b. Prove the elliptic law of sines in (P2k,Sk):

sin(√ka) sin(α) = sin(√kb) sin(β) = sin(√kc) sin(γ) .

### 7.3 Hyperbolic Geometry with Curvature

### k <

### 0

We may do the same gentle scaling of the Poincaré model of hyperbolic geometry as we did in the previous section to the disk model of elliptic geometry. In particular, for each negative numberk <0 we construct a model

for hyperbolic geometry with curvaturek.

We dene the space Dk to be the open disk of radius1/

p

|k| centered at the origin in C. That is,Dk consists of allz in Csuch that|z|<1/p|k|. In

this setting, the circle at innity is the boundary circle|z|= 1/p|k|.

The groupHkconsists of all Möbius transformations that sendDkto itself.

The geometry (Dk,Hk) with k < 0 is called hyperbolic geometry with

curvature k. Pushing analogy with the elliptic case, we may dene the group of transformations to consist of all Möbius transformations with this property: ifz andz∗ are symmetric with respect to the circle at innity then T(z)and

T(z∗)are also symmetric with respect to the circle at innity. Noting that the

point symmetric toz with respect to this circle is z∗= 1 |k|z =−

1

kz, we draw the satisfying conclusion thatT ∈ Hk if and only if the following holds:

ifz∗=−1

kz then T(z

∗_{) =}_{−} 1

kT(z).

Thus, the groupHkin the hyperbolic case has been dened precisely as the

groupSk in the elliptic case. Furthermore, one can show that transformations

inHk have the form

T(z) =eiθ z−z0

1 +kz0z

, wherez0 is a point inDk.

Straight lines in this geometry are the clines inC+orthogonal to the circle

at innity. By Theorem 3.2.8, a straight line in (Dk,Hk) is precisely a cline

with the property that if it goes throughzthen it goes through its symmetric point −1

kz.

The arc-length and area formulas also get tweaked by the scale factor, and now look identical to the formulas for elliptic geometry with curvaturek.

The arc-length of a smooth curve rin _{D}k is

L(r) =

Z b

a

2|r0_{(}_{t}_{)}_{|}

1 +k|r(t)|2 dt.

The area of a regionRgiven in polar coordinates is computed by the formula A(R) =

Z Z

R

4r

(1 +kr2_{)}2drdθ.

As in Chapter 5 when kwas xed at -1, the area formula is a bear to use, and one may convert to an upper half-plane model to determine the area of a 2

Section 5.5 to show that the area of a 2

3-ideal triangle in(Dk,Hk)(k <0) with

interior angleαis−1

k(π−α).

With this formula in hand, we can derive the area of any triangle inDk in

terms of its angles, exactly as we did in Chapter 5.

Lemma 7.3.1. In hyperbolic geometry with curvaturek, the area of a triangle with anglesα, β, andγ is

A= 1

k(α+β+γ−π).

### Observing negative curvature

Suppose we are located at a pointzin a hyperbolic universe with curvaturek.
We see in the distance a hyperbolic lineLthat seems to extend indenitely. We
might intuitively see the pointwon the line that is closest to us, as suggested
in Figure 7.3.2. Now suppose we look down the road a bit to a pointv. Ifv is
close towthe angle_{∠}wzvwill be close to 0. Asvgets further and further away
fromw, the angle will grow, getting closer and closer to the angleθ=_{∠}wzu,
whereuis an ideal point of the lineL.

z w u θ v d L

Figure 7.3.2: The angle of parallelismθof a point zto a lineL.
A curious fact about hyperbolic geometry is that this angle θ, which is
called the angle of parallelism of z to L, is a function of z's distance d
to L. In Section 5.4 we saw thatcosh(d) = 1/sin(θ) in(_{D},H). In particular,

one may deduce the distancedto the lineLby computingθ. No such analogy exists in Euclidean geometry. In a Euclidean world, if one looks farther and farther down the lineL, the angle will approach 90◦, no matter one's distance dfrom the line. The following theorem provides another formula relating the angle of parallelism to a point's distance to a line.

Theorem 7.3.3 Lobatchevsky's formula. In hyperbolic geometry with curvature k, the hyperbolic distance d of a point z to a hyperbolic line L is related to the angle of parallelismθ by the formula

tan(θ/2) =e− √

|k|d_{.}

Proof. For this proof, let s= √1

|k|. Note thats is the Euclidean radius of

the circle at innity in the disk model for hyperbolic geometry with curvature k. Since angles and lines and distances are preserved, assume z is the origin andLis orthogonal to the positive real axis, intersecting it at the pointx(with

7.3. Hyperbolic Geometry with Curvaturek <0 139
s
r
θ
x _{x}∗
L

Figure 7.3.4: Deriving Lobatchevsky's formula. Recall the half-angle formula

tan(θ/2) = tan(θ) sec(θ) + 1.

According to Figure 7.3.4,tan(θ) =r/s, whereris the Euclidean radius of the circle containing the hyperbolic lineL. Furthermore, sec(θ) = (x+r)/s, so

tan(θ/2) = r

x+r+s. (1)

We may expressxandrin terms of the hyperbolic distancedfrom 0 tox.
In Exercise 7.3.2 we prove the hyperbolic distance from 0 toxin(_{D}k,Hk)is

d=sln
_{s}_{+}_{x}
s−x
so that
x=s·e
d/s_{−}_{1}
ed/s_{+ 1}.

Express r in terms of d by rst expressing it in terms of x. Note that
segment xx∗ is a diameter of the circle containing L, where x∗ = −1_{kx} is the

point symmetric toxwith respect to the circle at innity. Thus,r is half the distance fromxtox∗:

r=−1 +kx

2

2kx . Replacingk with−1/s2, we have

r= s

2_{−}_{x}2

2x .

