half_plane - Math402 Upper Half—Plane Model of the...

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Unformatted text preview: Math402 Upper Half—Plane Model of the Hyperbolic Plane Consider Euclid/ Hilbert/ Birkhofl/ School axiomatic system of plane geometry without the Parallels Axiom (ale. without Euclid V or Playfair or Wallis Axiom). This axiomatic system has a non—standard model (hyperbolic plane) where the Parallels Axiom does not hold true. Hence the Parallels Axiom is independent from the rest of axioms of plane geometry. Historical Note: It had been believed since Euclid that the Parallels Axiom can be proven (tie. it is not really an axiom but rather a theorem). The reason was that the negation of this Axiom leads to very strange things, which seem to be readily disprovable (but in fact are not). Finally a model (hyperbolic plane) was produced (around 1830 independently by Bolyai. Lobachevski, and Gauss) which shows that Parallels Axiom is independent. There are several isomorphic models (or maps v see below) of the hyperbolic plane. Here is a descrip— tion of the upper half—plane model due to Poincare (around 1900). Recall that to describe a model is to give interpretation to all the undefined terms in the axiomatic system. 0 point : point in the upper half-plane of the usual plane (this half—plane is considered open, points on the boundary are not included): / "If; f,» Irv/1.x W; J42" ( l O M Se 1!; oi 1/ //n' [/1 1/,- ’ i’ if x // / / 0 there are two kinds of Zines: (1) semicircles centered on the boundary line (:perpendicular to the boundary line), and (2) vertical half—lines: e— limes not lines % flk ~-—v _ . _ _ -_ 0 point on line has the obvious meaning. 6 betweeness relation is the obvious order relation on semicircles (or halfilines). Note that, unlike the case of complete circles we deal with in spherical geometry, points on semicircles are naturally ordered. 0 congruence of angles or angle measure is the usual angle measure on the plane (ie the angle between two semicircles intersecting at a point P is the angle between their tangent lines at P). e congruence of segments or distance is harder to define and l postpone the definition to the later part of the course. In particular distances in hyperbolic geometry are not the usual distances on the half—plane unlike angles). For example, congruent segments (ale. segments of the same length) look shorter near the boundary line, in other words points are “crowded” near the boundary. I promise nevertheless that the distance/ congruence to be introduced will satisfy all the axioms. all {LEQQ {Viukfileg some ' angle?» these Sflfiwékix W9 ave magi/newt all (0‘45 Va em: (67%! leufilh) /l\ Kiwi. It is easy to see that these interpretations of geometric terms satisfy all the axioms except for Playfair. For example7 o (Pasch) a Semicircle partitions points into two subsets (plane separation by lines): /W I W x \x/ u t i? it? _ t s g 6”“. gthgg permits in u nhVfiafi x4: ’ / 9 /’%\i g; ' ‘3‘} f/ s , “\ " 3’3 a‘g r: as r a w (I, (a ( grast. riff g aw 9” i w. w. .w n. ,m g ,hi .i i ,1 nigger" Although formally correct (satisfy the axioms) the half—plane hyperbolic geometry looks strange. It is not clear why we call circles lines (they are not shortest distances like great circles on the sphere, nor are they symmetric with respect to themselves). Also there are different kinds of lines (semicircles of different radii and straight half—lines) — we expect all lines to be similar (this is true on the plane and on the sphere). The same applies to congruent triangles ~ they don’t look similar at all. The reason for these problems is that the half—plane is not the real thing - it is a map of the actual hyperbolic plane (a certain surface), like geographic maps of the Earth (sphere). As does any map, the half—plane distorts straight lines and distances (this particular map does not distort angles); on the actual hyperbolic plane lines are all similar, congruent triangles look the same, and segments are shortest distances between points. Here is an example of map distortions in spherical geometry: , in mi a {s‘ieaigwi has i té’éiifi'itiai} s sale; 3;; “merit ‘ : if? f} 3 gale: famagéés: One can ask why don’t we work with the hyperbolic plane directly, instead of using maps. The reason is that, unlike the sphere, the hyperbolic plane is very hard to imagine. Here is the picture of a hyperbolic plane (or rather its part): Note how points get crowded on the hyperbolic plane as compared to the usual flat plane (this is opposite to what happens on the sphere). Also on the sphere there are no parallel lines, on the plane there is one line parallel to a given line through a given point, and on the hyperbolic plane there are many. 80 in many senses a hyperbolic plane is the opposite of a Sphere ~ it is a sphere with a negative radius. And as there are spheres with various radii there are hyperbolic planes with various negative radii. Here is an example of a more curved hyperbolic plane: ...
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half_plane - Math402 Upper Half—Plane Model of the...

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