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2. Hyperbolic Geometry

Unlike the Euclidean plane and the sphere, the entire hyperbolic plane cannot be isometrically embedded in 3-dimensional Euclidean space. Therefore, any model of hyperbolic geometry in Euclidean 3-space must distort distance. The Poincaré circle model of hyperbolic geometry has two properties that are useful for artistic purposes:
      (1) it is conformal (i.e. the hyperbolic measure of an angle is equal to its Euclidean measure) - thus a transformed object has roughly the same shape as the original, and
      (2) it lies within a bounded region of the Euclidean plane - allowing an entire hyperbolic pattern to be displayed.
The "points" of this model are the interior points of a bounding circle in the Euclidean plane. The (hyperbolic) "lines" are interior circular arcs perpendicular to the bounding circle, including diameters. The sides of the hexagons of the {6,4} tessellation shown in Figure 2 lie along hyperbolic lines as do the backbone lines of the fish in Figures 3 and 4 .