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Conformal Maps

A conformal map is a complex function that preserves angles locally. If two smooth curves meet at a point and a function is complex differentiable there with nonzero derivative, then the images of the two curves meet at the same angle. Geometrically, near such a point the function acts like a rotation and a scaling, plus smaller errors that disappear as we zoom in.
Away from the critical point 00, the map z↦z2z\mapsto z^2 preserves the angle between two crossing line segments.

References

  • [Needham2023]Needham, Tristan. Visual Complex Analysis. 25th Anniversary Edition. Oxford University Press, 2023. First edition published in 1997.