4Hyperbolic geometry
IB Geometry
4 Hyperbolic geometry
At the beginning of the course, we studied Euclidean geometry, which was not
hard, because we already knew about it. Later on, we studied spherical geometry.
That also wasn’t too bad, because we can think of S
2
concretely as a subset of
R
3
.
We are next going to study hyperbolic geometry. Historically, hyperbolic
geometry was created when people tried to prove Euclid’s parallel postulate (that
given a line
`
and a point
P 6∈ `
, there exists a unique line
`
0
containing
P
that
does not intersect
`
). Instead of proving the parallel postulate, they managed to
create a new geometry where this is false, and this is hyperbolic geometry.
Unfortunately, hyperbolic geometry is much more complicated, since we
cannot directly visualize it as a subset of
R
3
. Instead, we need to develop the
machinery of a Riemannian metric in order to properly describe hyperbolic
geometry. In a nutshell, this allows us to take a subset of
R
2
and measure
distances in it in a funny way.