Parametrization in complex plane

Question

Parametrize the semicircle $| z-4-5i| = 3$ clockwise from $z=4+8i$ to $z=4+2i$
I know that $z(t)=r$ from $$ 0\leq t \leq \pi $$ Then I have the following inequality: $$3i\leq \sqrt{3}e^{-it}\leq -3i $$ What should I do next/have done wrong?

Answer 1

A circle with centre $\alpha$ and radius $r$ also has parameterization $$\gamma (t) = \alpha + r (\cos t + i \sin t)$$ where $t \in [0, 2\pi)$

From your question you know the radius and centre.

You finally need to determine the appropriate range of the parameter to describe the circular arc. In your case the real parts of the initial and final points of the path are equal and have the same value as the real part of the circle centre. This convenient arrangement may help.( In fact you even tell us it is a semi-circle.)

Note if you want to traverse the path clockwise you might have to take care with the parameterization and parameter range. The description above refers to anti-clockwise traversal.

For completeness, one solution for clockwise traversal is as follows.

$$\gamma (t) = \alpha + r (\cos(- t ) + i \sin( -t) )$$ where $t \in [-\pi/2, \pi/2]$ and using the symmetry of the trig. functions this becomes $$\gamma (t) = \alpha + r (\cos(t ) - i \sin(t) )$$ for $t \in [-\pi/2, \pi/2]$

Answer 2

If you know the endpoints $A (z=a)$ and $B(z=b)$ of a circular arc and the signed angle $\theta$ the segment $AB$ subtends, the parametric equation of the arc is $$\frac{ z-b}{ z-a} = t e^{i\theta}, t\ge0$$ In your case $\theta = \pi/2$, being a semicircle.