- #1

FQVBSina_Jesse

- 15

- 1

- TL;DR Summary
- Time derivative of quaternions can be easily found but I could not find any document on taking spatial derivative of quaternions, such as taking a curl of quaternions.

I have a 5x5x5 set of grid points in space. I can describe each point with p(x,y,z), and I can convert them to spherical or other coordinates. At each point, I have a quaternion assigned to it. So, numerically, I can describe a q(x,y,z) quaternion field. The goal is to obtain a functional form of the quaternion field, i.e. q(x,y,z) = Ax + By + Cz + Ax^2 + ... such that I can get the quaternion at any point given the coordinates. Then, I would be able to take a spatial derivative of it.

If I am dealing with a simple scalar, I can use the trilinear interpolation, described very well on Wikipedia. Then with a function f(x,y,z), I can take the spatial derivative of it to obtain quantities like curl and gradient. Even though many sources describe quaternions as just complex numbers, I feel like they carry more information than just numbers, and whether I can treat each part of the quaternion independently when taking spatial derivatives is unclear. Furthermore, if I just treat the quaternion like a number with a real part and 3 geometric parts, the real part of the quaternion is immediately gone upon taking any derivative, and that feels incorrect to me.

Does anyone have any extended reading on this topic, or have done it before and can give me some pointers?

Thanks in advance!

Jesse

If I am dealing with a simple scalar, I can use the trilinear interpolation, described very well on Wikipedia. Then with a function f(x,y,z), I can take the spatial derivative of it to obtain quantities like curl and gradient. Even though many sources describe quaternions as just complex numbers, I feel like they carry more information than just numbers, and whether I can treat each part of the quaternion independently when taking spatial derivatives is unclear. Furthermore, if I just treat the quaternion like a number with a real part and 3 geometric parts, the real part of the quaternion is immediately gone upon taking any derivative, and that feels incorrect to me.

Does anyone have any extended reading on this topic, or have done it before and can give me some pointers?

Thanks in advance!

Jesse