Notice: This material will be included in a forthcoming (summer 2000) book with the tentative title Experiencing Geometry in Two- and Three-Dimensional 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.
Let the following be postulated: to draw a straight line from any point to any point.
Euclid, Elements, Postulate 1 [A: Euclid]
At this point, you should be thinking intrinsically about the surfaces of spheres, cylinders, cones, and hyperbolic planes. In the problems to come you will have opportunities to apply your intrinsic thinking when you make your own definitions for triangle on the different surfaces and investigate congruence properties of triangles. First, we summarize what we know about geodesics and transformations, and then explore them further.
Geodesics are Locally Unique
In previous chapters we have studied geodesics, intrinsically straight paths. Our main criterion has been that a path is intrinsically straight (and thus a geodesic) if it has local intrinsic reflection-thru-itself symmetry. Using this notion we found, on a sphere, geodesics that are great circles and, on a hyperbolic space, geodesics that are represented by circles with centers on the x-axis in the upper half plane model. However, on more general surface which may have no (even local) reflections it is necessary to have a deeper definition of geodesic in terms of intrinsic curvature. (See for example, Chapter 3 of [DG: Henderson (1998)].) Then, to be precise, we must prove that the geodesics we found on spheres and hyperbolic planes are the only geodesics on these surfaces. It is easy to see that these geodesics that we have found are enough to have one geodesic proceeding from every point in every direction. To prove that these are the only geodesics it is necessary (as we have mentioned before) to involve some notions from Differential Geometry. In particular, one must first define a notion of geodesic that will work on general surfaces which have no (even local) intrinsic reflections. Then one shows that a geodesics satisfies a second order (nonlinear) differential equation (see Problem 8.4b of [DG: Henderson (1998)]). Thus it follows from the analysis theorem on the Existence and Uniqueness of Differential Equations, with the initial conditions being a point on the geodesic and the direction of the geodesic at that point, that:
For any given point and any direction at that point on a smooth surface there is a unique geodesic starting at that point and going in the given direction.
From this it follows that the geodesics which we have found are all the geodesics on a sphere and on a hyperbolic plane.
Problem 6.1. Properties of Geodesics
In the previous chapters you have studied much about geodesics on the plane, spheres, cylinders, cones, and hyperbolic spaces. In this problem we ask you to pull together a summary of the properties of geodesics on the plane, spheres, and hyperbolic planes. Mostly, you have already argued that these are true but we summarize the results here to remind us what we have seen and so that you can reflect again about why these are true. Remember that the cone and cylinder are locally the same geometrically as the plane; thus, for now, we will leave the cylinder and cone and return them later in Chapter 11.
a. For every geodesic on the plane, sphere, and hyperbolic plane there is a reflection of the whole space through the geodesic.
b. Every geodesic on the plane, sphere, and hyperbolic plane can be extended indefinitely (in the sense that the bug call walk straight ahead indefinitely along any geodesic).
c. Every pair of distinct points on the plane, sphere, and hyperbolic plane determines a (not necessarily unique) geodesic.
d. Every pair of distinct points on the plane or hyperbolic plane determines a unique geodesic segment joining them. On the sphere there are always at least two such segments.
e. On the plane or on a hyperbolic plane, two geodesics either coincide or are disjoint or they intersect in one point. On a sphere, two geodesics either coincide or intersect exactly twice.
[Note that for the plane and hyperbolic plane, Part e and Part f are equivalent in the sense that they each imply the other.]
Notice that these properties distinguish a sphere from both the Euclidean plane and from a hyperbolic plane; however, these properties do not distinguish the plane from a hyperbolic plane.
Problem 6.2. Transformations
An isometry of X is a transformation that takes X onto X and preserves all distances and angles.
A translation along l (a geodesic) is an isometry that takes each point on l to a different point on l and takes each point not on l to another point on the same side of l.
A rotation about P through the angle q, is an isometry f that leaves the point P fixed and is such that for every Q ¹ P the angle QPf(Q) = q.
A reflection through the line (geodesic) l is an isometry that fixes only those points that lie on l.
A glide reflection (or just plain, glide) along the geodesic l is an isometry that takes each point on l to a different point on l and takes each point not on l to a point of the other side of l.
The existence of translations, rotations, and glides is quite clear for the plane and a sphere, but is it not so clear for a hyperbolic plane. But in the hyperbolic it is clear that there are reflections about any geodesic (see 6.1a). Thus, for this problem assume only the results of Problem 6.1.
a. If g is a geodesic on the plane, sphere, or hyperbolic plane, and if A and B are two points on g, then there is a translation along g that takes A to B.
[Hint: Find a reflection that takes A to B (and takes the geodesic g to itself); and then use another reflection to move the points on g that the first reflection leaves fixed.]
b. If P is a point on the plane, sphere, or hyperbolic plane, and ÐAPB is any angle at P, then there is a rotation about P through the angle q = ÐAPB.
