What's the visualization of $z=\sec\theta+i\cdot\csc\theta$ in complex-plane?

Question

What would be the graph of $z=\sec\theta+i\cdot\csc\theta$ in complex plane? Since $z=\cos\theta+i\cdot\sin\theta$ is a circle.

Answer 1

With $x=sec \theta$ adn $y=csc \theta $ you get $$\frac {1}{x^2} + \frac {1}{y^2}=1$$ or $$y=\pm \sqrt{\frac {x^2}{x^2-1}}$$

The graph consists of four branches with two vertical asymptotes at $x=\pm 1$ and two horizontal asymptotes at $y=\pm 1$