In a conformal map, which property is preserved locally?

• A. Angles are preserved locally
• B. Areas are preserved locally
• C. Distances are preserved locally
• D. Volumes are preserved locally

Answer: (A)

Angles are preserved locally because a conformal map in the plane acts like a rotation and uniform scaling at each point. Locally, its effect on infinitesimal shapes is a linear transformation represented by multiplying by the derivative, which is a nonzero complex number. That multiplication rotates by the angle of the derivative and scales by its magnitude, so the angles between intersecting curves are kept. Distances and areas, on the other hand, can be stretched or compressed, so they aren’t preserved in general. The preservation applies in tiny neighborhoods around each point, not necessarily across the whole map. For example, a conformal map like z -> z^2 is locally a rotation and scaling away from critical points, preserving angles but changing sizes globally.