# 3 dimensional torque from angular velocity

• drfrev
In summary, the person is trying to find a way to calculate the torque that will make two rods become parallel. They have the rotation matrix and rotation axis, but need to determine the eigenvalue of the rotation matrix.
drfrev
Hello,

I am currently making a video game and have run into a bit of a physics problem. I have taken two semesters of physics in college (wasn't too great at it) so I know how to do simple things. I am currently trying to turn an angular velocity into a 3D torque. The angle is a Euler vector ex. (X,Y,Z) using a XYZ order, and I need the equivalent torque in 3 space. I only need the torque direction as the magnitude isn't that important. I have been trying to figure it out but with no luck.

Thanks for any help

It seems you want to find the rotation axis, but then do you have the rotation matrix? if yes, one of its eigenvectors corresponding the eigenvalue of 1.0 is the rotation axis.

I will be a bit more descriptive to flush out my questions better. I have two "rods" in three space. One rod is fixed, while the other is not. I need to apply a torque to the unfixed rod to make it become parallel to the fixed rod.

Okay, I am able to get the rotation matrix from each of the "rods", is there any way to "subtract" them to get the matrix rotation that separates them, because then I could use what you said. Also how would I go about calculating the eigenvalue of a rotation matrix. (I am not used to rotation matrices which is why I was using angular velocity above. After making this reply I will be looking up eigenvalue on google in case you don't want to go into that)

Each bar is represented by a vector. If you have the rotation matrix to multiply it by the vector of unfixed bar to become parallel the fix one, isn't your problem solve then? If so, I will explain how to construct the matrix from the two vectors.

I don't quite know what you mean but each bar is represented by a vector as the center and a rotation matrix representing orientation.

I hope you are familiar with the following programming language. The calculation is based on Euler–Rodrigues formula.
The matrix rotMat rotates every point of the object in 3D space by alpha radian around rotAx ( rotation axis). It can make the bars parallel.

public static Mat rotMat(Vect rotAx,double alpha)
{
double e1,e2,e3,e4;

e1=rotAx.el[0]*sin(alpha/2);
e2=rotAx.el[1]*sin(alpha/2);
e3=rotAx.el[2]*sin(alpha/2);
e4=cos(alpha/2);

Mat M=new Mat(3,3);
M.el[0][0]=pow(e1,2)-pow(e2,2)-pow(e3,2)+pow(e4,2);
M.el[0][1]=2*(e1*e2-e3*e4);
M.el[0][2]=2*(e1*e3+e2*e4);
M.el[1][0]=2*(e1*e2+e3*e4);
M.el[1][1]=-pow(e1,2)+pow(e2,2)-pow(e3,2)+pow(e4,2);
M.el[1][2]=2*(e2*e3-e1*e4);
M.el[2][0]=2*(e1*e3-e2*e4);
M.el[2][1]=2*(e2*e3+e1*e4);
M.el[2][2]=-pow(e1,2)-pow(e2,2)+pow(e3,2)+pow(e4,2);
return M;
}

I use it for the following problem:

I need to draw arrows showing the magnetic field in the space. Using Java3D, I can draw a 3D arrow in z direction and the arrow center is the origin but I need to draw the arrow at point P and in direction given by the field at point P. Using the matrix, the object is first rotated to become parallel with the field direction, then by a translation, moves to the point. I think it is kind of similar to your problem.

I need to use a torque on the object so that the physics engine can handle collisions. The engine will handle the actual vertex rotation. I have a function which can apply the torque to the object for the next physics tick. It can apply a local or global torque. (Though I do understand that language)

Sorry what i suggested was not physics by linear algebra. Out of curiosity, could you explain about the Euler angle you said. What are X,Y and Z ? I have heard of Euler angles but not Euler vector.

It's the same thing, (x,y,z) being the rotation acording to that axis. XYZ order means that it rotates on the x-axis first, then the y axis, then the z axis.

So you have the three angles and can construct the rotation matrix. From this matrix, you can obtain the rotation axis to apply torque to. This axis is the eigenvector of the matrix. see http://en.wikipedia.org/wiki/Rotation_matrix

Eigenvectors of a 3x3 matrix is, to my knowledge, found by iterative methods. However you can use a library for that purpose.

Thanks, I will try it out tomorrow. Thank you for all the help, I really do appreciate it.

## 1. What is 3 dimensional torque from angular velocity?

3 dimensional torque from angular velocity is a measure of the rotational force applied to an object in three dimensions. It takes into account both the magnitude and direction of the rotational force.

## 2. How is 3 dimensional torque from angular velocity calculated?

To calculate 3 dimensional torque from angular velocity, you need to multiply the angular velocity vector by the moment of inertia tensor. This will give you a vector representing the torque in three dimensions.

## 3. What is the relationship between angular velocity and torque?

Angular velocity and torque are directly proportional. This means that as the angular velocity increases, so does the torque. Similarly, a decrease in angular velocity will result in a decrease in torque.

## 4. How does 3 dimensional torque from angular velocity affect an object's motion?

3 dimensional torque from angular velocity can cause an object to rotate or change its rotational direction. It is also responsible for maintaining the object's angular momentum.

## 5. In what situations is understanding 3 dimensional torque from angular velocity important?

Understanding 3 dimensional torque from angular velocity is crucial in many scientific fields, such as physics, engineering, and biomechanics. It is especially important in the study of rotational motion and the design of rotating machinery.

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