Consequences of Girard's Theorem

Exercise: Distortion in maps. Everyone knows that a map from even a small portion of the sphere to the plane must involve some distortion. However Girard's Theorem makes that statement more precise. Let's call a map ideal if does two things.
Does an ideal map exist?

Are there maps that have one of the propeties?

It does not seem to be too much too ask for a map to be ideal. Think about these two questions. You can find the answers here.

Exercise: Similarity.

Suppose we have a triangle on the sphere with angles A, B, and C. Can we find a larger triangle with the same angles? In other words, do similar triangles exist on the sphere?

The answer is no! By the area formula, any triangle with these angles must have the same area, and therefore cannot be larger.

Exercise: Congruence theorems. In the plane there are a number of theorems aobut the congruence of triangles. They are usualy referred to by their acronyms, i.e., SSS, SAS, AAS, and ASA.

Can you modify the proofs of the planar theorems to prove these theorems on the sphere? It might be necessary to modify the statements of the theorems to take care of some special cases. (You should be warned that some of these are difficult to prove.)

For spherical triangles there is a new congruence theorem which states that any two triangles with the same angles must be congruent. So on the sphere we also have AAA. Can you prove this?

Exercise: Small triangles on large spheres. Let's look again at the formula for the sum of the angles of a spherical triangle.
When the radius R is very large, and the area of the triangle is small, the last term on the right hand side is extremely small. In such cases it would be difficult to distinguish the sum of the angles from 180 degrees. Thus the planar sum of the angles formula is almost true, perhaps to the limits of our capability to measure angles. For example, the radius of the earth is close to 4000 miles.
If we are looking at a triangle on the earth with area of 1 sq. mile, how much will the sum of the angles differ from 180 degrees? Do you think it likely that such a difference will be detectable?
Exercise: Spherical ploygons. Girard's theorem can easily be extended from triangles to spherical polygons. Of course a spherical polygon is a figure on the sphere which is bounded by segments of great circles.
Suppose that P is a spherical quadrilateral with angles a, b, c, and d. Show that


Suppose that P is a spherical polygon with n sides. Show that the sum of the angles is equal to


Does the result for polygons apply when the polygon is a lune?

There are some states, such as Utah and Colorado, which appear to be spherical polygons. Are they really? Look at these states on a map, and measure the angle sum. Compare it with the above formula, and decide whether they are actually spherical polygons. Can you explain the discrepancy?

In the next section we use Girard's theorem to give a proof of Euler's famous theorem relating the numbers of vertices, sides, and edges of polyhedron.
The previous section contains a proof of Girard's Theorem.
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John C. Polking <>
Last modified: Thu Apr 15 09:21:42 Central Daylight Time 1999