# Euclidean geometry and complex plane

Can someone please describe to me how Euclidean Geometry is connected to the complex plane? Angles preservations, distance, Mobius Transformations, isometries, anything would be nice.

Also, how can hyperbolic geometry be described with complex numbers?

Related Differential Geometry News on Phys.org
Perhaps you should read a good complex analysis book. Try to read "visual complex analysis" by Needham. A very easy book, but with a lot of intuition and connections to geometry.

Multiplying one complex number by another scales it and rotates it. Adding complex numbers give you translation. So, complex numbers can give you any dilation. So, if two figures are the similar, you can get from one to the other by multiplying and adding complex numbers to it.

There's a model for the hyperbolic plane which is the unit disc in the complex plane. Mobius transformations that map the unit disc to itself act as the rigid motions of the hyperbolic plane.

mathwonk