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The Poincaré disk is an example of a non-Euclidean geometry. It is a *hyperbolic geometry*. This means that given a line, and a point not on that line, there are infinitely many parallels through the point. What? Isn't there only supposed to be one parallel line through a point? To the non-mathematician, non-Euclidean geometries sound weird and fanciful. Their representations involve lines that don't look straight. What kind of line isn't straight? Well, it turns out, lines are never really straight in the geometry of our universe. If we consider a straight line to be the path taken by a ray of light, then what might look to be a perfectly straight line is actually bent by gravity. It is Euclidean geometry that is weird!

Unlike the typical Euclidean infinite plane, the Poincaré disk is represented by the interior of the unit circle. A *line* on this disk must either be a diagonal of the circle, or a circular arc that approaches the edge of the disk at right angles. You can explore this idea in desmos. The purple circle is the unit circle (centered at the origin, with radius of one). The red circle with equation will intersect the unit circle at right angles, so the portion of the red circle that is inside the purple circle is considered to be a straight line in the Poincaré disk. You can play with the values of and to create any straight line, except for the diagonals.

A little algebra shows that the radius of the red circle is The center of the circle is . We can find and with help from wikipedia. Another good source on the the mathematics of the disk is MathWorld.

While it may seem absurd to pretend the arc of a circle is a line, keep in mind that the disk is a representation embedded in something familiar. There are other distortions afoot that the disk does not make clear. For example, distance between points is not Euclidean distance. As you move towards the edge of the disk, your hyperbolic distance from the origin is increasing to infinity.

Below is a Processing sketch of the disk. Click inside the circle to designate one point, and a line is drawn through that point and the mouse pointer. Moving around the designated point shows you all lines that pass through that point. Latest source available on GitHub.