# Scale Factor

• Nov 8th 2009, 05:42 AM
stratovarius
Scale Factor
hello,

can someone explain me what a volume scale factor is and if I have a liner Transformation
T(v1) = av1 + bv2
T(v2) = cv1 + dv2

how can I find the volume scale factor of it?

Thank you
• Nov 8th 2009, 06:29 AM
Tuufless
You mean area factor? (Volume factors are for 3D...)

Think of it this way: if you have an enlargement matrix, say scale factor 2, then your matrix is $\displaystyle \begin{bmatrix}2 & 0\\0 & 2\end{bmatrix}$.

So, if we define a unit square (1x1 length) using four points, say $\displaystyle A = \begin{bmatrix}0\\0\end{bmatrix}, B = \begin{bmatrix}0\\1\end{bmatrix}, C = \begin{bmatrix}1\\1\end{bmatrix}$ and $\displaystyle D = \begin{bmatrix}1\\0\end{bmatrix}$.

Then if we put all those points into the matrix, the transformed points are $\displaystyle A' = \begin{bmatrix}0\\0\end{bmatrix}, B' = \begin{bmatrix}0\\2\end{bmatrix}, C' = \begin{bmatrix}2\\2\end{bmatrix}$ and $\displaystyle D' = \begin{bmatrix}2\\0\end{bmatrix}$.

These points define a matrix with an area of 4 square units, so the original unit square got enlarged by an area factor of 4. That is to say, if you used points to define a polygon with area n square units, then the transformed shape will have an area of 4n square units.

For any generic transformation matrix, the area factor is equal to the determinant.

i.e: For any matrix $\displaystyle M = \begin{bmatrix}a & b\\c & d\end{bmatrix}$, the area factor is $\displaystyle det(M) = ad-bc$ (for a 2x2 matrix).
• Nov 8th 2009, 07:01 AM
stratovarius
Thanks so much for the reply, i think I understood what you say , but no I meant volume scale factor or metric scale factor, that's how it was given to me.

Regards
• Nov 8th 2009, 07:42 AM
Tuufless
Oh, oops- my bad, I thought you were dealing with 2x2 matrices only since your original post had a system of two vectors.

In general, the volume factor is just the determinant of the matrix. (We say "volume factor" for anything above 2D.)