Section 7.3 Hyperbolic Geometry with Curvature k<0
Lemma 7.3.1.
In hyperbolic geometry with curvature k, the area of a triangle with angles α,β, and γ is
Subsection Observing negative curvature
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 parallelism θ by the formula
Proof.
For this proof, let \(s = \frac{1}{\sqrt{|k|}}\text{.}\) Note that \(s\) is the Euclidean radius of the circle at infinity in the disk model for hyperbolic geometry with curvature \(k\text{.}\) Since angles and lines and distances are preserved, assume \(z\) is the origin and \(L\) is orthogonal to the positive real axis, intersecting it at the point \(x\) (with \(0 \lt x \lt s\)).
Recall the half-angle formula
According to Figure 7.3.4, \(\tan(\theta) =r/s\text{,}\) where \(r\) is the Euclidean radius of the circle containing the hyperbolic line \(L\text{.}\) Furthermore, \(\sec(\theta) = (x+r)/s\text{,}\) so
We may express \(x\) and \(r\) in terms of the hyperbolic distance \(d\) from 0 to \(x\text{.}\) In Exercise 7.3.2 we prove the hyperbolic distance from 0 to \(x\) in \((\mathbb{D}_k,{\cal H}_k)\) is
so that
Express \(r\) in terms of \(d\) by first expressing it in terms of \(x\text{.}\) Note that segment \(x x^*\) is a diameter of the circle containing \(L\text{,}\) where \(x^* = \frac{-1}{kx}\) is the point symmetric to \(x\) with respect to the circle at infinity. Thus, \(r\) is half the distance from \(x\) to \(x^*\text{:}\)
Replacing \(k\) with \(-1/s^2\text{,}\) we have
One checks that after writing \(x\) in terms of \(d\text{,}\) \(r\) is given by
Substitute this expression for \(r\) into the equation labeled (1) in this proof, and after a dose of satisfactory simplifying one obtains the desired result:
Since \(s = 1/\sqrt{|k|}\) 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 star s is orthogonal to the plane of the Earth's orbit. The angle p is called the parallax, and in a Euclidean universe, p determines the star's distance from the Sun, D, by the equation D=d/tan(p), where d is the Earth's distance from the Sun.Exercises Exercises
1.
Prove that for k<0, any transformation in Hk has the form
where θ is any real number and z0 is a point in Dk.
2.
Assume k<0 and let s=1/√|k|. Derive the following measurement formulas in (Dk,Hk).
- The length of a line segment from 0 to x, where 0<x<s isdk(0,x)=sln(s+xs−x).Hint: Evaluate the integral by partial fractions.
- The circumference of the circle centered at the origin with hyperbolic radius r is c=2πssinh(r/s).
- The area of the circle centered at the origin with hyperbolic radius r is A=4πs2sinh2(r2s).
3.
Let's investigate the idea that the hyperbolic formulas in Exercise 7.3.2 for distance, circumference, and area approach Euclidean formulas when k→0−.
- Show that the hyperbolic distance dk(0,x) from 0 to x, where 0<x<s, approaches 2x as k→0−.
- Show that the hyperbolic circumference of a circle with hyperbolic radius r approaches 2πr as k→0−.
- Show that the hyperbolic area of this circle approaches πr2 as k→0−.
4.
Triangle trigonometry in (Dk,Hk).
Suppose we have a triangle in (Dk,Hk) with side lengths a,b,c and angles α,β, and γ as pictured in Figure 5.4.11. Suppose further that s=1/√|k|.
- Prove the hyperbolic law of cosines in (Dk,Hk):cosh(c/s)=cosh(a/s)cosh(b/s)−sinh(a/s)sinh(b/s)cos(γ).
- 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) approach those of Euclidean geometry. What happens to Lobatchevsky's formula as k approaches 0? What must be the angle of parallelism θ in the limiting case? Is this value of θ independent of d?
6.
Bessel determined a parallax of p=.3 arcseconds 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 find the smallest detected parallax to date, and use it to estimate a bound on k if the universe is hyperbolic.