One checks that after writingxin terms ofd,ris given by r= 2se

d/s

e2d/s_{−}_{1}.

Substitute this expression for rinto the equation labeled (1) in this proof, and after a dose of satisfactory simplifying one obtains the desired result:

tan(θ/2) =e−d/s. Sinces= 1/p|k|this completes the proof.

Parallax If a star is relatively close to the Earth, then as the Earth moves in its annual orbit around the Sun, the star will appear to move relative to the backdrop of the more distant stars. In the idealized picture that follows, e1 and e2 denote the Earth's position at opposite points of its orbit, and the

starsis orthogonal to the plane of the Earth's orbit. The anglepis called the parallax, and in a Euclidean universe,pdetermines the star's distance from the Sun, D, by the equation D = d/tan(p), where dis the Earth's distance from the Sun.

d D p e1 e2 Sun s

We may determine pby observation, and dis the radius of Earth's orbit
around the Sun (d is about 8.3 light-minutes.) In practice, p is quite small,
so a working formula is D = d_{p}. The rst accurate measurement of parallax

was recorded in 1837 by Friedrich Bessel (1784-1846) . He found the stellar
parallax of 0.3 arc seconds (1 arc second = 1/3600◦_{) for star 61 Cygni, which}

put the star at about 10.5 light-years away.

If we live in a hyperbolic universe with curvature k, a detected parallax puts a bound on how curved the universe can be. Consider Figure 7.3.5. As before,e1 and e2 represent the position of the Earth at opposite points of its

orbit, so that the distance between them is 2d, or about 16.6 light-minutes. Assume stars is on the positive real axis and we have detected a parallaxp, so that angle∠e2se1= 2p. e1 s e2 α 2p 2d a u

Figure 7.3.5: A detected parallax in a hyperbolic universe puts a bound on its curvature

The angle α=_{∠}e1e2sin Figure 7.3.5 is less than the angle of parallelism

θ = _{∠}e1e2u. Noting that tan(x) is an increasing function and applying

Lobatchevsky's formula it follows that

tan(α/2)<tan(θ/2) =e− √

|k|2d_{.}

We may solve this inequality for|k|: tan(α/2)< e−

√

|k|2d

7.3. Hyperbolic Geometry with Curvaturek <0 141 ln(tan(α/2)) 2d 2 >|k|. x2 is decreasing forx <0

To get a bound for|k|in terms ofp, note thatα≈π/2−2p(the triangles
used in stellar parallax have no detectable angular deviation from 180◦_{), so}

|k|<

_{ln(tan(}_{π/}_{4}_{−}_{p}_{))}

2d

2 .

We remark that for values ofpnear 0, the expressionln(tan(π/4−p))has

linear approximation equal to−2p, so a working bound fork, which appeared
in Schwarzschild's 1900 paper [27], is|k|<(p/d)2_{.}

Exercises

1. Prove that for k <0, any transformation inHk has the form

T(z) =eiθ z−z0

1 +kz0z

,

whereθis any real number andz0is a point inDk. Hint: Follow the derivation

of the transformations inHfound in Chapter 5.

2. Assume k < 0 and let s = 1/p|k|. Derive the following measurement
formulas in(_{D}k,Hk).

a. The length of a line segment from 0tox, where0< x < sis dk(0, x) =sln

_{s}_{+}_{x}

s−x

. Hint: Evaluate the integral by partial fractions.

b. The circumference of the circle centered at the origin with hyperbolic radiusrisc= 2πssinh(r/s).

c. The area of the circle centered at the origin with hyperbolic radius ris
A= 4πs2_{sinh}2

(_{2}r_{s}).

3. Let's investigate the idea that the hyperbolic formulas in Exercise 7.3.2 for distance, circumference, and area approach Euclidean formulas whenk→0−.

a. Show that the hyperbolic distancedk(0, x)from 0 tox, where0< x < s,

approaches2xask→0−.

b. Show that the hyperbolic circumference of a circle with hyperbolic radius rapproaches2πr ask→0−.

c. Show that the hyperbolic area of this circle approachesπr2 _{as}_{k}_{→}_{0}−_{.}

4. Triangle trigonometry in(_{D}k,Hk).

Suppose we have a triangle in (_{D}k,Hk)with side lengthsa, b, cand angles

α, β,andγas pictured in Figure 5.4.11. Suppose further thats= 1/p|k|.
a. Prove the hyperbolic law of cosines in(_{D}k,Hk):

cosh(c/s) = cosh(a/s) cosh(b/s)−sinh(a/s) sinh(b/s) cos(γ). b. Prove the hyperbolic law of sines in(Dk,Hk):

sinh(a/s) sin(α) = sinh(b/s) sin(β) = sinh(c/s) sin(γ) .

5. As k < 0 approaches 0, the formulas of hyperbolic geometry (Dk,Hk)

formula as k approaches 0? What must be the angle of parallelism θ in the limiting case? Is this value ofθindependent ofd?

6. Bessel determined a parallax ofp=.3arcseconds for the star 61 Cygni.

Convert this angle to radians and use it to estimate a bound for the curvature
constant k if the universe is hyperbolic. The units for this bound should be
light years−2 _{(convert the units for the Earth-Sun distance to light-years).}

7. The smallest detectable parallax is determined by the resolving power of our best telescopes. Search the web to nd the smallest detected parallax to date, and use it to estimate a bound onk if the universe is hyperbolic.