First of all, I refer you to D3DMatrixDecompose. If you want to break a standard 3D transformation matrix into its rotational, translational and scaling parts, without caring how it’s done, then look no further. If your needs are a little more specific and you’re sure you aren’t reinventing this wheel, then read on.

There is nothing clever about this decomposition, but it’s a question that comes up more often than I’d expect, so here’s the lowdown. I assume that your matrix is in DirectX-standard row-vector representation (so v_out = v_inM) and that the skew components are zero (M₁₄ = M₂₄ = M₃₄ = 0). You should visualise the matrix like this:

First, we extract the translation:

v_translation = (M₄₁, M₄₂, M₄₃)

Then the scaling factors:

s_x = √(M₁₁² + M₂₁² + M₃₁²)
s_y = √(M₁₂² + M₂₂² + M₃₂²)
s_z = √(M₁₃² + M₂₃² + M₃₃²)

The rotation matrix is then the upper-left 3×3 minor, after scaling back to unity.

M^rotation = M;

// Remove translation
M^rotation₄₁ = 0;
M^rotation₄₂ = 0;
M^rotation₄₃ = 0;

// Normalise
M^rotation₁₁ /= s_x;
M^rotation₂₁ /= s_x;
M^rotation₃₁ /= s_x;

M^rotation₁₂ /= s_y;
M^rotation₂₂ /= s_y;
M^rotation₃₂ /= s_y;

M^rotation₁₃ /= s_z;
M^rotation₂₃ /= s_z;
M^rotation₃₃ /= s_z;

Posted: Sunday, November 18th, 2007, 9:00 am
Filed under: Game Programming.