Complex exponential and lines

A vertical line in $\Bbb C$ is parallel to the $y$-axis; thus it is a set of points of the form

$a + it, \; a \in \Bbb R \; \text{fixed}, \; t \in \Bbb R, -\infty \to t \to \infty; \tag 1$

we have

$e^{a + it} = e^a e^{it} = e^a(\cos t + i \sin t), \tag 2$

which traces out a circle in $\Bbb C$ of radius $e^a$ as $-\infty \to t \to \infty$.

A horizontal line in $\Bbb C$ is parallel to the $x$-axis; thus it is a set of points of the form

$t + ib, \; b \in \Bbb R \; \text{fixed}, \; t \in \Bbb R, -\infty \to t \to \infty, \tag 3$

and thus,

$e^{t + ib} = e^t e^{ib} = e^t(\cos b + i\sin b); \tag 4$

this traces out the half-ray of points of the form (in polar corrdinates) $re^{i \theta}$, where $0 < r < \infty$ and $\theta \in [0, 2\pi)$ is that unique angle such that

$\cos \theta = \cos b, \; \sin \theta = \sin b. \tag 5$

We note in closing that $a + 0i$ is a point on the real axis $\Bbb R$, not a line, and that $0 + bi$ is a point on the imaginary axis $i\Bbb R$, and that neither represents a line in $\Bbb C$.