The Lumenary · Early undergraduate

Non-Euclidean geometry: spherical and hyperbolic

Name: ______________________

Date: ______________________

You will learn to

Work with great circles, spherical triangles and the spherical law of cosines, and prove Girard's theorem that a spherical triangle has area $R^2(\alpha + \beta + \gamma - \pi)$
Describe lines, distances and angles in the upper half-plane and Poincaré disc models, and prove that Möbius transformations are hyperbolic isometries
Prove that through a point not on a hyperbolic line there are infinitely many lines that do not meet it, and that a hyperbolic triangle has area $\pi - (\alpha + \beta + \gamma) < \pi$
Explain what it means for the parallel postulate to be independent of Euclid's other axioms, and how models establish this

1.
A pilot flies from the North Pole due south to the equator, turns to fly a quarter of the way around the equator, and then flies due north back to the pole. What is the angle sum of this triangle, in degrees?
2.
A triangle on a sphere of radius $2$ has angles $\frac{\pi}{2}$ , $\frac{\pi}{2}$ and $\frac{\pi}{3}$ . Its area is $\frac{k\pi}{3}$ . What is $k$ ?
3.
A triangle in the hyperbolic plane (curvature $-1$ ) has three angles of $\frac{\pi}{6}$ each. What is its area?
- A. $\frac{\pi}{2}$
- B.It cannot exist: the angles of a triangle add up to $\pi$
- C. $-\frac{\pi}{2}$
- D.It depends on the lengths of the sides
4.
Which of these curves are lines (geodesics) of the upper half-plane model $\mathbb H$ ? (Choose all that apply.)
- A.The vertical half-line $\operatorname{Re} z = 3$ , $\operatorname{Im} z > 0$
- B.The upper half of the circle $|z - 2| = 1$
- C.The horizontal line $\operatorname{Im} z = 1$
- D.The upper half of the circle $|z - (1 + i)| = 2$
- E.The circle $|z - 2i| = 1$
5.
What is the hyperbolic distance in $\mathbb H$ between $i$ and $e^3 i$ ?
6.
In the theorem that $g(z) = \frac{az + b}{cz + d}$ is an isometry of $\mathbb H$ we required $a, b, c, d$ real with $ad - bc = 1$ . What goes wrong for $g(z) = -z$ , which has $a = -1$ , $b = c = 0$ , $d = 1$ and $ad - bc = -1$ ?
- A.It sends $\mathbb H$ to the lower half-plane, because $\operatorname{Im} g(z) = (ad - bc)\operatorname{Im} z / |cz + d|^2 < 0$
- B.It does not preserve Euclidean lengths
- C.It does not preserve angles
- D.Nothing: it is a hyperbolic isometry
7.
The Lean theorem many_parallels states: for c : ℝ with 1 ≤ c and y : ℝ with 0 < y, normSq (I * y - c) ≠ normSq (1 + I - c). What does it say geometrically?
- A.No point $iy$ of the line $L = \{\operatorname{Re} z = 0\}$ is at the same Euclidean distance from $c$ as $1 + i$ , so the semicircle centred at $c$ through $1 + i$ misses $L$
- B.The hyperbolic distance from $iy$ to $1 + i$ is never $c$
- C.Every line through $1 + i$ misses $L$
- D.There is exactly one parallel to $L$ through $1 + i$
8.
Why does the upper half-plane model show that the parallel postulate cannot be proved from the other axioms of plane geometry?
- A.All the other axioms hold in the model and the parallel postulate fails there; a proof from the other axioms would make it hold in every model
- B.It shows that the parallel postulate is false
- C.Because the lines in the model are curved
- D.It shows that Euclidean geometry is inconsistent
9.
Write hyp_dist(z, w) for points of the upper half-plane (Python complex numbers with positive imaginary part), using $\cosh d = 1 + \frac{|z - w|^2}{2\operatorname{Im} z \operatorname{Im} w}$ , and defect(z1, z2, z3), the defect $\pi - (\alpha + \beta + \gamma)$ of the triangle, computing each angle from the side lengths by the hyperbolic law of cosines $\cos\gamma = \frac{\cosh a \cosh b - \cosh c}{\sinh a \sinh b}$ (where $c$ is the side opposite $\gamma$ ).
Computational — Practice

Answer key — Non-Euclidean geometry: spherical and hyperbolic

1.
All three angles are right angles, so the sum is $270^\circ$ . The excess $90^\circ = \pi/2$ times $R^2$ is the area, $\frac{\pi}{2}R^2 = \frac18 \cdot 4\pi R^2$ : one eighth of the sphere.
2.
The excess is $\frac\pi2 + \frac\pi2 + \frac\pi3 - \pi = \frac\pi3$ , so the area is $R^2 \cdot \frac\pi3 = \frac{4\pi}{3}$ : $k = 4$ .
3.
Area $= \pi - 3 \cdot \frac\pi6 = \frac\pi2$ . Shrinking the angles towards $0$ makes the triangle larger, with area approaching $\pi$ , the area of an ideal triangle.
4.
The vertical half-line and the semicircle centred at $2$ are lines. The horizontal line, the off-centre circle and the circle inside $\mathbb H$ are not (the last is a hyperbolic circle).
5.
$d(i, e^3 i) = \int_1^{e^3} \frac{dy}{y} = \ln e^3 = 3$ . Check with the formula: $1 + \frac{(e^3 - 1)^2}{2e^3} = \frac{e^3 + e^{-3}}{2} = \cosh 3$ .
6.
$\operatorname{Im} g(z) = (ad - bc)\operatorname{Im} z/|cz + d|^2$ , so with $ad - bc < 0$ the upper half-plane goes to the lower one: $-i = g(i) \notin \mathbb H$ . The hypothesis $ad - bc > 0$ (normalised to $1$ ) is exactly what keeps $\mathbb H$ in place.
7.
It says that the circle centred at $c$ through $1 + i$ contains no point $iy$ with $y > 0$ : the hyperbolic line through $1 + i$ with centre $c$ never meets the imaginary axis. Since this holds for every $c \ge 1$ , there are infinitely many parallels.
8.
A proof of $P$ from axioms $\Sigma$ shows $P$ in every model of $\Sigma$ . The half-plane is a model of the other axioms in which $P$ fails, so no such proof exists — provided the real numbers, from which the model is built, are consistent.

import math

def hyp_dist(z, w):
    return math.acosh(1 + abs(z - w) ** 2 / (2 * z.imag * w.imag))

def defect(z1, z2, z3):
    a, b, c = hyp_dist(z2, z3), hyp_dist(z1, z3), hyp_dist(z1, z2)

    def angle(opp, s1, s2):
        x = (math.cosh(s1) * math.cosh(s2) - math.cosh(opp)) / (
            math.sinh(s1) * math.sinh(s2))
        return math.acos(max(-1.0, min(1.0, x)))

    return math.pi - (angle(a, b, c) + angle(b, a, c) + angle(c, a, b))

By the area theorem, the defect is the hyperbolic area. The nearly ideal triangle has defect close to, but below, $\pi$ .