Dilation is a type of transformation that changes the size of a figure without altering its shape. In mathematical terms, dilation scales the coordinates of a figure by a certain factor, known as the scale factor. This transformation can be represented using a transformation matrix.
Understanding Dilation
Dilation involves a scale factor, denoted by $k$. If $k > 1$, the figure enlarges, and if $0 < k < 1$, the figure shrinks. For example, if a point $(x, y)$ is dilated by a scale factor of $k$, its new coordinates become $(kx, ky)$
Transformation Matrix for Dilation
A transformation matrix is a 2×2 matrix used to perform linear transformations on vectors. For dilation, the transformation matrix is:
$begin{bmatrix} k & 0 \ 0 & k end{bmatrix}$
When this matrix is multiplied by a coordinate vector, it scales the vector by the factor $k$. For instance, consider a point $(x, y)$ represented as a vector $begin{bmatrix} x \ y end{bmatrix}$. The dilation transformation is performed as follows:
$begin{bmatrix} k & 0 \ 0 & k end{bmatrix} begin{bmatrix} x \ y end{bmatrix} = begin{bmatrix} kx \ ky end{bmatrix}$
This results in the point $(kx, ky)$, which is the dilated version of $(x, y)$
Example of Dilation
Let’s consider a simple example. Suppose we have a point $(2, 3)$ and we want to dilate it by a factor of 2. The transformation matrix for this dilation is:
$begin{bmatrix} 2 & 0 \ 0 & 2 end{bmatrix}$
Multiplying this matrix by the vector representing the point $(2, 3)$, we get:
$begin{bmatrix} 2 & 0 \ 0 & 2 end{bmatrix} begin{bmatrix} 2 \ 3 end{bmatrix} = begin{bmatrix} 4 \ 6 end{bmatrix}$
So, the point $(2, 3)$ is transformed to $(4, 6)$, effectively doubling its distance from the origin.
Impact on Figures
When dilation is applied to a geometric figure, every point of the figure is scaled by the same factor. For instance, if we have a triangle with vertices at $(1, 1)$, $(2, 1)$, and $(1, 2)$, and we apply a dilation with a scale factor of 3, the vertices will move to $(3, 3)$, $(6, 3)$, and $(3, 6)$, respectively. The shape of the triangle remains the same, but its size increases.
Conclusion
Dilation affects a transformation matrix by scaling the coordinates of a figure uniformly. The transformation matrix for dilation is a diagonal matrix where the diagonal elements are the scale factor. This matrix, when multiplied by a coordinate vector, scales the vector by the factor, resulting in a dilated figure. Understanding how dilation affects transformation matrices is crucial in fields like computer graphics, where scaling objects is a common operation.