Download Understanding Alternative Geometries: Spherical, Hyperbolic, and Taxicab - Prof. Jeremy Ma and more Study notes Mathematics in PDF only on Docsity!
CHAPTER 1
Other Geometries:
A Computational
Introduction
In order to provide a better perspective on Euclidean geometry, three alternative geometries are described. These are the geometry of the surface of the sphere, hyperbolic geometry, and taxicab geometry.
1. Spherical Geometry
Due to its relationship with geography and astronomy, spherical geometry was studied extensively by the Greeks as early as 300 BC. Menelaus (circa 100) wrote the book Spherica on spherical trigonometry which was greatly extended by Ptolemy (100-178) in his Almagest. Many later mathematicians, including Leonhard Euler (1707-1783) and Carl Friedrich Gauss (1777-1855) made substantial contributions to this topic. Here it is proposed only to compare and contrast this geometry with that of the plane. Because the time to develop spherical geometry in the same manner as will be done with Euclidean geometry is not available, this discussion is necessarily informal and frequent appeals will be made to the readers’ visual intuition.
Strictly speaking there are no straight lines on the surface of a sphere. Instead it is both customary and useful to focus on curves that share the "shortest distance" property with the Euclidean straight lines. The following thought experiment will prove instructive for this purpose. Imagine that two pins have been stuck in a smooth sphere in points that are not diametrically opposite and that a (frictionless) rubber band is held by the pins in a stretched state. Rotate this sphere until one of the two pins is directly above the other right in front of your mind's eye. It is then hard to avoid the conclusion that the rubber band will be stretched out along the sphere in the plane formed by the two pins and the eye - the plane of the book's page in Figure 1.1. The inherent symmetry of the sphere dictates that this plane should cut the sphere into two identical hemispheres, in other words, that this plane should pass through the center of the sphere. It is also clear
Figure 1.1 A geodesic on the sphere.
that the tension of the stretched rubber band forces it to describe the shortest curve on the surface of the sphere that connects the two pins. The following may therefore be concluded.
PROPOSITION 1.1.1 (Spherical geodesics). If A and B are two points on a
sphere that are not diametrically opposite, then the shortest curve joining A and B on
1.1 SPHERICAL GEOMETRY
Figure 1.2 The lune α.
angle (Fig. 1.2). The measure of the spherical angle is defined to be the measure of the angle between their tangent lines at A (or at B ). Alternately, this equals the measure of the angle formed by the radii from the center of the sphere to the midpoints of the bounding great semicircles. For example, each meridian forms a 90 o^ angle with the equator at their point of intersection. In the Euclidean plane the relationships between lengths of straight line segments and measures of angles are given by well known trigonometric identities. Some fundamental theorems of spherical trigonometry are now stated without proof. Any three points A, B, C on the sphere no two of which are diametrically opposite, constitute the vertices of a spherical triangle denoted by Δ ABC. The three sides of this triangle are the geodesic segments that join each pair. The sides opposite the vertices A, B, C (and their lengths) are denoted a, b, c respectively. The interior angle α at the vertex A is the lune between AB and AC. The interior angles β and γ at B and C are defined in a similar manner.
PROPOSITION 1.1.2 (Spherical trigonometry). On a sphere of radius R = 1 , let
Δ ABC be a spherical triangle with sides a, b, c and interior angles α, β, γ. Then ,
i) cos α = cos^ a sin^ -^ b cos sin^ b^ c cos^ c i') cos a = cos b cos c + cos α sin b sin c
ii) cos a = cos^ α sin^ + cos β sin^ β γ^ cos^ γ ii') cos α = cos a sin β sin γ - cos β cos γ
iii) sin sin α a = sin sin β b = sin sin γ c.
1.1 SPHERICAL GEOMETRY
These are known as the first spherical law of cosines, the second spherical law of cosines, and the spherical law of sines. It should be noted that i and i' are really the same equation as are ii and ii', although, as will be demonstrated by the examples below, their uses are different. The solution of a triangle consists of the lengths of its sides and the measures of its interior angles.
