Yearning for the Impossible by Stillwell John;

Author:Stillwell, John;
Language: eng
Format: epub
Publisher: CRC Press LLC
Published: 2015-06-22T04:00:00+00:00

Figure 5.24: Geodesic‐preserving map of the pseudosphere.

Clearly the wedge does not fill the open disk, but it has a natural extension to the open disk. The pseudosphere can be “unrolled” all the way along the dotted ellipses (Beltrami did this by imagining a surface wrapped infinitely many times around the pseudosphere, and mapping the wrapping surface onto the disk), and each line segment can be extended backward to the other side of the disk.

The “distance” between points in the wedge is taken to be the distance between the corresponding points on the pseudosphere, and this notion of distance also extends naturally to the whole open disk. Let us call it pseudodistance. Since the line segments in the disk are images of geodesic segments on the pseudosphere, each line segment on the disk gives the shortest pseudodistance between its endpoints. Also, the pseudodistance from any point in the open disk to the boundary circle is infinite, because the length of the pseudosphere is infinite.

The open disk can thus be interpreted as an infinite “plane,” whose “points” are the points of the disk, whose “lines” are line segments joining boundary points of the disk, and whose “distance” between “points” is pseudodistance. Each “line” is infinite and there is a unique “line” through any two “points.” But the “plane” is non‐Euclidean, as Figure 5.25 shows. For any “line” , and a “point” P outside it, there are many lines through P that do not meet

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.