Up: Models of Hyperbolic Geometry

The Klein-Beltrami Model

The Klein-Beltrami model of a hyperbolic geometry is defined to be all points in the open disk D. The "lines" of this geometry are open chords - Euclidean line segments with end points on the boundary of the disk. (Note: As the model excludes the boundary of the disk, the end points of the segments are also excluded from the open chords - hence the name open chord.) In the illustration below l, m and n are open chords.

Notice that l and n are not parallel - they intersect. Lines m and n are divergently parallel - they have no points in common. The open chords l and m are asymptotically parallel. That is, the open chords intersect on the boundary. (Recall that the disk is open so the boundary is not included within the model.) Thus l and m do not intersect in the hyperbolic space - and by definition they are parallel.

Direct measurement of angles in the Klein-Beltrami model is very difficult. The Klein-Beltrami model is conformal only at the origin. That is, only at the origin can we measure angles with a protractor (a Euclidean method). At all other points of the model measurement with a protractor would yield an incorrect angle.

We can simplify the angle measurement by utilizing an isomorphism between the Poincaré and Klein-Beltrami models. There exists an isomorphism between the Klein-Beltrami model and the Poincaré Disk that takes angles in the Klein-Beltrami model to congruent angles in the Poincaré Disk. So, to measure angles in the Klein-Beltrami model we can map the two intersecting lines to the Poincaré disk using the isomorphism and measure the angles in the Poincaré disk by taking the Euclidean angle of the tangents at the point of intersection.

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Created: Jul 15 1996 --- Last modified: Jul 15 1996