Suppose $ h : =z\mapsto \dfrac{az+b}{cz+d} $ is an homography of the complex plane preserving a vertical line.
What is the general form of $ h $? Is $ c $ necessarily equal to $ 0 $?
2026-03-29 22:06:44.1774822004
Homography preserving a vertical line
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Presumably by "complex plane" you really mean the Riemann sphere, and "vertical line" includes the point at $\infty$. Then besides a suitable linear function, you could take an inversion about a point on the line.