matematicas visuales visual math
Inversion: an anticonformal mapping

Inversion in a circle is a transformation that preserves the magnitude of angles but each angle is mapped to an angle of opposite sign. The sense or direction of each angle is reversed. Inversion in a circle is an anticonformal mapping (like a reflection in a line).

In particular, inversion is a mapping that preserves the angle of intersection of two circles. Orthogonal circles (or lines) invert into orthogonal circles (or lines). A circle orthogonal to the circle of inversion is mapped into itself.

Inversion transforms each circe ortogonal to the circle of inversion into itself | matematicasvisuales

All the lines through a point and all the circles concentric with the same point as a center can be seen as two particular cases of coaxal sistem of circles. These two coaxal sistems are orthogonal. The lines form a coaxal system of intersecting type and the concentric circles are a non-intersecting coaxal system. These coaxal systems are mapped into two orthogonal coaxal systems:

Inversion of two orthogonal coaxal systems | matematicasvisuales

Every "small" rectangle is transformed into an "small" rectangle:

Every small rectangle is transformed into an small rectangle | matematicasvisuales

In this picture we can see how the order of colors is reversed:

Colors shows how the sense of the angle is reversed | matematicasvisuales


Hilber and Cohn-Vossen - Geometry and the Imagination (pag. 253) - Chelsea Publishing Company
Coxeter - Introduction to Geometry - Wiley and Sons.
Pedoe - Circles, a Mathematical View - Dover
Tristan Needham - Visual Complex Analysis. (pag. 124) - Oxford University Press
Rademacher and Toeplitz - The Enjoyment of Mathematics
Ogilvy - Excursions in Geometry (pag. 24)- Oxford University Press


Complex Polynomial Functions(1): Powers with natural exponent
Complex power functions with natural exponent have a zero (or root) of multiplicity n in the origin.
Complex Polynomial Functions(2): Polynomial of degree 2
A polynomial of degree 2 has two zeros or roots. In this representation you can see Cassini ovals and a lemniscate.
Complex Polynomial Functions(3): Polynomial of degree 3
A complex polinomial of degree 3 has three roots or zeros.
Complex Polynomial Functions(4): Polynomial of degree n
Every complex polynomial of degree n has n zeros or roots.
Complex Polynomial Functions(5): Polynomial of degree n (variant)
Every complex polynomial of degree n has n zeros or roots.
Cero and polo (Spanish)
Podemos modificar las multiplicidades del cero y del polo de estas funciones sencillas.
The Complex Exponential Function
The Complex Exponential Function extends the Real Exponential Function to the complex plane.
The Complex Cosine Function
The Complex Cosine Function extends the Real Cosine Function to the complex plane. It is a periodic function that shares several properties with his real ancestor.
The Complex Cosine Function: mapping an horizontal line
The Complex Cosine Function maps horizontal lines to confocal ellipses.
Multifunctions: Two branch points
Multifunctions can have more than one branch point. In this page we can see a two-valued multifunction with two branch points.
Taylor polynomials: Complex Exponential Function
The complex exponential function is periodic. His power series converges everywhere in the complex plane.
Taylor polynomials: Complex Cosine Function
The power series of the Cosine Function converges everywhere in the complex plane.
Taylor polynomials: Rational function with two complex singularities
We will see how Taylor polynomials approximate the function inside its circle of convergence.