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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 halfplane 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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 Spring '09
 malkin

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