MAT334--2020F > Chapter 1

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RunboZhang:
Hi guys, we talked about inversion during the previous lecture and I am a bit confused by the last slide. So by definition, we have
z-> w=z^-1, and then we can calculate the inversion of any point either by its inverse or its polar form. We have also proved that the inversion of a circle is a vertical straight line in the same slide. But I am a bit confused by the red highlighted part, "inversion is self-inverse". My thought is that an inversion of a circle is a straight line and correspondingly the inversion of that straight line is the original circle. And this property thus makes it self-inverse. I don't know if my understanding is correct so I am writing this post to look for some help. And one more brief question, do we have any restriction on inversion/self-inversion?

(btw slide pic is attached below)

Victor Ivrii:

--- Quote from: RunboZhang on September 16, 2020, 04:11:52 PM ---Hi guys, we talked about inversion during the previous lecture and I am a bit confused by the last slide. So by definition, we have
$z\to w=z^{-1}$, and then we can calculate the inversion of any point either by its inverse or its polar form. We have also proved that the inversion of a circle is a vertical straight line in the same slide. But I am a bit confused by the red highlighted part, "inversion is self-inverse". My thought is that an inversion of a circle is a straight line and correspondingly the inversion of that straight line is the original circle. And this property thus makes it self-inverse. I don't know if my understanding is correct so I am writing this post to look for some help. And one more brief question, do we have any restriction on inversion/self-inversion?

(btw slide pic is attached below)

--- End quote ---

First of all only circle, passing through origin becomes a straight line, and this straight line is vertical only if the center of this circle, passing through origin, is on $x$-axis.

Inversion is self-inverse means that the double inversion is identical (so, operation, inverse to inversion, is inversion itself). Mirror-reflections have the same property as well.