# Why are conformal transformations so prevalent in physics?

What is it about conformal transformations that make them so widely applicable in physics?

These preserve angles, in other words directions (locally), and I can understand that might be useful. Also, I gather this is equivalent to scale invariance, which seems like another handy feature.

Are those the main properties that make them useful, or are they incidental features and there are other (differential?) aspects that are more the determining factor in their use?

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Conformal mappings are very useful, for example, to solve the Laplace equation in an area with a complicated boundary. Typically, there is always a conformal mapping transforming such an area into an area with a simpler boundary, say, into a unit disk. Then you may use the inverse mapping to get a solution for the initial area from a solution for the area with a simpler boundary. On the other hand, the solutions of the Laplace equation are very important for, say, electrostatics and theory of incompressible liquid.

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