Parametrizing shapes, curves, lines in $\mathbb{C}$ plane I've been struggling with parametrizing things in the complex plane.
For example, the circle $|z-1| = 1$ can be parametrized as $z = 1 + e^{i\theta}$. I'm not sure how this was done. I understand how we can now write it as $z = 1+\cos t + i\sin t$. 
In general, how do I parametrize things in the complex plane?
 A: Consider the equation of a circle,
$$
x^2 + y^2 = 1.\tag{1}
$$
The parametrization of equation (1) is simply
\begin{align}
x(t) &= \cos(t)\\
y(t) &= \sin(t)
\end{align}
for $t\in[0,2\pi)$.
From your question, we have
$$
\lvert z -1\rvert = \lvert x-1 + iy\rvert = \sqrt{(x-1)^2 + y^2} = 1\tag{2}
$$
where I let $z= x+iy$.  By squaring both sides of equation (2), we have
$$
(x-1)^2 + y^2 = 1
$$
In $\mathbb{R}^2$, this would be a circle with center $(1,0)$ or radius $1$.  We have already seen the parametrization of circle with the center at the origin of radius $1$.  If we can shift that $x(t)$ to the right by $1$, we will be good to go; that is,
\begin{align}
x(t) &= 1 + \cos(t)\\
y(t) &= \sin(t)
\end{align}
Now $z(t) = x(t) + iy(t) = 1 + \cos(t) + i\sin(t) = 1 + e^{it}$.

Your second question was how does one go about parametrizing in the Complex plane.

  
*
  
*Draw the plot.
  
*Determine the direction of motion--counter clockwise or clockwise.
  
*Pick a starting point.
  

As example, consider a square with in the Complex plane with vertices $(0, 0), (1, 0), (1, 1), (0, 1)$.  The direction will be counter clockwise.
\begin{align}
\gamma_1: & \text{ will go from } (0,0)\to(1,0)\\
\gamma_2: & \text{ will go from } (1,0)\to(1,1)\\
\gamma_3: & \text{ will go from } (1,1)\to(0,1)\\
\gamma_4: & \text{ will go from } (0,1)\to(0,0)
\end{align}
At $t=0$, we will start at $(0,0)$, and at $t=1$, we need to be at $(1,0)$, and so and so on for the other points.
\begin{alignat}{2}
\gamma_1(t) &= t &&\quad 0\leq t\leq 1\\
\gamma_2(t) &= 1 + i(t - 1) &&\quad 1\leq t\leq 2\\
\gamma_3(t) &= (3 - t) + i &&\quad 2\leq t\leq 3\\
\gamma_1(t) &= i(4 - t) &&\quad 3\leq t\leq 4
\end{alignat}
