Does $\gamma(t)=e^{it^2} , t \in [0,\sqrt{2\pi}]$ represent the unit circle?

Question

$$ \begin{align} \gamma(t)=e^{it^2} , t \in [0,\sqrt{2\pi}] \end{align} $$

Is my thinking correct that $\gamma$ represents the unit circle correct?

Answer 1

As commenters said, you are right: this is a circle

More generally, if $\Gamma:[a,b]\to \mathbb C$ is a curve, and $f:[c,d]\to [a,b]$ is a strictly increasing continuous function such that $f(x)=a$ and $f(d)=b$, then the composition $\Gamma\circ f$ is a different parametrization of the same geometric object.

In your case, $f(t)=t^2$ and $[c,d] =[0,\sqrt{2\pi}]$; also, $\Gamma(t)=e^{it}$ and $[a,b]=[0,2\pi]$.