= Power maps of a reciprocal-coordinate hyperbola
{title2=$z=t+ia/t,\quad\operatorname{Re}(z^4)=[\operatorname{Im}(z^4)]^2/(16a^2)-4a^2$}
Let $a>0$, $t>0$ and $z=t+ia/t$. The real and imaginary coordinates obey $XY=a$, a <hyperbola>. Put $s=t^2-a^2/t^2$, which is strictly increasing. Then $z^2=s+2ia$ traces a horizontal line, and $z^4=s^2-4a^2+4ias$ traces a <parabola>. Eliminating $s$ gives the displayed quadratic relation between <real part> and <imaginary part>. The value of $t$ fixes an orientation and determines which segment is traversed; the full real parabola is obtained as $t$ ranges over $(0,\infty)$.
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