![]() ![]() However, there is a revolution model called Beltrami's pseudosphere, which requires us only to regard the shortest distance (geodesic) as a straight line. We have called this hyperboloid a model, but, needless to say, it is not a model unless we have the agreement or definition (metric) of distance for the model. Then we can get the corresponding straight line (red arc) on Poincare's disk. : Also we put a disk at the origin O, and make a sweeping ray from the point -1 on the axis of revolution to the brown straight line on the hyperboloid as shown. We put a disk on the bottom of the hyperboloid, which allows us to get the corresponding straight line (green) on Klein's disk as the line of intersection with the plane. : A hyperbolic straight line (brown) on the hyperboloid is the line of intersection with a plane (pink) that goes through the origin O. Though the model is actually stretched up to infinity, it is cut for easy observation. The bottom of the hyperboloid is at distance 1 from the origin O. It represents the surface of revolution of a rectangular hyperbola. Within about 0.5 hyperbolic distance, the ratio of visual distances in the two disks is roughly 1 to 2.įig. Points P and K in this figure correspond to those of Fig. 4 shows distances from the origin on Poincare's disk and Klein's disk. But the diameter of disk is twice as big as the above one, so we have to observe it accordingly.įig. It directly changes a straight line to an arc. 3' shows a simple way to get Poincare's model without going up and down. That is, points N, O, P, K and t are coplanar.įig. Point t is on the southern hemisphere, and the straight line tK is vertical. Point N is the north pole and point S is the south pole, supposing that the disk is an equatorial plane. Point P is on Poincare's model and point K is on Klein's, and they correspond each other. The disk is used for both Poincare's and Klein's models. Point O is the common center of the sphere and the disk with blue edge. 3 shows the relation between Poincare's disk and Klein's disk. But on Klein's disk we have to compute or convert the green lines to red lines (Poincare's).įig. As you can see, both centers are not at the visual centers of the circle or the ellipse.Īngles and are hyperbolically the same. A circle is drawn as a circle (red) on Poincare's disk, but as an ellipse (green) on Klein's disk. Their centers are located at a hyperbolic distance of 1.5 from the origin. : Hyperbolic circles have a hyperbolic radius 1. The points on these lines correspond to each other as shown. :The red and green curves are the same hyperbolic straight line. But on Klein's disk, a straight line is straight (green). On Poincare's disk model, a straight line looks like a circular arc (red) that is perpendicular to the disk edge (circumference at infinity). The radius of a disk model depends on what is called curvature or metric, but we choose to make it 1 in Euclidean measurement in order to keep things simple.īoth types of disks, Poincare's and Klein's, are superimposed on each other. ![]() Here we look at the two disk models, Poincare's and Klein's, and the hyperboloid model in connection with the two disk models. But only the two disk models show the entire world. There are several kinds of models for the Hyperbolic Non-Euclidean World, such as Poincare's disk, Klein's disk, the hemisphere model, the upper half plane, the hyperboloid model, the dual graph, Beltrami's Pseudo-sphere, and so on. ![]()
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