The complex linear function [math]f(z)=Az[/math] is crucial in understanding the complex derivative.[br]The Geometry of the magnification by [math]|A|[/math] leads to a cone in the mapping diagram with its vertex determined by [math]|A|[/math], the twist that is connected to [math]A[/math] giving a swirl like curve on the cone that is close to a "geodesic" curve on the cone.

Move A and z# to change the parameter A and the point z#.[br]Click on the boxes to show mapping diagram arrows and/or mapping diagram "geodesic" curves on the magnification cone.[br]Change r on the slider to change the domain circle "|z| = r".