Also known as the a Lorentz transformation or Procrustian stretch, a hyperbolic transformation leaves each branch of the hyperbola 
invariant and transforms circles
into ellipses with the same area.
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(1)
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(2)
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Hyperbolic Rotation
See also
Crossed Hyperbolic RotationExplore with Wolfram|Alpha
References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 101, 1967.Referenced on Wolfram|Alpha
Hyperbolic RotationCite this as:
Weisstein, Eric W. "Hyperbolic Rotation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HyperbolicRotation.html