= Reciprocal transformation of a generalized circle
{title2=$(a,b,c)\mapsto(c,\bar b,a)$}
Write a <generalized circle> equation as $a|z|^2+b\bar z+\bar bz+c=0$, where $a,c$ are real. Under the reciprocal <Möbius transformation> $z=1/w$, with $w\ne0$, multiply by $|w|^2$ to obtain
$$
c|w|^2+\bar b\bar w+bw+a=0.
$$
Thus the coefficient triple transforms as $(a,b,c)\mapsto(c,\bar b,a)$. A nondegenerate <circle> through zero has $c=0$ and becomes a <straight line>; a <straight line> missing zero becomes a <circle> through zero on the <Riemann sphere>. When working only with finite nonzero coordinates, retain the excluded point $w=0$ even if it satisfies the multiplied equation.
This reciprocal transformation is different from <inversion in a circle>, whose unit-circle formula is $z\mapsto1/\bar z$. The latter reverses <orientation>; the reciprocal transformation is holomorphic away from its pole.
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