Proposition

Find a parametrization for each of the following paths:

The circle $C[1 + i, 1]$, oriented counter-clockwise.
The line segment from $-1 - i$ from $2i$.
The top half of the circle $C[0, 34]$, oriented clockwise.
The rectangle with vertices $\pm 1 \pm 2i$, oriented counter-clockwise.
The ellipse $\{ z \in \mathbb{C}: \abs{z - 1} + \abs{z + 1} = 4 \}$, oriented counter-clockwise.

Solution

1

$\gamma(t) = 1 + i + e^{2\pi it}$ with $0 \leq t \leq 1$.

2

$\gamma(t) = t + (3t + 2)i$ with $-1 \leq t \leq 0$.

3

$\gamma(t) = 34e^{-2\pi it}$ with $1/2 \leq t \leq 1$.

4

\[\begin{align*} \gamma(t) &= \begin{cases} 1 + (t - 2)i & (0 \leq t \leq 4) \\ 1 - (t - 4) + 2i & (4 \leq t \leq 6) \\ -1 + (2 - (t - 6))i & (6 \leq t \leq 10) \\ -1 + (t - 10) - 2i & (10 \leq t \leq 12). \end{cases} \end{align*}\]

5

$\gamma(t) = 2\cos(t) + i\sqrt{3}\sin(t)$.

Justification:

\[\begin{align*} \abs{\gamma(t) - 1} + \abs{\gamma(t) + 1} &= \sqrt{(2\cos t - 1)^2 + 3\sin^2t} + \sqrt{(2\cos t - 1)^2 + 3\sin^2t} \\ &= \sqrt{\cos^2 t - 4\cos t + 4} + \sqrt{\cos^2 t + 4\cos t + 4} \\ &= \abs{\cos t - 2} + \abs{\cos t + 2} \\ &= 2 - \cos t + \cos t + 2 \\ &= 4. \end{align*}\]