Hyperbolic Geometry


In elementary books (like this one) the development of hyperbolic geometry is often based on a stronger version of ∼ Playfair’s Postulate: Postulate H. If P is any point and A B is any straight line not passing through P (even if produced), then through P there are straight lines YPZ and WPX such that
  1. (1)

    YPX is not a single straight line

  2. (2)

    YPZ and WPX are each parallel to AB, and

  3. (3)

    no straight line through P entering ∠ YPX is parallel to AB. (See Figure 146.)



Acute Angle Euclidean Geometry Hyperbolic Geometry Small Triangle Equal Angle 
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© Birkhäuser Boston 2008

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