Notice: This material will be included in a forthcoming (summer 2000) book with the tentative title Experiencing Geometry in Euclidean, Spherical, and Hyperbolic Spaces. This new book will be an expanded and updated version of Experiencing Geometry on Plane and Sphere. This material is in draft form and may not be duplicated or quoted without the author's written permission, except for purposes of review or trying out the material with students. As always comments are welcome and will affect the final draft. Send comments to email@example.com.
Geography is a representation in picture of the whole known world together with the phenomena which are contained therein.
... The task of Geography is to survey the whole in its proportions, as one would the entire head. For as in an entire painting we must first put in the larger features, and afterward those detailed features which portraits and pictures may require, giving them proportion in relation to one another so that their correct measure apart can be seen by examining them, to note whether they form the whole or a part of the picture. ... Geography looks at the position rather than the quality, noting the relation of distances everywhere, ...
It is the great and the exquisite accomplishment of mathematics to show all these things to the human intelligence ...
Claudius Ptolemy, Geographia, Book One, Chapter I
A major problem for map makers (cartographers) since Ptolemy (approx. 85-165 a.d., Alexandria, Egypt) and before is how to represent accurately a portion of the surface of a sphere on the plane. It is the same problem we have been having when making drawings to accompany our discussions of the geometry of the sphere. We shall use the terminology used by cartographers and differential geometers to call any one-to-one function from a portion of a sphere onto a portion of a plane a chart. As Ptolemy states in the quote above, we would like to represent the sphere on the plane so that proportions (and thus angles) are preserved and the relative distances are accurate. It is impossible to make a chart without some distortions. Which results that you have studied so far show that there must be distortions when attempting to represent a portion of a sphere (or a hyperbolic plane) on the plane? For a history and mathematical descriptions of map projections of the sphere, see [Hi: Snyder].
Nevertheless, there are projections (charts) from a portion of a sphere to the plane which take geodesics to straight lines, that is, which preserve the shape of straight lines. There are other projections which preserve all areas. There are still other projections which preserve the measure of all angles. In this chapter, we will study these three types of projections for spheres and then after that look at the situation for hyperbolic planes. We already seen one chart for the hyperbolic plane, the upper half plane model. For hyperbolic planes it is usual to use the term "model" instead of "projection" or "chart".
Problem 16.1. Gnomic Projection
Figure 16.1. Gnomic projection.
Imagine a sphere resting on a horizontal plane. A gnomic projection is obtained by projecting from the center of a sphere onto the plane. Note that only the lower open hemisphere is projected onto the plane; that is, if X is a point in the lower open hemisphere, then its gnomic projection is the point, g(X), where the ray from the center through X intersects the plane.
Show that a gnomic projection takes the portions of great circles in the lower hemisphere onto straight lines in the plane. (Because of this, a gnomic projection is said to be a geodesic mapping.)
Gnomic projection is often used to make navigational charts for airplanes and ships. Why would this be appropriate?
[Hint: Start with our extrinsic definition of great circle.]
Problem 16.2. Cylindrical Projection
Imagine a sphere of radius r, but this time center it in a vertical cylinder of radius r and height 2r. The cylindrical projection is obtained by projecting from the axis of the cylinder which is also a diameter of the sphere; that is, if X is a point (not the North or South Poles) on the sphere and O(X) is the point on the axis at the same height as X, then X is projected onto the intersection of the cylinder with the ray from O(X) to X.
Show that cylindrical projection preserves areas.
Figure 16.2. Cylindrical projection.
Geometric Approach: Look at an infinitesimal piece of area on the sphere bounded by longitudes and latitudes. Check that when it is projected onto the cylinder that the horizontal dimension becomes longer but the vertical dimension becomes shorter. Do these compensate for each other?
Analytic Approach: Find a function f from a rectangle in the (z,q)-plane onto the sphere and a function h from the same rectangle onto the cylinder such that c(f(z,q)) = h(z,q). Then use the techniques of finding surface area from vector analysis. (For two vectors A, B, the magnitude of the cross product |A x B| is the area of the parallelogram spanned by A and B. An element of surface area on the sphere can be represented by | fz x fq | dz dq, the cross product of the partial derivatives.)
We can easily flatten the cylinder onto a plane and find its area to be 4pr2. We thus conclude:
The area of a sphere of radius r is 4pr2.
Problem 16.3. Stereographic Projection
Imagine the same sphere and plane, only this time project from the uppermost point (North Pole) of the sphere onto the plane. This is called stereographic projection.
a. Show that stereographic projection preserves the sizes of angles. (Such mappings are variously called angle-preserving, isogonal, or conformal.)
