Angle-preserving (isogonal) map

Preserving the size of angles between curves, without necessarily preserving their orientation.

A map is angle-preserving, or isogonal, at a point if it keeps the magnitude of the angle between any two crossing curves the same, whether or not it also keeps the angle's orientation (clockwise vs. counterclockwise). Conformal maps are angle-preserving, but not every angle-preserving map is conformal: complex conjugation, $f(z)=\bar z$ , is isogonal everywhere -- it is a reflection, so angle sizes survive -- yet it reverses every angle's orientation, so it is not conformal. "Conformal" is the stronger and far more commonly used of the two terms in this course.

Examples

Reflection across the real axis, $f(z)=\bar z$ , preserves angle magnitudes but reverses their orientation, so it is isogonal but not conformal.
Every conformal map is automatically angle-preserving, since preserving orientation as well as magnitude is a strictly stronger condition.