Finding Components of Rotated Ellipsoid using Euler Angles

In summary, the conversation discusses finding an algorithm to solve an overlap problem between polyhedra in 3D space. The focus is on an Ellipsoid characterized by its radii and Euler angles, and the question is how to determine the components of these radii in an arbitrary direction after rotation. The proposed solution involves using the inverse rotation and spherical coordinates equations, and using the Pythagorean theorem to calculate the distance from the center to the surface of the ellipsoid. The speaker expresses uncertainty about their approach and seeks feedback.
  • #1
TheDestroyer
402
1
Hello guys,

I'm trying to find an algorithm to solve an overlap problem between many polyhedra in space, so here this question appears.

Imagine an Ellipsoid rotated by Euler angles in 3D space. This Ellipsoid could be characterised by its 3 radii Rx,Ry and Rz, and by 3 Euler angles in space Ix,Iy and Iz.

The question is, how do I get the components of those radii in an arbitrary direction in space after it being rotated by Euler angles? is this problem easy to solve? I don't know where to start.

Thank you
 
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  • #2
I think I got it.

One has to do the inverse rotation for the arbitrary direction by the angles -Ix,-Iy,-Iz. Then use the spherical coordinates equations to get the distances from the center.

x = Rx sin(theta) cos(phi)
y = Ry sin(theta) sin(phi)
z = Rz cos(theta)

then one could take use the Pythagorean to the distance from the center to the surface of that ellipsoid.

Is this approach correct?

Thanks
 
  • #3
Oh my god! not a single comment? am I in the wrong place?
 

1. What are Euler angles components and how are they used in science?

Euler angles components are a set of three angles used to describe the orientation of a rigid body in three-dimensional space. They are commonly used in scientific fields such as mechanics, robotics, and aerospace engineering to represent the rotation of an object.

2. How do Euler angles differ from other methods of representing rotations?

Euler angles are different from other methods such as quaternions and rotation matrices in that they are based on three separate rotations around the axes of a coordinate system, rather than a single rotation around a specific axis.

3. What is the order of rotations used in Euler angles?

The order of rotations in Euler angles can vary depending on the convention used, but the most commonly used convention is known as "yaw, pitch, and roll", where the rotations are applied in the order of z-axis (yaw), y-axis (pitch), and x-axis (roll).

4. Can Euler angles lead to gimbal lock?

Yes, Euler angles can lead to gimbal lock in certain situations. Gimbal lock occurs when two of the three axes align, resulting in the loss of one degree of freedom and a singularity in the orientation representation. This can cause issues in control systems and must be carefully considered when using Euler angles.

5. Are there alternative methods for representing rotations besides Euler angles?

Yes, there are alternative methods such as quaternions, rotation matrices, and axis-angle representations. Each method has its own advantages and disadvantages, and the choice of which one to use depends on the specific application and requirements.

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