Last updated: June 27, 1999. Updates will be posted as they become available.

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

Chapter 17

Duality and Trigonometry

After we have found the equations [The Laws of Cosines and Sines for a Hyperbolic plane] which represent the dependence of the angles and sides of a triangle; when, finally, we have given general expressions for elements of lines, areas and volumes of solids, all else in the [Hyperbolic] Geometry is a matter of analytics, where calculations must necessarily agree with each other, and we cannot discover anything new that is not included in these first equations from which must be taken all relations of geometric magnitudes, one to another. ... We note however, that these equations become equations of spherical Trigonometry as soon as, instead of the sides a, b, c we put ...

N. Lobachevsky, quoted in [NE: Greenberg]

Problem 17.1. Circumference of a Circle

a. Find a simple formula for the circumference of a circle on a sphere in terms of its intrinsic radius and make the formula as intrinsic as possible.

[Hint: We suggest that you make an extrinsic drawing (similar to Figure 17.1) of the circle, its intrinsic radius, its extrinsic radius, and the center of the sphere. You may well find it convenient to use trigonometric functions to express your answer. Note that the existence of trigonometric functions for right triangles follows from the properties of similar triangles that were proved in Problem 14.3.]

Figure 17.1. Intrinsic radius r.

In Figure 17.1, rotating the segment of length r´ (the extrinsic radius) through a whole revolution produces the same circumference as rotating r, which is an arc of the great circle as well as the intrinsic radius of the circle on the sphere.

Even though the derivation of the formula this way will be extrinsic, it is possible, in the end, to express the circumference only in terms of intrinsic quantities. Thus, also think of the problem:

b. How could our 2-dimensional bug derive this formula?

[Hint: By looking at very small circles, the bug could certainly find uses for the trigonometric functions that they give rise to. Then the bug could discover that the geodesics are actually (intrinsic) circles, but circles which do not have the same trigonometric properties as very small circles. And then what? Use your experience from Chapter 12.]

Using the expressions of trigonometric and hyperbolic functions in terms of infinite series, it is proved (in [NE: Greenberg], page 337) that

Theorem 17.1. In a hyperbolic plane of radius 1, a circle with intrinsic radius r has circumference c equal

c = 2p sinh(r).

c. Use the theorem to show that on a hyperbolic plane of radius r, a circle with intrinsic radius r has circumference c equal

c = 2p R sinh(r/R).

[Hint: When going from a hyperbolic plane of radius 1 to a hyperbolic plane of radius r, all lengths scale by a factor of r. Why?]

The formula in Part c should look very much like your formula for Part a (possibly with some algebraic manipulations). This is precisely what Lobachevsky was talking about in the quote at the beginning of this chapter. Check in out in:

d. Show that, if you replace R by iR in the formula of Part c, then you will get the formula in Part a.

[Hint: Look up the definition of sinh (hyperbolic sine) and express it as a Taylor series.]

Problem 17.2. Law of Cosines

If we know two sides and the included angle of a (small) triangle, then according to SAS the third side is determined. If we know the lengths of the two sides and the measure of the included angle, how can we find the length of the third side? The various formulas that gives this length are called the Law of Cosines.

a. Find a law of cosines for triangles in the plane.

Figure 17.2. Law of Cosines.

You have learned in school (but perhaps forgotten) the Law of Cosines on the plane: cab- 2ab cos(q). For a geometric proof of this "law," look at the pictures in Figure 17.3. These pictures show the squares as rigid with hinges at all the points marked . Note that in the middle picture q is greater than p/2. You must draw a different picture for q less than p/2. Prove the Law of Cosines on the plane using the pictures in Figure 17.3, or in any other way you wish.

)2 = a2 + b2 + 2ab c2 = a2 + b2 + 2ab(-cos (q) )

c2 = a2 + b2

Figure 17.3. Three related geometric proofs.1

b. Find a law of cosines for small triangles on a sphere with radius R.

On the sphere there are various versions of the Law of Cosines that you can find. One approach that will work is to project the triangle by a gnomic projection onto the plane tangent to the sphere at the vertex of the given angle. This projection will preserve the size of the given angle (Why?) and, even though it will not preserve the lengths of the sides of the triangle, you can determine what effect it does have on these lengths. Now apply the planar Law of Cosines to this projected triangle. It is very helpful to draw a 3-D picture of this projection.

