Isometries in The Complex Plane
In terms of complex numbers, the isometries of the plane either of the form
or of the form
for some complex numbers a and ω with |ω| = 1. This is easy to prove: if a = f(0) and ω = f(1) − f(0) and if one defines
then g is an isometry, g(0) = 0, and g(1) = 1. It is then easy to see that g is either the identity or the conjugation, and the statement being proved follows from this and from the fact that f(z) = a + ωg(z).
This is obviously related to the previous classification of plane isometries, since:
- functions of the type z → a + z are translations;
- functions of the type z → ωz are rotations (when |ω| = 1);
- the conjugation is a reflection.
Read more about this topic: Euclidean Plane Isometry
Famous quotes containing the words complex and/or plane:
“When distant and unfamiliar and complex things are communicated to great masses of people, the truth suffers a considerable and often a radical distortion. The complex is made over into the simple, the hypothetical into the dogmatic, and the relative into an absolute.”
—Walter Lippmann (1889–1974)
“At the moment when a man openly makes known his difference of opinion from a well-known party leader, the whole world thinks that he must be angry with the latter. Sometimes, however, he is just on the point of ceasing to be angry with him. He ventures to put himself on the same plane as his opponent, and is free from the tortures of suppressed envy.”
—Friedrich Nietzsche (1844–1900)