EXAMPLE 1.1.3. Solve the spherical triangle with sides a = 1, b = 2, and c = π/2.
It follows from the first spherical law of cosines that
cos α = cos 1 sin 2 sin^ -^ cos 2 cos π/2 π/2 = cos 1 sin 2
so that
α = cos-^1 (cos 1 sin 2 ) ≈ 53.54o.
The angles β and γ are similarly shown to have measures 119.64o^ and 72.91o^.
EXAMPLE 1.1.4. On a sphere of radius 4000 miles, solve the triangle in which an
interior angle of 50 o^ lies between sides of lengths 7000 miles and 9000 miles respectively. Since the radius is the unit it follows that we may set
b = 70004000 = 1.75 c = 90004000 = 2..
Hence, from the first law of cosines,
1.1 SPHERICAL GEOMETRY
angle sum near 540 o^ is described in Figure 1.4. where A, B, C are points that are equally spaced along a great circle. As A ′, B ′ , C ′ approach A, B, C respectively, the angles they form are flattened out and come arbitrarily close to 180o^ each. For example, since the spherical distance between any two of the points A, B, C is 2 π/3 it might be assumed that the spherical distance between any two of the points A', B', C' is a = 2 π/3 - 0.00001, in which case each of the angles of Δ A'B'C' is
Figure 1.4 A nearly maximal spherical triangle.
cos-^1 ( cos^ a sin^ -^ cos a sin^ a^ a cos^ a ) ≈ 179.52o
and their sum is 538.56o. Since, by definition, each of the interior angles of the spherical triangle is less than 180 o, it follows that the sum of these angles can never equal 540 o. Similarly, as will be shown momentarily, the sum of these angle cannot equal the lower bound of 180o either. The area of the spherical triangle is also of interest. An elegant proof of this formula is offered in Section 3.2.
1.1 SPHERICAL GEOMETRY
PROPOSITION 1.1.6. If a triangle on a sphere of radius R has angles with radian measures α, β, γ then it has area ( α + β + γ − π )R^2.
For example, the spherical triangle formed by the equator, the Greenwich meridian and the 90o^ East meridian has all of its angles equal to π/2 and hence its area is
(^ π 2 +^ π 2 +^ π 2 - π) R^2 =^ π R
2
This answer is consistent with the fact that the said triangle constitutes one fourth of a hemisphere. Since the surface area of the sphere is 4 π R^2 , this triangle has area
1 4
4 π R^2 2 =
π R^2 2
which agrees with the previous calculation. The quantity α + β + γ − π is called the excess of the spherical Δ ABC. The above theorem in effect states that the area of a spherical triangle is proportional to its excess. This assertion is supported by the triangle below which has excess π/2 + π/2 + α − π = α and whose area is clearly proportional to α as long as A and B vary along the equator and C remains at the north pole (see Fig. 1.5).
1.1 SPHERICAL GEOMETRY
a) SSS b) SAS c) ASA d) SAA e) AAA. 9*. Prove that the three angles α, β, γ are the three interior angles of a spherical triangle if and only if they satisfy all of the following conditions: α + β + γ > π, α + π > β + γ, β + π > α + γ, γ + π > α + β.
2. Hyperbolic Geometry
Imagine a two dimensional universe, with a superimposed Cartesian coordinate system, in which the x - axis is infinitely cold. Imagine further, that as the objects of this universe approach the x - axis, the drop in temperature causes them to contract (see Fig. 1.6). Thus,
Figure 1.6 The shrinkage that defines the hyperbolic plane.
the inhabitants of this fictitious land will find that it takes them less time to walk along a horizontal line from A (0, 1) to B (1, 1) (Fig. 1.7) than it takes to walk along a horizontal line from C (0, .5) to D (1, .5). Since their rulers contract just as much as they do, this observation will not seem at all paradoxical to them. If it is assumed that the contraction
1.2 HYPERBOLIC GEOMETRY
is such that the outside observer sees the length of any object as being proportional to its distance from the x - axis, then the inhabitants will find that walking from C (0, 0.5) to D (1, .5) takes twice as long as walking from A (0, 1) to B (1, 1) and one fifth of the time
Figure 1.7 Paths of unequal hyperbolic lengths.
of walking from E (0, .1) to F (1, .1). To differentiate between the Euclidean length of such a segment and its length as experienced by these fictitious beings, it is customary to refer to the latter as the hyperbolic length of the segment. Accordingly, the hyperbolic lengths of the segments AB, CD, and EF of Figure 1.7 are 1, 2, and 10 respectively. In general, the hyperbolic length of a horizontal line segment at distance y from the x - axis is given by the formula
hyperbolic length = Euclidean length y (1).
