EggMath:
Spherical Geometry
We want to demonstrate that geometry on the
surface of an egg would be different from the
Euclidean geometry you are used to in a (flat)
plane. An egg is not quite shaped like a sphere,
but its geometry has many features in common with
that of a sphere. A symmetric, round sphere is
easier to work with to demonstrate these
features.
The following applet demonstrates certain
features of spherical geometry, in particular, the
fact that there is no welldefined way to keep
track of directions. See the instructions below
it.
But before you try the applet, stand up and do
the following experiment:
 Curl the fingers
of your right hand into a fist, but leave your
thumb sticking straight out.
 Hang your right
arm down with your thumb pointing forward.
 Keep your arm
straight throughout, and never twist your wrist
unnecessarily.
 Swing your arm up
and out to the side. (Your thumb will still point
forward.)
 Next swing your
arm forwards, until it points straight
ahead. (Since your remembered not to twist your
wrist, your thumb now points left.)
 Finally swing
your arm down, until it rests at your side
again.
Notice that your thumb now points to the left.
Your arm is back where you started, and you never
twisted your wrist, but your thumb has ended up
twisted by 90^{o}.
In that experiment, your hand is a point moving
around a sphere. (Since your arm remains
straight, your hand is always a fixed distance
from your shoulder.) Your thumb always points
perpendicular to your arm. Since your thumb
points in a possible direction for your hand to
move, along the sphere, it is called a tangent
vector to the sphere.
This experiment has demonstrated that when we
transport a tangent vector around a closed path on
the sphere, it will come back twisted. This
happens even though locally we never twist it 
in particular, if we move along a straight path (a
geodesic or great circle on the sphere) we keep
the tangent vector at a a constant angle to our
direction of motion). The twist we end up with
anyway, the socalled holonomy, illustrates
that the surface of the sphere is curved and
indeed measures exactly the curvature of the
region our path encloses.
You might use this as follows:
First draw a triangle (either select this mode,
or use the right mouse button) by clicking three
times. Note two things: You can drag a vertex
after you click to create it. Also, between
vertices, you can rotate the sphere (selecting
that mode or using the middle mouse button) for a
better view of where you want to place the next
vertex.
Now drag the arrow around the triangle (by
selecting again the original drag mode). Click it
to one vertex, and then press the save arrow
button to hold a shadow there in the original
position. Now either drag the arrow around the
triangle, or if you want to make sure it gets
dragged along the straight edges just click to
take it directly from one vertex to the next.
It never twists as it's being dragged: it
always keeps its orientation in space as closely
as possible subject to always remaining tangent to
the sphere. You can see that if it is dragged
along a great circle it keeps a constant angle
with that line of motion. (A great circle, like
the equator or the edges of our spherical
triangle, is the analog of a straight line on the
sphere.)
However, when the arrow is transported all the
way around a triangle, back to its starting point,
it is in a different orientation. The amount by
which it twists is exactly equal to the amount by
which the angle sum of the triangle exceeds
180^{o}. This
quantity is called the
(total) curvature of the triangular region.
If you look at the purple triangles drawn on
the sphere, you see each of them has three
90^{o} angles.
So each triangular region
has curvature 90^{o}; the whole sphere,
made up of eight such triangles, has total
curvature 720^{o}.
Because of the symmetry of a round sphere,
every piece looks like every other piece. The
amount of curvature in any region is proportional
to its area. If our sphere has total area 720 (in
some units) then the area of any triangle (in
those units) equals its angle excess (in degrees).
Small triangles on the sphere look almost like
triangles in a flat plane. Their angle sum is
hardly more than 180^{o}, so their area
and curvature are almost 0.
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