Figure 16.3. Stereographic projection is angle-preserving.
There are several approaches for exploring this problem. Using a purely geometric approach requires visualization but only very basic geometry. An analytic approach requires knowledge of the differential of a function from R2 into R3.
Geometric Approach: An angle at a point X on the sphere is determined by two great circles intersecting at X. Look at the two planes that are determined by the North Pole N and vectors tangent to the great circles at X. Notice that the intersection of these two planes with the horizontal image plane determines the image of the angle. Since the 3-dimensional figure is difficult for many of us to imagine in full detail, you may find it helpful to consider what is contained in various 2-dimensional planes. In particular, consider the plane determined by X and the North and South Poles, the plane tangent to the sphere at X, and the planes tangent to the sphere at the North and South Poles. Determine the relationships among these planes.
Analytic Approach: Introduce a coordinate system and find a formula for the function s-1 from the plane to the sphere which is the inverse of the stereographic projection s. Use the differential of s-1 to examine the effect of s-1 on angles. You will need to use the dot (inner) product and the fact that the differential of s-1 is a linear transformation from the (tangent) vectors at s(X) to the tangent vectors at X.
b. Show that stereographic projection takes circles to circles. (Such mappings are called circle-preserving.)
Figure 16.4. Stereographic projection is circle-preserving.
Let g be a circle on the sphere with point A and B and let g¢, A¢, B¢ be their images under stereographic projection. Form the cone which is tangent to the sphere along the circle g and let P be its cone point (note that P is not on the sphere). See Figure 16.4. Thus the segments BP and AP are tangent to the sphere and have the same length r. Look in the plane determined by N, A, and P and show that ÐAPP¢ is congruent to ÐA¢P¢P. You probably have already proved this is Part b; if not, look at the intersection of the plane determined by N, A, and P and the plane tangent to the North Pole N. Then use similar triangles to show that
and thus g¢ is a circle with center at P¢.
Problem 16.4. Poincaré Disk Model
You showed in Problem 5.2a that the coordinate map x from a hyperbolic plane to the upper half plane preserves angles (is conformal); this we called the upper half plane model. Now we will study other models of the hyperbolic plane.
Let z: H2 ® R2+ be the coordinate map defined in Problem 5.2 that defines the upper half plane model. We will now transform the upper half plane model to a disk model that was first discussed by Poincaré in 1882.
a. Show that any inversion through a circle whose center is not on the boundary of the upper half plane will transform the upper half plane onto an open (without its boundary) disk. Show that the hyperbolic geodesics in the upper half plane are transformed by this inversion into circular arcs (or line segments) perpendicular to the boundary of the disk.
[Hint: Review the material on inversions discussed in Problem 4.2.]
b. If w: R2+ ® D2 be a map from the upper half plane to a (open) disk from Part a, then show that the composition
is conformal. We call this the (Poincaré) disk model, after Henri Poincaré (1854-1912, French).
[Hint: Review the material on inversions in 4.2 and the upper half plane model in 5.2.]
c. Show that any inversion through a circular arc (or line segments) perpendicular to the boundary of D2 takes D2 to itself. Show that these inversions correspond to isometries in the (annular) hyperbolic plane. Thus, we call these circular arcs (or line segments) hyperbolic geodesics and call the inversions hyperbolic reflections in the Poincaré disk model.
[Hint: Review Problem 5.3.]
Problem 16.4. Projective Disk Model
Let D2 be the disk model of a hyperbolic plane and assume its radius is 2. Then place a sphere of radius 1 tangent to the disk at its center. Call this point of tangency the South Pole S. See Figure 16.5.
Figure 16.5. Obtaining the projective disk model of hyperbolic plane.
Now let g be the stereographic projection from the sphere to the plane containing D2, and note that s(equator) is the boundary of D2, and thus g takes the Southern Hemisphere onto the D2. Now let h be the projection of the Southern Hemisphere onto the disk, B2, of radius 1.
a. Show that the mapping takes D2 to B2 and takes each circle (or diameter) of D2 to a (straight) cord of B2. Thus
is a map from the hyperbolic plane to B2 which takes geodesics to straight line segments (cords) in B2. We call this the projective disk model, but it is also in the literature called the Beltrami/Klein model or the Klein model, named after Eugenio Beltrami (1835-1900, Italian) who described the model in 1868, and Felix Klein (1849-1925, German) who fully developed it in 1871.