Other curves also have a hyperbolic length and a method for computing this is given in Exercise 16 below. Not surprisingly, perhaps, the Euclidean straight line segment joining two points does not constitute the curve of shortest hyperbolic length between them. When setting out from A (0, 1) to B (1, 1) the inhabitants of this strange land may find that if they bear a little to the north their journey will be somewhat shorter because, unbeknownst to them,
1.2 HYPERBOLIC GEOMETRY
joins the points A (0, 1) and X (100, 1) has hyperbolic length 9.21, a mere 9% of the hyperbolic length of the segment AX. Given any two points, their hyperbolic distance is the minimum of the hyperbolic lengths of all the curves joining them. As was the case for spherical geometry, the geodesic segments of hyperbolic geometry are those curves that realize the hyperbolic distance between their endpoints. The hyperbolic plane consists of the portion of the Cartesian coordinate system that lies above the x - axis.
PROPOSITION 1.2.1 (Hyperbolic geodesics). The geodesic segments of the
hyperbolic plane are arcs of circles centered on the x-axis and Euclidean line segments that are perpendicular to the x-axis.
Figure 1.10 Six hyperbolic geodesics.
The geodesics of the first variety are called bowed geodesics, whereas the vertical ones are the straight geodesics (see Fig. 1.10). This distinction is only meaningful to the outside observer. The inhabitants of this geometry perceive no difference between these two kinds of geodesics. It so happens that as the inhabitants of the hyperbolic plane approach the x - axis they shrink at such a rate as to make the x - axis unattainable. Technically speaking, the hyperbolic lengths of all of the geodesic segments in Figure 1.10 diverge to infinity as
1.2 HYPERBOLIC GEOMETRY
the endpoints Q approach the x-axis. This claim will be given a quantitative justification at the end of this section. A hyperbolic angle is the portion of the hyperbolic plane between two geodesic rays (Fig. 1.11). The measure of the angle between two geodesics is, by definition, the
Figure 1.11 Three hyperbolic angles.
measure of the angle between the tangents to the geodesics at the vertex of the angle. Accordingly, two geodesics are said to form a hyperbolic right angle if and only if their tangents are perpendicular to each other as Euclidean straight lines (Fig. 1.12). Given any three points that do not lie on one hyperbolic geodesic, they constitute the vertices of a
Figure 1.12 Hyperbolic right angles.
hyperbolic triangle formed by joining the vertices, two at a time, with hyperbolic geodesics (Fig. 1.13).
1.2 HYPERBOLIC GEOMETRY
and so α ≈ cos-^1 (.9461) ≈ 18.89o. Similarly β ≈ 87.67o^ and γ ≈ 39.34o.
EXAMPLE 1.2.4. Solve the hyperbolic triangle with two sides of lengths 2, 3
respectively, if they are to include an angle of 30o. Set α = 30 o, b = 2, c = 3. It follows from Formula i’) of hyperbolic trigonometry that
a = cosh-^1 (cosh b cosh c - cos α sinh b sinh c ) = 2.545...
Now that all three sides of the triangle are known, the method of the previous example yields
β = cos-^1 (cosh^ a sinh^ cosh a^ sinh c^ -^ cosh c^ b ) ≈ 16.64o
γ = cos-^1 (cosh^ a sinh^ cosh a^ sinh b^ -^ b cosh^ c ) ≈ 52.28o
As for the sum of the angles of a hyperbolic triangle, the situation is diametrically opposite to that on the sphere.
PROPOSITION 1.2.5. The sum of the angles of every hyperbolic triangle is less
than 180o.
