Parametric curves including complex numbers

Question

$$z = te^{it}$$

This is a parametric curve, but I am not sure on how to exactly plot it. Can I confirm that it is a anticlockwise spiral from the origin? Or am I incorrect?

Answer 1

If you take the imaginary part as the y-axis and the real part as the x-axis, then you actually get a counter-clockwise spiral.

Rewrite

You can put the formula in "radius of angle"-notation $r(\phi)$, just like you do to plot circles: $$ \begin{align*} z &= t \cdot e^{t \cdot \mathrm{i}}\\ r(\phi) &= \left| \phi \cdot e^{\phi \cdot \mathrm{i}} \right|\\ r(\phi) &= | \phi | \cdot \left| e^{\phi \cdot \mathrm{i}} \right|\\ r(\phi) &= | \phi | \cdot \left| \cos(\phi) + \sin(\phi) \cdot \mathrm{i} \right|\\ r(\phi) &= | \phi | \cdot \sqrt{ \cos(\phi)^{2} + \sin(\phi)^{2} }\\ r(\phi) &= | \phi | \cdot \sqrt{ 1 }\\ r(\phi) &= | \phi | \cdot 1\\ r(\phi) &= | \phi |\\ \end{align*} $$ Where $r$ is the radius and $\phi$ is the angle.

Plotting

Plotting point by point

You can use $r$ as the distance from the point to plot to the origin and $\phi$ as the the angle from the real axis to the point across the origin aka $\arg(z)$. If you simply insert a couple of values, you get points which, as t grows, get further and further away from the origin or "always" change their "angle".

Now you can connect the points with a curve and you get the plot.

Plotting via Wolfram|Alpha

You can just type this in Wolfram|Alpha:

plot r = t from t = a to t = b

For Wolfram|Alpha, "plot" means that Wolfram|Alpha is a graphic representation for an equation, inequality, function, line, point, ...

• "r = t" implies for what we call $r(\phi) = | \phi |$.
• "from t = a" tells Wolfram|Alpha what is the lowest value of t that Wolfram|Alpha should plot.
• "to t = b" tells Wolfram|Alpha what is the hugest value of t that Wolfram|Alpha should plot.
• "from t = a to t = b" tells Wolfram|Alpha that Wolfram|Alpha can / may only use values ​​of $a \leq t \leq b$.

E.G. If you use $a = 0 \text{ and } b = 16 \cdot \pi$, you will get: