinate basis.

EDIT: What happened to the link? It showed up in the preview...
Well, here it is in text:
http://mathworld.wolfram.com/EulerAngles.html
 
  • #10
D H said:
This is still a non sequitur. You are correct in the sense that it does indeed take two parameters to describe the rotation of a single vector in 3 space. However, a transformation matrix (or a rotation matix) describes the transformation (or rotation) of an entire space, not just one vector.

It does take just 2 independent parameters to describe the rotation of a single vector in three space. This means that it takes 2 independent parameters to describe the rotation of a plane (sans coordinates on the plane) in three space. Rotating a normal to the plane will do it. However, this does not give the requisite information for a coordinate transform in three space. The rotated plane needs to be rotated about its normal. Thus, three rotations are required: Two to properly orient a reference plane, and one more to properly align the reference plane.

Finally, a ray of light! I think I see this now. The rotational matrix does not describe the rotation of one specific vector, but the entire vector space, which introduces an extra DoF. Right? Thanks a lot. It feels great when the necker cube suddenly falls into place.

Molu
 

What is a determinant?

A determinant is a mathematical value that can be calculated for a square matrix. It represents the scaling factor of the matrix and is used to solve systems of linear equations and determine the invertibility of a matrix.

How is the determinant of a rotation matrix calculated?

The determinant of a 2x2 rotation matrix can be calculated by taking the product of the elements on the main diagonal (top left to bottom right) and subtracting the product of the elements on the other diagonal. For larger rotation matrices, the determinant can be calculated using the Laplace expansion method or by using a computer algorithm.

What does the determinant of a rotation matrix represent?

The determinant of a rotation matrix represents the scaling factor of the matrix, which determines the effect of the rotation on the area and volume of a shape. For a 2D rotation matrix, the determinant is always equal to 1, indicating that the rotation does not change the area of a shape. In 3D, the determinant can be equal to 1, -1, or 0, which respectively represent no change in volume, a reversal of orientation, or a collapse of dimensionality.

How does the determinant affect the properties of a rotation matrix?

The determinant of a rotation matrix affects its properties in several ways. The determinant determines whether the matrix is invertible, with a non-zero determinant indicating invertibility. It also determines the orientation of a shape after rotation, with a positive determinant indicating a counterclockwise rotation and a negative determinant indicating a clockwise rotation. Additionally, the determinant affects the scaling of a shape after rotation, with a determinant of 1 indicating no scaling and a determinant of -1 indicating a reflection.

Can the determinant of a rotation matrix be negative?

Yes, the determinant of a rotation matrix can be negative. This occurs when the rotation causes a reversal of orientation, such as in a 3D rotation that flips an object over. In 2D, the determinant of a rotation matrix is always positive, but in 3D it can be either positive or negative depending on the direction of rotation.

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