# Detect non-uniform scaling in matrix

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Does anyone know of a fast way to tell if a matrix has non-uniform scaling?

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You can probably get the length of the basis vectors of the matrix (for example the length of the first, second and third row). This gives you the x, y and z scale. If they are not equal to each other there is non-uniform scaling.

Not sure if this is the best/safest or fastest method though, but it is definitely faster than doing full polar/spectral decomposition.

Edited by Buckshag

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If a matrix is the composition of a rotation and a uniform scaling, its three columns should be orthogonal to each other and have the same length.

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Thanks for the answer. This is the formula I came up with finally:

// note: this is opengl type matrix
bool Matrix::IsUnformScaling() const
{
const float THRESHOLD = 0.01f;
float xLen = m[0]*m[0] + m[1]*m[1] + m[2]*m[2];
float yLen = m[4]*m[4] + m[5]*m[5] + m[6]*m[6];
float zLen = m[8]*m[8] + m[9]*m[9] + m[10]*m[10];
return fabsf(xLen - yLen) < xLen*THRESHOLD && fabsf(xLen - zLen) < xLen*THRESHOLD;
}


I tested it and it seems to work ;)

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I can construct a matrix that has non-uniform scaling in it and passes your test, though...

EDIT: More explicitly,
  m[0] = 0.8; m[1] = 0.6; m[2] = 0.0;
m[4] = 0.6; m[5] = 0.8; m[6] = 0.0;
m[8] = 0.0; m[9] = 0.0; m[10] = 1.0;


Edited by Álvaro

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I can construct a matrix that has non-uniform scaling in it and passes your test, though...

EDIT: More explicitly,

  m[0] = 0.8; m[1] = 0.6; m[2] = 0.0;
m[4] = 0.6; m[5] = 0.8; m[6] = 0.0;
m[8] = 0.0; m[9] = 0.0; m[10] = 1.0;



do you have a solution perchance?

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I posted that already, although perhaps my description was too succinct. You also need to verify that those three vectors are orthogonal to each other (check all three pairs).
bool are_orthogonal(Vector3 v, Vector3 w) {
static float const tolerance = 1e-6;
float dp = dot_product(v,w);
return dp * dp < dot_product(v,v) * dot_product(w,w) * tolerance;
}


EDIT: To be complete, you should also verify that the transformation preserves orientation (its determinant is positive). Edited by Álvaro