[Hint: Find a reflection that takes A to B (and keeps the point P fixed); and then apply another reflection through the line BP. Remember to show that the composition of these two reflections has the desired angle property.]
c. If g is a geodesic on the plane, sphere, or hyperbolic plane and if A and B are two points on g, then there is a glide along g that takes A to B.
[Hint: Combine Parts a and b.]
Later in Chapter 18 we will show that these are the only isometries of the plane, sphere, and hyperbolic plane.
Problem 6.3. Side-Angle-Side (SAS)
We will now look at triangles on all the surfaces that you have studied: plane, sphere, cone, cylinder, and hyperbolic space. [If you skipped any of these surfaces, you should still find that this and the succeeding chapters will still make sense but you will want to limit your investigations to triangles on the surfaces which you studied.]
In order that we may start out with some common ground, let us agree on some common terminology: A triangle has three points (vertices) which are joined by three straight line (geodesic) segments (sides). A triangle divides the surface into two regions (the interior and exterior). The (interior) angles of the triangle are the angles between the sides in the interior of the triangle as we will discuss below, on a sphere you must decide which region you are going to call the interior.
Are two triangles congruent if two sides and the included angle of one are congruent to two sides and the included angle of the other?
Figure 6.1. SAS.
In some textbooks SAS is listed as an axiom; in others it is listed as the definition of congruency of triangles, and in others as a theorem to be proved. But no matter how one considers SAS, it still makes sense and is important to ask: Why is SAS true on the plane? One can also ask: Is SAS true on spheres, cylinders, cones, and hyperbolic space?
If you find that SAS is not true for all triangles on a sphere or another surface, is it true for sufficiently small triangles? Come up with a definition for "small triangles" for which SAS does hold.
Be as precise as possible, but use your intuition. In trying to prove SAS on a sphere you will realize that SAS does not hold unless some restrictions are made on the triangles. Keep in mind that everyone sees things differently, so there are many possible definitions of "small." Some may be more restrictive than others (that is, they don't allow as many triangles as other definitions). Use whatever definition makes sense for you.
Remember that it is not enough to simply state what a small triangle is; you must also prove that SAS is true for the small triangles under your definition explain why the counterexamples you found before are now ruled out and explain why the condition(s) you list is (are) sufficient to prove SAS. Also, try to come up with a basic, general proof that can be applied to all surfaces.
And remember what we said before: By "proof" we mean what most mathematicians use in their everyday practice, i.e., a convincing communication that answers Why? We do not ask for the usual two-column proofs from high school (unless, of course, you find the two-column proof sufficiently convincing and that it answers Why?). Your proof should convey the meaning you are experiencing in the situation. Think about why SAS is true on the plane think about what it means for actual physical triangles then try to translate these ideas to the other surfaces.
Let us clarify some terminology that we have found to be helpful for discussing SAS and other theorems. Two triangles are said to be congruent if through a combination of translations, rotations, and reflections, one of them can be made to coincide with the other. If no reflections are needed, then the triangles are said to be directly congruent. In this course we will focus on congruence and not specifically on direct congruence; however, some students may wish to keep track of the distinction as we go along.
Figure 6.2. Direct congruence and congruence.
In Figure 6.2, DABC is directly congruent to Da'b'c' but Dabc is not directly congruent to Da"b"c". However, Dabc is congruent to both Da'b'c' and Da"b"c" and we write: Dabc @ Da'b'c' @ Da"b"c".
So, why is SAS true on the plane? We will now illustrate one way of looking at this question. Referring to Figure 6.3, suppose that Dabc and Da'b'c' are two triangles such that Ðbac @ Ðb'a'c', ab @ a'b' and ac @ a'c'. Translate Da'b'c' along aa' so that a' coincides with a. Since the sides ac and a'c' are congruent we can now rotate Da'b'c' (about a=a') until c' coincides with C. If after this rotation B and B' are not coincident, then a reflection (about AC = A'C') will complete the process and all three vertices, the two given sides, and the included angle of the two triangles will coincide.
So, why is it that, on the plane, the third sides (BC and B'C') must now be the same? Since the third sides (BC and B'C') coincide, DABC is congruent to DA'B'C'. (In the case that no reflection is needed, the two triangles are directly congruent.)
Figure 6.3. SAS on plane.
The proof of SAS on the plane is not directly applicable to the other surfaces because properties of geodesics differ on the various surfaces and translations and rotations are not so clear. In particular, the number of geodesics joining two points varies from surface to surface and is also relative to the location of the points on the surface. On a sphere, for example, there are always at least two straight paths joining any two points. As we saw in Chapter 1, the number of geodesics joining two points on a cylinder is infinite. On a cone the number of geodesics is dependent on the cone angle, but for cones with angles less than 180° there is more than one geodesic joining two points. It follows that the argument made for SAS on the plane is not valid on cylinders, cones, or spheres. The question then arises: Is SAS ever true on those surfaces?
