LAS 100 --- Freshman Seminar --- Fall 1996

Reasoning with shape and quantity

The geometry of maps

Instructor: Professor John Wood

What would a straight line on a curved surface be? If you say, "as straight as it can be and still stay on the surface," then you can create these lines with masking tape. It bends easily to stay on the surface, but it puckers if you try to turn "to the side." Here are some lines on a water melon.

The lines formed several triangles. We measured the angles.

The three angles added to more than a straight angle, and to different sums for different triangles. To consider this further we went outside...

...and ate the melon.

To continue our investigation we chose a more regular surface, a beach ball.

We marked triangles with masking tape and extended their sides to be great circles.

There is another triangle of the same size on the opposite side of the sphere, the antipodal triangle.

Two great circles enclose a lune (or biangle). Our triangle defines three pairs of lunes which cover the sphere. Points outside the original triangle and its antipodal twin are covered once. Points inside the two triangles are covered three times.

We measured the interior angles of the triangles and computed their sums. Larger triangles have larger sums.

Professor Wood recording data at the board.

Students considering the consequences of these measurements.

Putting together facts about the area of a lune and the way the lunes lie on the sphere leads to Girard's formula for the area of a spherical triangle.

The area of a lune with angle A is (A/360) x Area of the sphere

The sum of the areas of the six lunes is 2(A + B + C)/360 x Area of the sphere

It also equals the area of the sphere + 4 x Area of the triangle

So 4 x Area of the triangle = (A + B + C - 180) x (Area of the sphere)/180

This means triangles on the sphere always have a larger angle sum than 180 degrees. And this leads to the conclusion that maps on flat paper of the any part of the spherical earth must always have some distortion of distance.

For more pictures of maps and spheres and more details on Girard's theorem see the

Geometry of the Sphere

page by John Polking at Rice University.