Figure 1.14 A hyperbolic triangle with three small angles.
1.2 HYPERBOLIC GEOMETRY
This proposition is borne out by the above two examples. Figure 1. demonstrates that this sum can be quite small. In fact, in the hyperbolic triangle with sides a = b = c = 10 each angle equals
α = cos-^1 (cosh 10 cosh 10 sinh 10 sinh 10^ -^ cosh 10 ) ≈ .77o.
EXAMPLE 1.2.6. Solve the hyperbolic triangle with a = 2, β = γ = 60o. By
formula ii' of hyperbolic trigonometry
cos α = cosh 2 sin 60osin 60o^ - cos 60o^ cos 60o^ ≈ 2..
Since the cosine of an angle cannot exceed 1, such a hyperbolic triangle does not exist. Note that a Euclidean triangle with the same specifications does exist. Exercises 4, 5 contain some related information.
The area of the hyperbolic triangle is of course of interest too. Its formula is quite surprising.
PROPOSITION 1.2.7. The area of the hyperbolic triangle whose angles have
radian measures α, β, γ is π − α − β − γ.
This formula is given some support by Figure 1.15. Note that the sum of the angles of the larger hyperbolic Δ ABC is less than the sum of the angles of the smaller hyperbolic Δ AB'C'. The quantity π − α − β − γ is called, by analogy with its spherical counterpart, the defect of the hyperbolic triangle. Thus, the above theorem asserts that the area of a hyperbolic triangle is equal to its defect.
1.2 HYPERBOLIC GEOMETRY
It follows from this independence of scale that in hyperbolic geometry it is not necessary to specify units of length. It might be instructive to show that this independence holds in the vertical direction as well. The hyperbolic length of a vertical segment of the hyperbolic plane is easily computed with the aid of calculus. For, if dy denotes the Euclidean length of the
Figure 1.17 Hyperbolic length along the y - axis.
(infinitesimally) small vertical line segment at height y above the x - axis, then its hyperbolic length is dy/y (see Fig. 1.17). Consequently the total hyperbolic length of the segment PQ is
⌡^
a
b dy y =^ ln^ b^ -^ ln^ a^ =^ ln^
b a.^ (2).
In particular, if a = 1 and b = e = 2.718..., then the hyperbolic length of the y - axis between P (0, 1) and Q (0, e ) is
ln 1 e = ln e = 1.
1.2 HYPERBOLIC GEOMETRY
Moreover, if the scale on the axes is changed by a factor of s , then P = (0, s ) and Q = (0, es ) and again their hyperbolic distance is
ln es s = ln e = 1.
Thus, if P and Q are points with coordinates (0, 1) and (0, e ) relative to some unit of the Cartesian coordinate system, then the line segment joining P and Q has hyperbolic length 1 regardless of the scale that is actually used. Moreover, this Euclidean line segment PQ , which has hyperbolic length 1, is also a hyperbolic geodesic in contrast with the aforementioned segment AB which also has hyperbolic length 1 but is not a hyperbolic geodesic. Consequently, this geodesic can be taken as the natural unit , or absolute unit of length, of hyperbolic geometry. Both spherical and hyperbolic geometry look very different from Euclidean geometry. Nevertheless, it is well known that on a large sphere, a small portion of the surface may be practically indistinguishable from a piece of a plane. This resemblance accounts for the fact that people first thought that the world was flat and small children still do so today. The same confusion could occur in the hyperbolic plane. If the portion of the hyperbolic plane that is subject to the direct experience and observation of its inhabitants is sufficiently small, their geometry would appear to them as practically indistinguishable from that of the Euclidean plane. This affinity between the hyperbolic and Euclidean planes is a topic that will be revisited many times in the subsequent discussion. The explanation of how the trigonometry of a small portion of the hyperbolic plane may be confusable with Euclidean trigonometry can be found in the references. At this point it will be demonstrated that, just like the Euclidean plane and in contrast with the sphere, hyperbolic geometry extends indefinitely in all directions. In other words, the inhabitants of the hyperbolic plane have no reason to suspect that part of their
