Symmetry Definition and Equation

Question

I need some help to understand Inversian Symmetry, Conformal Symmetry, and Scale Symmetry. Could you give me some guideline?

Answer 1

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.