Triangles in the Poincaré Disk
This Demonstration gives a representation of hyperbolic geometry showing a hyperbolic triangle. You can drag the vertices.
Poincaré used the points of the unit disk without its boundary as a model of hyperbolic geometry. A hyperbolic straight line is represented as an arc of a circle that is perpendicular to the disk edge. If a hyperbolic line passes through the center, it is straight. Angles are preserved so they can be measured directly from the figure. The angles of a hyperbolic triangle do not sum to 180°; the sum is zero for the limiting case of all vertices on the edge and close to 180° for vertices close to center or for small triangles, whose sides are almost straight.
Poincaré rediscovered this model in 1882; the conformal disk realization of hyperbolic geometry was found by Eugenio Beltrami and published in 1868 but does not carry his name.