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Unit 8

Geometries Beyond Euclid

8.5 Shapes on a Plane: Making Maps

DISTORTION

If you were to take a globe and compare it to a flat map of Earth, you would find some great discrepancies. Greenland, for instance, appears much larger on a map than on a globe. The reason for this is that whoever designed the map was faced with the tricky task of using a flat surface to portray a piece of the surface of a sphere. Anyone who has ever tried to wrap an unboxed basketball or soccer ball as a gift will have encountered a similar problem—it's difficult to take a flat surface and attempt to form it into a sphere.

world map Item 2877 / Giraldon, MAPPE-MOND ES SUR DIVERSES PROJECTIONS (WORLD MAP)(Ca. 1815). Courtesy of Kathleen Cohen.

For our purposes, a map is a flat representation of a curved space.


Orange

Likewise, if you've ever peeled an orange, especially if you've accomplished this with just one strand of peel, you've found that you can't lay it on a flat surface without tearing it. This is actually just the opposite problem of gift-wrapping the ball—here we're trying to take a spherical surface and turn it into something flat.

Orange Peel

This is also exactly the problem that the mapmaker faces. Fortunately, there are a variety of techniques for translating an image of a curved surface onto a flat piece of paper. What's even better is that, once we have a general technique, we can use it to make maps not only of spheres, but also of pseudospheres and other curved surfaces that defy our intuition. The technique that we will focus on in this discussion is called a "stereographic projection."

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PROJECTING THE SPHERE

Let’s first see how stereographic projection works for a sphere.

Projecting Sphere

Take a sphere and place it on a plane. Let's call the exact point where the sphere touches the plane the south pole. The plane is tangent to this point. The north pole would then be the point antipodal to the south pole, or in other words, the point directly across the interior of the sphere.

Projecting Sphere

To perform a stereographic projection, we draw straight lines from the north pole to the plane, rather like staking down a tarp or a tent.

Projecting Sphere

Now, each of these lines will intersect the sphere and continue on to the plane. This is how we will map each unique point on the sphere to a unique point on the plane.

Notice that every point on the sphere will be uniquely mapped onto the plane except for the north pole. Where should it go? If we notice that points arbitrarily close to the north pole get mapped further and further out on the plane, it makes sense to define the north pole to be mapped to infinity.

What happens to geodesics in this mapping? To answer this question, we can start by looking at the equator, which is a special case of geodesic. Notice that the equator gets mapped to a circle on the plane.

Projecting Sphere

We call this the "equatorial circle." All other geodesics on the sphere get mapped to circles and lines in the plane that intersect the equatorial circle at two opposite (called "antipodal") points.

What determines whether or not a geodesic on the sphere gets mapped to a circle or a line on the plane? Recall that the north pole gets mapped to infinity, so any geodesic on the sphere that passes through the north pole will, when mapped to the plane, extend to infinity, forming what we normally think of as a line.

Projecting Sphere

A great thing about the stereographic projection is that it is conformal, which means that it preserves the angles between geodesics. In other words, the shape of an object, such as a triangle, is preserved because its angles are, but its size is not preserved. This is because in order to preserve angle, we must distort lengths.

Projecting Sphere

Notice that triangle A, in the southern hemisphere, looks more or less the same size in its projection, whereas triangles B and C, each located progressively further north on the sphere, get larger and larger on the plane. The triangles are still three-sided figures, but their sizes have changed. Also, notice that they appear "fat" on this map. This is a consequence of the fact that the three angles of a triangle can add up to more than 180 degrees in spherical geometry. The only way to represent this on the map is to replace the straight geodesics of the plane with the curved geodesics of the sphere.

Triangles

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THE POINCARÉ DISK

disk Multiple parallel lines through a given point. Note that the edges of the disk represent infinity.

Now that we have seen a map of spherical space, let's look at a map of hyperbolic space. We saw earlier that the pseudosphere is a good model of hyperbolic space. In a process similar to the one we used with the sphere, we can make a map of the pseudosphere.

This map is called a Poincaré disk, in honor of Henri Poincaré, the great French mathematician, who was its initial creator. The boundary of the disk is mapped to infinity. Most geodesics are represented on this map as semicircles that make right angles with the boundary, signifying that lines in hyperbolic space both "begin" and "end" at infinity. Geodesics that pass through the center of the disk, however, are represented as straight lines, connecting antipodal points.

Antipodal Points

Like the spherical map we just saw, this map is conformal—it preserves the angles between geodesics. We can see quite easily how Saccheri's quadrilateral, if mapped on a Poincaré disk, would have acute summit angles.

Triangles in hyperbolic space appear, on this map, to be "skinny" or "cuspy," showing that their angles add up to less than 180 degrees.

hyp Item 1725 / Jos Leys, HYP023 (2002). Courtesy of Jos Leys.

A flap map of hyperbolic space tiled with quadrilaterals.


We can see that the "many-parallels" version of Euclid's fifth postulate is obeyed. If we draw a geodesic, we will get a rainbow shape. If we then choose a point not on that line, we will be able to draw as many parallel lines as we choose. In other words, this proves that we do indeed have a map of hyperbolic space.

Hyperbolic Space

We see from these examples that it is possible to make maps of the different geometries discussed so far. It should be evident from exploring the nature of these geometries and their maps that, to be comfortable "getting around" in these new multi-dimensional realms, we are going to have to understand curves as well as we understand straight lines. To further that understanding, we now turn to the subject of how to measure curvature.

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Next: 8.6 Curvature


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