Dilation - crowlogic/arb4j GitHub Wiki

A dilation, also referred to as a homothety, is a specific type of geometric transformation that enlarges or shrinks objects while preserving their shape. In the context of complex analysis, a dilation can be thought of as a transformation that scales the size of a figure in the complex plane by a fixed factor and may change its orientation.

A dilation in the complex plane can be defined by a complex function of the form:

$$ f(z) = \alpha z $$

Here, $z$ is a complex number, and $\alpha$ is a nonzero complex constant called the dilation factor. The magnitude of the dilation factor $|\alpha|$ determines the scaling factor, while the argument of the dilation factor $\arg(\alpha)$ determines the rotation angle.

When $|\alpha| > 1$, the dilation enlarges the figure in the complex plane, and when $0 < |\alpha| < 1$, the dilation shrinks the figure. The orientation of the figure will be changed if $\arg(\alpha) \neq 0$, resulting in a rotation of the figure in addition to the scaling.

In summary, a dilation is a geometric transformation that scales the size of a figure in the complex plane by a fixed factor, preserves its shape, and may change its orientation, depending on the argument of the dilation factor.