Question

How do I convert a 2×2 rotation matrix into a Euler angle? The rotation matrix is:

{{.46, .89}, {.89, -.46}}

Wikipedia instructs me that a 2D rotation matrix takes the form:

{{cos(a), -sin(a)}, {sin(a), cos(a)}}

Knowing that

{{cos(a), -sin(a)}, {sin(a), cos(a)}} = {{.46, .89}, {.89, -.46}}

I computed

{{inverseCos(a), -invereSin(a)}, {invereSin(a), inverseCos(a)}}

to get (these values were converted to degrees)

{{62.3, -62.3}, {62.3, 117.8}}

What am I supposed to do with these numbers? Aren’t they supposed to be equal? The universe no longer makes sense to me.

Answer 1

Your matrix is not a pure rotation. The first and last entries are the negative of each other and so can’t possibly both be equal to the cosine of some angle.

Looks like you actually have a scale matrix multiplied in there too of the form

{{1, 0}, {0, -1}}

Edit:

From wikipedia:

In this case, the diagonal matrix Σ is uniquely determined by M (though the matrices U and V are not).

In your wolfram example both U and V have a scale matrix of {{1,0},{0,-1}} in them which of course cancel out to leave pure rotation matrices. My understanding is that these would be a valid choice of U and V too i.e. the decomposition is not unique.