The hyperbolic plane

Author:
a.zampa
This applet represents the Poincaré model of the hyperbolic plane, which corresponds to the white interior of the pictured circle.
You can explore many aspects of hyperbolic geometry, e.g.:
  • examine the sum of the interior angles of triangles observing, in particular, what happens when the sides of the triangle become very small;
  • given a point exterior to a line , construct the perpendicular to passing through , and then the perpendicular to passing through (this is the Euclidean construction of a parallel);
  • given a point exterior to a line , construct as before the perpendiculars and , select any point on , draw the perpendicular to passing through , and the circle with center and radius , where is the intersection of and . Name the point of intersection of and which lies between and , and draw the line passing through and . What can you say about ?
How many lines exist that are parallel to a given line and pass through an exterior point ? The latter is the János Bolyai construction of the asymptotic parallel line (there are two such lines for any and .