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:
“We must open our eyes and see that modern civilization has become so complex and the lives of civilized men so interwoven with the lives of other men in other countries as to make it impossible to be in this world and out of it.”
—Franklin D. Roosevelt (1882–1945)
“Even though I had let them choose their own socks since babyhood, I was only beginning to learn to trust their adult judgment.. . . I had a sensation very much like the moment in an airplane when you realize that even if you stop holding the plane up by gripping the arms of your seat until your knuckles show white, the plane will stay up by itself. . . . To detach myself from my children . . . I had to achieve a condition which might be called loving objectivity.”
—Anonymous Parent of Adult Children. Ourselves and Our Children, by Boston Women’s Health Book Collective, ch. 5 (1978)