There are two kinds of absolute crocheting adventures with hyperbolic planes pdf, Euclidean and hyperbolic. Single lines in hyperbolic geometry have exactly the same properties as single straight lines in Euclidean geometry.
For example, two points uniquely define a line, and lines can be infinitely extended. Two intersecting lines have the same properties as two intersecting lines in Euclidean geometry. When we add a third line then there are properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given 2 intersecting lines there are infinitely many lines that do not intersect either of the given lines. In hyperbolic geometry, there is no line that remains equidistant from another. Through every pair of points there are two horocycles. The arc-length of a hypercycle connecting two points is longer than that of the line segment and shorter than that of a horocycle, connecting the same two points.
The arclength of both horocycles connecting two points are equal. The arc-length of a circle between two points is larger the arc-length of a horocycle connecting two points. 180 degrees and the apeirogon approaches a straight line. Also the midpoint of the side segments are all equidistant to the same axis. This results in some formulas becoming simpler. The area of a horocyclic sector is equal to the length of its horocyclic arc.
360 degrees, and hyperbolic rectangles differ greatly from Euclidean rectangles since there are no equidistant lines, so a proper Euclidean rectangle would need to be enclosed by two lines and two hypercycles. These all complicate coordinate systems. There are however different coordinate systems for hyperbolic plane geometry. Other coordinate systems use the Klein model or the Poincare disk model described below, and take the Euclidean coordinates as hyperbolic. Construct a Cartesian-like coordinate system as follows. Their works on hyperbolic geometry had a considerable influence on its development among later European geometers, including Witelo, Gersonides, Alfonso, John Wallis and Saccheri.
Lobachevsky realized they had discovered a new geometry. Gauss did not publish his work. Euclidean geometry” and “hyperbolic geometry” to be synonyms. Lobachevsky published in 1830, while Bolyai discovered it independently and published in 1832. Hyperbolic geometry was finally proved consistent and is therefore another valid geometry.