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June 20th, 2019, 07:42 AM  #1 
Senior Member Joined: Jun 2019 From: USA Posts: 203 Thanks: 80  3D projection problem (pesky angles)
(Note: changed Greek variables to Latin for simplicity typing; "." represents the dot product) I ran a quick static calibration experiment on some accelerometers. Now I'm trying to apply the results correctly. Problem statement: I have a fixed RH Cartesian unit vector coordinate system x,y,z (though I only care about z), and another RH Cartesian unit vector coordinate system X,Y,Z attached to a rigid object. I have measured two angles. a is the angle between the zX plane and the ZX plane. b is the angle between the Yz plane and the YZ plane. I don't care about the third angle (basically "yaw"), because I am only interested in projections on the z axis. a and b are defined such that: If b = 0, then {z.X = 0 z.Y = sin(a) z.Z = cos(a)} If a = 0, then {z.X = sin(b) z.Y = 0 z.Z = cos(b)} Problem is, I have some cases where both a and b are unavoidably nonzero. How do I find the projections (z.X, z.Y, z.Z) as a function of both a and b? If anyone can help lead me to the answer (or give it with an explanation), I would be extremely grateful. I have tried looking at z cross X and z cross Y unit vectors, even pulled out some of my index notation vector identity tricks, but so far I haven't found anything helpful. BTW, I am aware that when a = +/ pi/2, b may be undefined, and vice versa, but it seems like this shouldn't matter in terms of the projections on z. Also, if it turns out I could have solved this as the product of two trivial rotation matrices, I'm going to be mad, lol. 
June 24th, 2019, 03:24 PM  #2 
Senior Member Joined: Jun 2019 From: USA Posts: 203 Thanks: 80 
Alright, getting warmer and managed to draw the picture correctly. $\displaystyle \alpha$ is the angle between the XZ plane (green) and the Xz plane (yellow) as shown. $\displaystyle \beta$ is the angle between the YZ plane (red) and the Yz plane (magenta) as shown. Looking for the projections of z onto X, Y, and Z. It looks like it's just: $\displaystyle sin(\beta)$ $\displaystyle sin(\alpha)$ $\displaystyle cos(\alpha)cos(\beta)$ Was it really that simple? Can someone verify, please? 

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angles, pesky, problem, projection 
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