A circular path connecting two complex numbers

Question

For any two complex numbers $z_1$ and $z_2$, $f(t)$= $z_1+t(z_2-z_1) $is a path in $ℂ$ where $t∊[0,1]$. The image of this path is a line segment. Is there a way of getting a similar path but to connect two given complex numbers using a circular path, i.e. an explicit function to connect two complex numbers that lie on a circle centered at 0 (for simplicity). All I can think of is to maybe write $z_1=z_2e^{iθ}$ but I am not sure whether this is write or whether there is a better way to do this. Any help will be much appreciated.

Answer 1

Let $t_1=\arg(z_1/|z_1|)$ and $t_2=\arg(z_2/|z_2|)$. If $z_1,z_2$ are in the same circle, with radius $r$ and center in the origin then, $$f(t)=re^{it},\ t\in [0,2\pi],$$

is a circular path with $f(t_1)=z_1$ and $f(t_2)=z_2$.

Remark: Given any two distinct points in the plane, there is a alway a circular path containing these points.