How do I solve a matrix for it's variables?

I defined this matrix

function [euler]=eu(z,y,x)
[euler]=[cosd(z)*cosd(y) cosd(z)*sind(y)*sind(x)-sind(z)*cosd(x) cosd(z)*sind(y)*cosd(x)+sind(z)*sind(x); sind(z)*cosd(y) sind(z)*sind(y)*sind(x)+cosd(z)*cosd(x) sind(z)*sind(y)*cosd(x)-cosd(z)*sind(x); -sind(y) cosd(y)*sind(x) cosd(y)*cos(x)]
end

I wish to calculate z, y, x when euler=[0,0,-1;0,-1,0;-1,0,0].

I tried

>> solve(eu(z,y,x)==[0,0,-1;0,-1,0;-1,0,0])

What command should I use?

Answer 1

[Roger Stafford]

You cannot determine x, y, and z uniquely from your given value of 'euler'. For this value they all reduce to three simple equalities:

cosd(x-z) = -1
sind(x-z) = 0
sind(y) = 1

from which you can only conclude that y is equal to 90 plus or minus some multiple of 360, and that x and z differ by 180 plus or minus some multiple of 360.

[Katrine Rav]

Allright, bad example. What if I want to solve another value of euler, where euler is a 3x3 matrix? How would I write that?

[Walter Roberson]

9 nonlinear equations in 3 unknowns seldom has a solution, especially since you are using floating point arithmetic to define the values.

But more of a problem is that you are trying to find the unique Euler angles that will bring about a particular 3D rotation. There are always at least two sets of angles that bring about the same 3D rotation.

Answer 2

[Katrine Rav]

I know it's not guaranteed that it will work. But what do I write to check if it works?

The rotationmatrix [0.447314,-0.358681,0.852032;0.447214,0.879974,0.179045;-0.774597,0.300969,0.491919] should generate the angles x=-20, y=60 and z=45.

How do I test that?