I need some help to understand Inversian Symmetry, Conformal Symmetry, and Scale Symmetry. Could you give me some guideline?
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You should make your question more specific. – superAnnoyingUser Apr 03 '14 at 07:54
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I mean that I don't understand what is the definition of those symmetries. I have try to look it up online but I found none. I want to know about what they are. – TBBT Apr 03 '14 at 08:04
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So are you asking what the analytical deffinitions are, or are you asking about their geometrical sense? – superAnnoyingUser Apr 03 '14 at 08:06
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Both would help me a lots. I need to understand it before I can do my homework which is about those kinds of symmetries. – TBBT Apr 03 '14 at 08:07
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The inversion symmetry is inversion with respect to a circle, which maps its inside to the outside and vice versa. You can read about it here. The inversion symmetry is a conformal symmetry. In the complex line, an inversion is a Mobius transformation.
The scaling symmetry is just scaling invariance or invariance under maps of the kind $x\to ax, a\not=0$. When you consider negative constants they also realize a reflection (i.e. change the orientation). Scalings are conformal symmetries.
The conformal symmetries preserve oriented angles between curves. They are all scalings, translations, rotations, special conformal symmetries or a combination of those. All isometries are conformal. Analytically, a transformation $x\to y$ is conformal if $x\to y=a(x)^2x$ where $a$ is a smooth function. In the complex line, they are scalings, inversions, rotations and translations. In other words the conformal group of the complex line coincides with the Mobius group.
superAnnoyingUser
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You are wellcome. If you liked my answer you can also upvote it. – superAnnoyingUser Apr 03 '14 at 09:51
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I would like to vote for you. But I need to have at least 15 reputation. Now I have 3. Sorry about that. – TBBT Apr 03 '14 at 10:19