Solving for Euler angles and 3-D coordinate Rotations.

[vtm6001]

Hi,

(attachment with visuals is included)

I have a 3-D vector dataset that is measured in a reference frame (measurement reference frame) that is oriented relative to a horizontal coordinate system. In this dataset I have x-y- and z-component data for the vectors relative to a coordinate system on a flat surface (+ve x is to the right, +ve y is to the top, and +ve z is toward the screen/you).

I have an irregular surface, for which I have calculated slope in the x- and y- directions (sign (+/-) follows the right hand rule). The slopes are therefore measured in the x-z and y-z planes. Slopes usually range between -45 to 45 degrees.

I want to solve for the x'-, y'- and z'- component data for the vectors (which remain fixed) in a reference frame where the z component is normal to the irregular surface (and not in the measurement reference frame).

I've been crunching the 2-D examples from my texts, but something does not seem right when I put them together. On the other hand, I know 3-D rotations involve having the Euler angles as an input, but I have no clue how to obtain them using the data I have that characterizes the surface (x- and y- component slopes relative to the measurement reference frame). I also know that the order of the rotations affect the outcome and have no idea how to approach this part of the problem either.

Anyone willing to pitch in? Let me know if I haven't explained the problem clearly.

See attached for the pdf with images showing a) the irregular surface, b) x-slope from irregular surface, c) y-slope from irregular surface and d) vector field on that surface. Note that vector magnitudes and directions remain the same in the measured and rotated reference frames, but it's their components that change in the rotated frame (x', y' and z').

 

[sunjin09]

first off, you figure out the unit vectors x', y', z' for every point on your irregular surface, in terms of x, y, z, then project your vector field v onto these unit vectors by taking the dot product between v and x', y', z', both of which are given in terms of x,y,z, to obtain those coefficients of v in terms of x',y',z'

 

[vtm6001]

first off, you figure out the unit vectors x', y', z' for every point on your irregular surface, in terms of x, y, z, then project your vector field v onto these unit vectors by taking the dot product between v and x', y', z', both of which are given in terms of x,y,z, to obtain those coefficients of v in terms of x',y',z'

I attempted something like that, which looks like this (i/i', j/j' and k/k' have hats over them):

x' = r.i' = x(i.i') + y(j.i') + z(k.i')

y' = r.j' = x(i.j') + y(j.j') + z(k.j')

z' = r.k' = x(i.k') + y(j.k') + z(k.k')

For z' however, I do not know the angles to use between the i-k', j-k' and k'k' vector components.