direction cosine matrix to axis angle of rotation

The angle of rotation can be found from the trace of the direction cosine matrix to axis angle of rotation matrix

A + A + A = 3cos (α ) + (1 − cos(α ))(e2 + e2+ e2) 11 22 33 1 2 3

Noting that the axis of rotation is a unit vector and has a length of 1 means

2 2 2 e1 + e2 + e3 = 1

therefore

A11 + A22 + A33 = 1 + 2cos(α )

rearranging gives

−1 1- α = cos (2 (A11 + A22 + A33 − 1))

(1)

Inverse cosine is a multivalued function and there are 2 possible solutions for α. Normally, the convention is to choose the principle value such that 0 < α < π

As long as α is not zero, the unit vector is given by

⌊ ⌋ (A23-−-A32)- ⌊ ⌋ || 2sin(α) || e1 | (A31 − A13) | ⌈ e2 ⌉ = || --2sin(α)---|| e3 |⌈ (A − A ) |⌉ --12-----21-- 2sin(α)

(2)

Above equation should be proved at some time...

"direction cosine matrix to axis angle of rotation" is owned by bloftin.
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