Figure 17.4 . Radian measure of lengths.

It is often convenient to measure lengths of great circle arcs on the sphere in terms of the radian measure of the angle which the arc subtends at the pole of the great circle. In particular:

radian measure of the arc = (length of the arc)/R,

where R is the radius of the sphere. For example, the radian measure of 1/4 great circle would be p/2 and the radian measure of half a great circle would be p. In Figure 17.4, the segment a is subtended by the angle at the pole and by the same angle a at the center of the sphere. (Do you see why these angles are congruent? If not, imagine looking straight down on the sphere from above the pole.) The radian measure of a is the radian measure of a.

Pause, explore, and write about this problem before you read further.

If we measure lengths in radians, then one possible formula for the spherical triangle of radius R in Part b is:

cos c = cos a cos b + sin a sin b cos q.

(This is not the only such formula.)

For a right triangle (p/2) the above formula becomes

cos c/R  =  cos a/R cos b/R,

which can be considered as the spherical equivalent of the Pythagorean Theorem.

Theorem 17.2. A Law of Cosines for triangles on a hyperbolic plane with radius R is

cosh c/R  =  cosh a/R cosh b/R  +  sinh a/R sinh b/R cos q.

This theorem is proved in [NE: Stahl], page 125, using analytic techniques and is proved in [NE: Greenberg] using infinite series representations. This is one of the "equations" that Lobachevsky is talking about in the quote at the beginning of the chapter. It is important to realize that a comprehensive study of hyperbolic trigonometric functions was published by J.H. Lambert in 1768 well before their use in hyperbolic geometry by Lobachevsky.

Problem 17.3. Law of Sines

Closely related to the Law of Cosines is the Law of Sines.

a. If DABC is a triangle on the plane with sides, a, b, c, and corresponding opposite angles, a, b, g, then

a/sin a = b/sin b = c/sin g.

Figure 17.5. Law of Sines.

The standard proof for the Law of Sines is to drop a perpendicular from the vertex to the side c and then to express the length of this perpendicular as both (sin a) and (sin b). See Figure 17.6. From this the result easily follows. Thus, on the plane the Law of Sines follows from an expression for the sine of an angle in a right triangle.

Figure 17.6. Standard proof of Law of Sines on plane.

b. What is an analogous property on the sphere?

For triangles on the sphere we can find a very similar result. If DACD is a triangle on the sphere with the angle at being a right angle, then use gnomic projection to project DACD onto the plane which is tangent to the sphere at A. Since the plane is tangent to the sphere at A, the size of the angle is preserved under the projection. In general, angles on the sphere not at will not be projected to angles of the same size, but in this case the right angle at will be projected to a right angle. (Be sure you see why this is the case. A good drawing and the use of symmetry and similar triangles will help.) Now express the sine of in terms of the sides of this projected triangle.

Pause, explore, and write about this problem before you go to the next page.

When we measure the sides in radians, on the sphere the Law of Sines becomes:

(sin a)/sin = (sin b)/sin b = (sin c)/sin g.

For a right triangle this becomes:

sin = (sin a)/(sin c).

Figure 17.7. Law of Sines for right triangles on a sphere.

Duality on a Sphere

We can now ask: Is it possible to find expressions corresponding to the other triangle congruence theorems which we proved in Chapters 6 and 9. Let us see how we can be helped by a certain concept of duality which we will not develop.

When we were looking at SAS and ASA, we noticed a certain duality between points and lines (geodesics). SAS was true on the plane or open hemisphere because two points determine a unique line segment and ASA was true on the plane or open hemisphere because two (intersecting) lines determine a unique point. In this section we will make this notion of duality broader and deeper and look at it in such a way that it applies to both the plane and the sphere.

On the whole sphere two distinct points determine a unique straight line (great circle) unless the points are antipodal. In addition, two distinct great circles determine a unique pair of antipodal points. Also, a circle on the sphere has two centers which are antipodal. Remember also that in most of the triangle congruence theorems, we had trouble with triangles which contained antipodal points. So our first step is to consider not points on the sphere but rather point-pairs, pairs of antipodal points. With this definition in mind, check the following:

Now, we can make the duality more definite: Some books use the term "polar" in place of "dual."