Look for triangles for which SAS is not true. Some of the properties that you found for geodesics on spheres, cones, and cylinders will come into play. As you look closely at the features of triangles on those surfaces, you may find that they challenge your notions of triangle. Your intuitive notion of triangle may go beyond what can be put into a traditional definition of triangle. When you look for a definition of small triangle for which SAS will hold on these surfaces, you should try to stay close to your intuitive notion. In the process of exploring different triangles you may come up with examples of triangles that seem very strange. Let's look at some unusual triangles.
For instance, keep the example in Figure 6.4 in mind:
Figure 6.4. SAS is false on sphere.
All the lines shown in Figure 6.4 are geodesic segments of the sphere. The two sides and their included angle for SAS are marked. As you can see, there are two possible geodesics that can be drawn for the third side the short one in front and the long one that goes around the back of the sphere. Remember that on a sphere, any two points define at least two geodesics (an infinite number if the points are at opposite poles). Look for similar examples on a cone and cylinder. You may decide to accept the smaller triangle into your definition of "small triangle" but to exclude the large triangle from your definition. But what is a large triangle? To answer this, let us go back to the plane. What is a triangle on the plane? What do we choose as a triangle on the plane?
On the plane, a figure that we want to call a triangle has all of its angles on the "inside." Also, there is a clear choice for inside on the plane; it is the side that has finite area. See Figure 6.5. But what is the inside of a triangle on a sphere? The restriction that the area on the inside has to be finite doesn't work for the spherical triangles because all areas on a sphere are finite. So what is it about the large triangle that challenges our view of triangle? You might try to resolve the triangle definition problem by specifying that each side must be the shortest geodesic between the endpoints. However, be aware that antipodal points (that is, a pair of points that are at diametrically opposite poles) on a sphere do not have a unique shortest geodesic joining them. On a cylinder we can have a triangle for which all the sides are the shortest possible segments, yet the triangle does not have finite area. Try to find such an example. In addition, a triangle on a cone will always bound one region that has finite area. Look at some of these ornery examples of triangles. A triangle that encircles the cone point may cause problems. Covering spaces can help you in your investigation of these triangles. For example, what happens when we try to unwrap or lift one of these triangles onto a covering space?
Figure 6.5. Insides of a plane triangle.
Problem 6.4. Angle-Side-Angle (ASA)
Are two triangles congruent if one side and the adjacent angles of one are congruent to one side and the adjacent angles of another?
Figure 6.6. ASA.
This problem is similar in many ways to the previous one. As before, look for counterexamples on all surfaces, and if ASA doesn't hold for all triangles, see if it works for small triangles. If you find that you must restrict yourself to small triangles, see if your previous definition of "small" still works; if it doesn't work here, then modify it.
There are also a few things to keep in mind while working on this problem. First, when considering ASA, both of the angles must be on the same side the interior of the triangle. For example, see Figure 6.7.
Figure 6.7. Angles of a triangle must be on same side.
Let us look at a proof of ASA on the plane:
Figure 6.8. ASA on the plane.
The planar argument for ASA does not work on spheres, cylinders, and cones because, in general, geodesics on these surfaces intersect in more than one point. But can you make the planar work on a hyperbolic plane? (You may want to modify the planar proof to use only reflections.)
As was the case for SAS, we must ask ourselves if we can find a class of small triangles on each of the different surfaces for which the above argument is valid. You should check if your previous definitions of small triangle are too weak, too strong, or just right to make ASA true on spheres, cylinders, and cones. It is also important to look at cases for which ASA does not hold. Just as with SAS, some interesting counterexamples arise.
In particular, try out the configuration in Figure 6.9 on a sphere. To see what happens you will need to try this on an actual sphere. If you extend the two sides to great circles, what happens? You may instinctively say that it is not possible for this to be a triangle, and on the plane most people would agree, but try it on a sphere and see what happens. Does it define a unique triangle? Remember that on a sphere two geodesics always intersect twice.
Figure 6.9. Possible counterexample to ASA.
Finally, notice that in our proof of ASA on the plane, we did not use the fact that the sum of the angles in a triangle is 180°. We avoided this for two reasons. For one thing, to use this "fact" we would have to prove it first. As we have already discussed in Chapter 4, this is both time consuming and unnecessary. We will prove it later (Problem 10.4). The other reason is that a proof using the fact that the angles sum to 180° will not work on a sphere or on a hyperbolic plane because there are at least some triangles on these surfaces whose angles sum to other than 180°. A common example is the triple-right triangle on a sphere, depicted in Figure 6.10, which you may have seen before. In hyperbolic triangles the sum of the angles appears to be less than 180°, see Figure 6.11 for an example.
Figure 6.10. Triple-right triangle on a sphere.
Figure 6.11. Hyperbolic triangle.
Remember that it is best to come up with a proof that will work for all surfaces because this will be more powerful, and, in general, will tell us more about the relationship between the plane and the other surfaces.