NON-EUCLIDEAN 
                   GEOMETRY RENDERING
                   Diana Algma
                           M.C. Escher –
                           “Relativity”
                                                  I WILL TALK ABOUT…
      • MATH
          • What is Euclidean Geometry?
          • What is Non-Euclidean Geometry?
             • Elliptic
             • Hyperbolic
      • COMPUTER GRAPHICS
          • Hyperbolic planes
          • Ray tracing – space warping
          • Portals
                                                   M.C. Escher –
                                                   “Balcony”
                                                       EUCLIDEAN GEOMETRY
        • The “normal” geometry
        • Euclid’s 5 postulates (“The Elements” ~300 BC) (by Sir Thomas Heath):
            1.  "To draw a straight line from any point to any point."
            2.  "To produce [extend] a finite straight line continuously in a straight line."
            3.  "To describe a circle with any center and distance [radius]."
            4.  "That all right angles are equal to one another."
            5.  Theparallel postulate: "That, if a straight line falling on two straight lines make 
                the interior angles on the same side less than two right angles, the two straight 
                lines, if produced indefinitely, meet on that side on which are the angles less 
                than the two right angles."
                                                       EUCLIDEAN GEOMETRY
        • The “normal” geometry
        • Euclid’s 5 postulates (“The Elements” ~300 BC) (by Sir Thomas Heath):
            1.  "To draw a straight line from any point to any point."
            2.  "To produce [extend] a finite straight line continuously in a straight line."
            3.  "To describe a circle with any center and distance [radius]."
            4.  "That all right angles are equal to one another."
            5.  Theparallel postulate (Playfair): "In a plane, through a point not on a given 
                straight line, at most one line can be drawn that never meets the given line."