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 dwh2@cornell.edu.

Chapter 16

Hyperbolic Planes.

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.

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

*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.]

**P**roblem** 16.2. Cylindrical Projection**

Imagine a sphere of radius *r*, but this time center it in a vertical cylinder of radius *r* and height 2*r*. The ** cylindrical projection** is obtained by projecting from the axis of the cylinder which is also a diameter of the sphere; that is, if

* 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

We can easily flatten the cylinder onto a plane and find its area to be 4p*r*^{2}. We thus conclude:

* The area of a sphere of radius r is *4p*r*^{2}*.*

**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**,

**Figure 16.3. Stereographic projection is angle-preserving.**

*Suggestions*

**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 ****R**^{2} into **R**^{3}.

** Geometric Approach**: An angle at a point

** Analytic Approach**: Introduce a coordinate system and find a formula for the function

**b.** *Show that stereographic projection takes circles to circles*.* *(Such mappings are called ** circle-preserving**.)

**Figure 16.4. Stereographic projection is circle-preserving.**

*Suggestions*

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 Ð*APP*¢ is congruent to Ð*A*¢*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

Let ** z**:

**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*:

*is conformal. *We call this the (*Poincar*** é**)

[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 D*

[Hint: Review Problem **5.3**.]

**Problem 16.4. Projective Disk Model**

Let *D*^{2} 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 *D*^{2}, and note that ** s**(equator) is the boundary of

**a.** *Show that the mapping takes D*

*is a map from the hyperbolic plane to B*