Enlargements and Stretches
Enlargements
- \(\begin{pmatrix}a & 0 \\ 0 & b\end{pmatrix}\) represents a stretch of scale factor \(a\) parallel to the x-axis and a stretch of scale factor \(b\) that is parallel to the y-axis.
- When \(a=b\), this is an enlargement.
Stretch of factor 2 in the x axis
\[\begin{pmatrix}2 & 0 \\ 0 & 1\end{pmatrix}\]
Stretch of factor 4 in the x direction and stretch of factor 16 in the y direction
\[\begin{pmatrix}4 & 0 \\0 & 16\end{pmatrix}\]
Using the derivative to calculate areas
\(A(1, 1), B(1, 2), C(2, 2)\) are points on a triangle. The transformation of matrix \(\mathbf{M}=\begin{pmatrix}4 & 0 \\ 0 & 3\end{pmatrix}\) is applied to the triangle.
1. New coords are \(A(4, 3), B(4, 6), C(8, 6)\)
2. The area of the original triangle is 1/2
3. The area of the new triangle is 6
4. The determinant of the original matrix gives you the scale factor that you will enlarge the area by when you complete the transformation. The determinant of the matrix is \(12\) and \(\frac{1}{2}\times 12=6\)
This rule is written as: Area of image = Area of object \(\times |\det(\mathbf{M})|\)