Angle Preservation

Complex Analysis · Axiom Academy

Why an analytic map with keeps the angle between any two curves exactly the same. 1. The Map Multiplies Every Tangent by f'(z_0) Send a smooth curve through z_0 into its image . By the chain rule, the tangent of the image is the original tangent multiplied by the constant f'(z_0) : image tangent = f'(z_0) × original tangent Watch two curves cross at z_0 = i under w = z^2 . Here f'(i) = 2i , so every tangent is turned by and stretched by |2i| = 2 . The blue and orange tangents both swing by the same 90°. Multiplying by f'(z_0) adds to the argument of every tangent. When you subtract two arguments to get the angle between the curves, that common rotation cancels: So the whole fan of directions at z_0 is rigidly rotated by and uniformly scaled by |f'(z_0)| . A rigid rotation keeps every pairwise angle fixed — that is exactly the conformal property. Every direction turns by the same angle . Every length stretches by the same factor |f'(z_0)| . A common rotation cancels in the difference, so angles are preserved. Uniform scaling keeps small shapes similar — circles stay circles. Near z_0 , — a translation, a rotation, and a scaling. That is a similarity transformation , and similarities preserve angles by definition. 3. Where It Breaks: Critical Points The argument only works when . If f'(z_0) = 0 and the first nonzero derivative is the n -th, then near z_0 the map looks like (z - z_0)^n instead of a single multiplication:

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