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 well-defined 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 90o.

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 so-called holonomy, illustrates that the surface of the sphere is curved and indeed measures exactly the curvature of the region our path encloses.

If Java worked here, you'd see something like this.
Image of Cassini Applet

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 180o. 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 90o angles. So each triangular region has curvature 90o; the whole sphere, made up of eight such triangles, has total curvature 720o.

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 180o, so their area and curvature are almost 0.

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