A differentiable map between domains that preserves oriented angles at every point where it is differentiable and has nonzero derivative; infinitesimal shapes are scaled and rotated but not sheared.

Principle

Local similarity: at each point the map acts as a similarity transformation (a local scaling and rotation) so that the angle between any two differentiable curves is preserved.

Demonstration

A Möbius transformation sending the unit disk to the upper half-plane is conformal on the complex plane away from its poles: it maps intersecting curves to curves with the same intersection angles.

Misapplication

Calling a homeomorphism conformal because it preserves some angles at isolated points, or confusing global area preservation with conformality.

Consequence

Infinitesimal geometric structure is preserved up to scale and orientation, so harmonic functions pull back to harmonic functions and local shape features are maintained under the map.

Reversal

An area-preserving diffeomorphism that does not preserve angles (e.g., a shear) — it keeps area but alters local angles and infinitesimal shapes.

Boundary

Requires differentiability and nonvanishing derivative where the property is asserted; in dimensions greater than two conformal maps are much more rigid (compositions of similarities and inversions), and discontinuous or merely topological maps are excluded.

Semantic Tension

Isometry versus conformal map: an isometry preserves lengths and angles globally, while a conformal map preserves angles but may rescale lengths nonuniformly.

Synthesis

A conformal mapping is a differentiable, locally similarity transformation between domains that preserves oriented angles and infinitesimal shapes up to a position-dependent scale and rotation.