Other curves: hypercycles

Another curve of importance is the hypercycle.
This is one of a pair of curves  each of which is  equidistant from a given d-line. The segment MP has constant length as M is moved on the d-line.

So we see that in the Poincaré model a Euclidean circle represents: 
 

  • a hyperbolic circle if it is entirely inside the disc; 
  • a horocycle if it is inside  except for one point where it is tangent to the disc; 
  • an hypercycle if it cuts the boundary non-orthogonally in two points; 
  • a hyperbolic line if it cuts the boundary orthogonally at two points. 

Other curves; horocyclesReflection isometry

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