OPEN UNIVERSITY Intro to Hyperbolic Geometry Shirleen Stibbe www.shirleenstibbe.co.uk My badge for this session M203 Pure Mathematics Summerschool
OPEN UNIVERSITY
Intro to Hyperbolic Geometry
Shirleen Stibbe www.shirleenstibbe.co.uk
My badge for this session
M203 Pure Mathematics Summerschool
Euclid's Parallel Postulate
Given any line l and a point P not on l, there is a unique line which passes
through P and does not meet l.
How to lose it
Define a non-Euclidean geometry, where either:
1) there are lots of lines through P
or
2) there are no lines through P
which don't meet l.
Hyperbolic geometry is the first kind. (The other is Elliptic.)
2
Poincarre model of Hyperbolic Geometry
Unit disc: D = {z : |z | < 1} = {(x, y): x2 + y2 < 1}
Unit circle: C = {z : |z | = 1} = {(x, y): x2 + y2 = 1}
Boundary point: a point on C [not in N-E geometry]
d-point: a point in D
d-line: part of a generalised circle which meets C at right angles, and lies in D
d-lines
boundary points
d-point
(x, y) or x + iy
C
D
3
Hyperbolic Parallelism
l1 l2
l3
Parallel lines: meet on C [not in D]
Ultra-parallel lines: do not meet at all
l2 & l3 are parallel
l1 & l2 are ultra-parallel
l1 & l3 are ultra-parallel
4
Some jaunty figures in D
(with some interesting properties)
Sum of angles < π
Sum of angles < 2π
Sum of angles = 0
Each side is parallel to the
other two
!
5
Given a d-line l and a d-point p not on l, there are:
a) exactly two d-lines through p which are parallel to l
b) infinitely many d-lines through p which are ultra-parallel to l.
Given a line l and a point p not on l, there is exactly one line through p which is parallel to l
p
p l1
l2
l p l
NB: l1 and l2 are both parallel to l but not to each other
E vs Non-E Parallel lines
E
Non-E
l
6
Parallel d-lines have none
E Parallel lines have lots
Non-E
?
Not possible
Angle sum of triangle would be π
Ultra-parallel d-lines have one
? Not possible
Angle sum of quadrilateral would be 2π
E vs Non-E Common Perpendiculars
7
Non-Euclidean reflections
Reflect in diameter l Reflect in d-line l
l
Properties:
1 the d-line l is left fixed by the non-E reflection in l. 2 d-lines are mapped to d-lines
3 magnitudes of angles are preserved, but orientation is reversed
4 reflections are indirect transformations
l
8
Direct Transformations
Reflection in l1 followed by reflection in l2
A
Lines meet at A ∈ D Rotation about A A is a fixed point
Lines are parallel Limit rotation No fixed points
1 Rotations
2 Translations
Lines are ultra-parallel
9
Direct azbbaz)z(M
++
= a, b ∈ C, |b| < |a|
Indirect azbbza)z(M
++
= a, b ∈ C, |b| < |a|
Matrix ⎟⎠
⎞⎜⎝
⎛=
abba
A
Why in this form?
Transformations must map the unit disc to itself
Therefore restricted to a subgroup of
the Mobius transformations that fix D.
Mobius transformations [General]
10
Direct zm1mzK)z(M
−−
= |K| =1, |m| < 1
Indirect
Matrix ⎟⎠
⎞⎜⎝
⎛
−
−=
1mm1
A
mz1mzK)z(M
−−
=
M(m) = 0 M(0) = -Km
|K| =1, |m| < 1
M(m) = 0 M(0) = -Km ̅
Origin lemma - very NB
Given any point A in D, there exists a non-E transformation sending A to 0.
Mobius transformations [Canonicall]
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Non-Euclidean Distance
★★ Key idea: non-E transformations should preserve non-E distance!
Distance function: d(z, w) denotes the non-E distance between the points z and w in D.
If M is a non-E transformation, then:
d(z, w) = d(M(z), M(w))
distance between the points
distance between their images
Definition:
d(0, z) = tanh-1(|z|), z ∈ D
distance between z and the origin
Alternative form:
d(0, z) = ⎟⎟⎠
⎞⎜⎜⎝
⎛
−
+
z1z1
log21
e
12
13
Calculate d(v, w)
Example: Find d(i/2, -i/2)
1
2
3 Then d(i/2, -i/2) = d(0, -4i/5)
= tanh-1(4/5)
= 1.0986
z)2/i(12/iz)z(M
+−
=
5/i44/11i
)2/i(2/i12/i2/i)2/i(M −=
+−
=−+−−
=−
1 Find a transformation taking v to 0
v) m ,1K(zv1vz)z(M ==
−−
=
2 Calculate M(w)
3 Then d(v, w) = d(M(v), M(w))
= d(0, M(w))
= tanh-1(|M(w)|)
14
Going round in circles
Some useful notations for circles
In R2: K1 has equation: (x - a)2 + (y - b)2 = r12
Centre = (a, b), Radius = r1
In C: K2 = {z: |z - α| = r2} , z, α ∈ C, r2 ∈ R
Centre = α, Radius = r2
Note: if K1 = K2, then
α = a + ib, r1 = r2
In D: K3 = {z: d(z, β) = r3}, z, β ∈ C, r3 ∈ R
Centre = β, Non-E radius = r3
Note: r3 ≠ tanh-1(r2)
β ≠ α
15
Example: K is the non-E circle K = {z: d(z - 0.34i) = 0.61}
Find the Euclidean centre and radius of K.
Non-E: d(O, a) = 0.61 + 0.35 = 0.96 = tanh-1(|a|)
Euclidean: |a| = tanh(0.96) = 0.75 = 3/4
⇒ a = 3i/4 (imaginary, above the origin)
Non-E: d(O, b) = 0.61 - 0.35 = 0.26 = tanh-1(|b|)
Euclidean: |b| = tanh(0.26) = 0.25 = 1/4
⇒ b = -i/4 (imaginary, below the origin)
E-centre: c = 1/2(a + b) = 1/2(3i/4 – i/4) = i/4
E-radius: r = 1/2|a - b| = 1/2|3i/4 + i/4| = 1/2
E-circle: K = {z: |z - i/4| = 1/2
Draw a picture
Non-E centre: m = 0.34i, so |m| = 0.34
Non-E distance: d(O, m) = tanh-1(0.34) = 0.35 (< 0.61)
Non-E distances:
Origin to m: 0.35
Radius: 0.61
a, b on opposite sides of the origin
m
a
b
0.61
0.61 0.35
Other views 1: (Lobachevsky)
Unit Disk viewed as a Pseudosphere
16
Other views 2:
Project the unit disc onto a dome world
17
L
tanh-1(L)
Non-Euclidean length: d(0, z) = tanh-1(|z|)
1,1212log2
12
12log2)(
21
2
2
212
<⎟⎟⎠
⎞⎜⎜⎝
⎛
−+
+−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−−
+−−−=
t
t
tttf
e
e
Formula for dome outline:
Other views 2:
Lengths: Unit disc vs projection onto dome world
18
M.C. Escher's Circle Limit III, 1959
19
Other views 3: Escher