matematicas visuales visual math

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

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

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

Every small rectangle is transformed into an small rectangle

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

Colors shows how the sense of the angle is reversed

REFERENCES

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

LINKS

Inversion
Inversion
Inversion is a plane transformation that transform straight lines and circles in straight lines and circles.
Moebius transformations (Spanish)
The Complex Exponential Function
The Complex Exponential Function
The Complex Exponential Function extends the Real Exponential Function to the complex plane.
More about complex functions
More about complex functions
Examples of complex functions: polynomials, rationals and Moebius Transformations.