Spherical Geometry Demo

John M Sullivan, University of Illinois
Image of parallel transport
appletIf you see this message, your browser doesn't support Java, or Java isn't enabled. Thus the image here is not a working applet, but merely a snapshot.

This applet demonstrates certain features of spherical geometry, in particular, the parallel transport of tangent vectors.

Below are some suggestions for using the applet. Note that there are three modes for what the mouse does: drag arrow, rotate sphere, and draw triangle. The menu lets you select which of these corresponds to the standard (left) mouse button. If you have a 3-button mouse, you can instead just use the three different buttons. For instance, when the menu selection is "drag arrow", the middle button will rotate the sphere and the right button will draw the triangle.

First draw a triangle by clicking three times in "draw triangle" mode. Note that you can drag a vertex after you click to create it. Also, between vertices, you can rotate the sphere (by clicking in "rotate sphere" mode) for a better view of where you want to place the next vertex.

Now drag the arrow around the triangle. Click it to one vertex, and then press the save arrow button to hold a copy there in the original position. Now you can either drag it 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 straight line (meaning of course a great circle on the sphere, like the edges of the triangle), it keeps a constant angle with that line of motion.

However, when it 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 (or total curvature, 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.

The code for the applet above is online. Thanks to Stuart Levy for adding the display of the triangle's angles.