2Spherical geometry
IB Geometry
2 Spherical geometry
The next thing we are going to study is the geometry on the surface of a sphere.
This is a rather sensible thing to study, since it so happens that we all live on
something (approximately) spherical. It turns out the geometry of the sphere is
very different from that of R
2
.
In this section, we will always think of
S
2
as a subset of
R
3
so that we can
reuse what we know about R
3
.
Notation. We write
S
=
S
2
⊆ R
3
for the unit sphere. We write
O
= 0 for the
origin, which is the center of the sphere (and not on the sphere).
When we live on the sphere, we can no longer use regular lines in
R
3
, since
these do not lie fully on the sphere. Instead, we have a concept of a spherical
line, also known as a great circle.
Definition (Great circle). A great circle (in
S
2
) is
S
2
∩
(
a plane through O
).
We also call these (spherical) lines.
We will also call these geodesics, which is a much more general term defined
on any surface, and happens to be these great circles in S
2
.
In
R
3
, we know that any three points that are not colinear determine a
unique plane through them. Hence given any two non-antipodal points
P, Q ∈ S
,
there exists a unique spherical line through P and Q.
Definition (Distance on a sphere). Given
P, Q ∈ S
, the distance
d
(
P, Q
) is the
shorter of the two (spherical) line segments (i.e. arcs)
P Q
along the respective
great circle. When
P
and
Q
are antipodal, there are infinitely many line segments
between them of the same length, and the distance is π.
Note that by the definition of the radian,
d
(
P, Q
) is the angle between
−−→
OP
and
−−→
OQ, which is also cos
−1
(P ·Q) (where P = OP , Q = OQ).