The Three Reflections Theorem

The Three Reflections Theorem

Christopher Tuffley

Institute of Fundamental SciencesMassey University, Palmerston North

24th June 2009

Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 1 / 19

Outline

1 The Three Two-dimensional GeometriesEuclideanSphericalHyperbolic

2 The Three Reflections TheoremStatementProof

3 Orientation preserving isometries


The Three Two-dimensional Geometries Euclidean

The Euclidean plane

The Euclidean plane is

E2 = {(x , y)|x , y ∈ R},

with the Euclidean distance

d((x1, y1), (x2, y2)

)=

√(x1 − x2)2 + (y1 − y2)2.

(x1, y1)

(x2, y2)

d



Arc length

If γ : [a, b] → E2 is a smooth curve then

length(γ) =

∫ b

ads,

where ds2 = dx2 + dy2 is the infinitesimal metric.

γ(a)

γ(b)

|γ′(t)| =

√(dx

dt

)2+

(dydt

)2

The distance from P to Q is the infimum of

{length(γ)|γ a curve from P to Q}.



Euclidean isometries

DefinitionAn isometry is a distance preserving map.

Euclidean examples

Orientation preserving:

Translations

Rotations

Orientation reversing:

Reflections

Glide reflections





Euclidean examples


Translations

Rotations


Reflections

Glide reflections





Euclidean examples


Translations

Rotations


Reflections

Glide reflections





Euclidean examples


Translations

Rotations


Reflections

Glide reflections





Euclidean examples


Translations

Rotations


Reflections

Glide reflections


The Three Two-dimensional Geometries Spherical

Spherical geometry

Restrict the 3-dimensional Euclidean metric

ds2 = dx2 + dy2 + dz2

to the unit sphere S2 in R

3.

Arc length on S2 is given by (3d) Euclidean arc length.



Lines in spherical geometry

Lines in spherical geometry are great circles: the intersection of aplane through the origin with S

2.

Great circles are geodesics: locally length minimising curves.

Any two lines (great circles) intersect in a pair of antipodal points.



Lines in spherical geometry

Lines in spherical geometry are great circles: the intersection of aplane through the origin with S

2.

Great circles are geodesics: locally length minimising curves.

Any two lines (great circles) intersect in a pair of antipodal points.



Spherical isometries

Spherical isometries include

rotations about a diameter

reflections in a plane through the origin.

A reflection in a plane through the origin may be regarded asa reflection in the corresponding great circle, i.e. spherical line.


The Three Two-dimensional Geometries Hyperbolic

Hyperbolic geometry: the upper half plane model

Hyperbolic geometry may be modelled by the upper half plane

H2 = {(x , y) ∈ R

2|y > 0},

with metric

ds2 =dx2 + dy2

y2 .

The vectors shown all have the samehyperbolic length.

Hyperbolic angle in H2 co-incides

with Euclidean angle.��

��

x

y

Other models exist, including the conformal disc model.



Lines in the upper half plane model

A line in H2 is

a vertical ray x = constant, or

a semi-circle with centre on the x-axis.

There is a unique line through any pair of distinct points.

��

��

Disjoint lines may be asymptotic or ultraparallel.The x-axis together with ∞ forms the circle at infinity.




A line in H2 is




��

��





A line in H2 is




��

��




Hyperbolic isometries

The metric ds2 =dx2 + dy2

y2 is preserved by

Horizontal translationsz 7→ z + c, c real

Euclidean dilationsz 7→ ρz, ρ > 0

Reflections in vertical rayse.g. z 7→ −z̄

Inversions in semi-circular linese.g. z 7→ 1/z̄

��

��1

i





y2 is preserved by





��

��1

i





y2 is preserved by





��

��1

i





y2 is preserved by





��

��1

i





y2 is preserved by





��

��1

i


The Three Reflections Theorem Statement

The Three Reflections Theorem

The following hold in each of the three geometries E2, S

2 and H2.

Theorem (Characterisation of lines)

The set of points equidistant from a pair ofdistinct points P and Q is a line. Reflection inthis line exchanges P and Q.

P

QConversely, every line is the set of points equidistant from a suitablychosen pair of points P, Q.

Corollary (The Three Reflections Theorem)

Any isometry is a product of at most three reflections.


The Three Reflections Theorem Proof

Step 1: three points determine an isometry

Lemma

Any point P is uniquely determined by itsdistances to three non-collinear points A, B,C.

Consequently, any isometry is completelydetermined by the images of any threenon-collinear points.

A

B C

P

Proof.

Suppose Q has the same distances to A, B, C.Then A, B, C must lie on the line equidistant from P and Q,contradicting the fact they are not collinear.




Lemma



A

B C

P

Q

Proof.





Lemma



A

B C

P

Q

Proof.




Step 2: decompose isometries into reflections.

Given an isometry φ, let A, B, C be non-collinear.

1 If A 6= φ(A), reflect in the line equidistant from A and φ(A).2 If B′ 6= φ(B), reflect in the line equidistant from B′ and φ(B).3 If C′′ 6= φ(C), reflect in the line equidistant from C′′ and φ(C).

A

BC

φ(A)φ(B)

φ(C)

The product of these reflections must be φ, because it co-incideson A, B, C.






A

BC B′ C′

φ(A)φ(B)

φ(C)







A

BC B′ C′

C′′

φ(A)φ(B)

φ(C)







A

BC B′ C′

C′′

φ(A)φ(B)

φ(C)



Orientation preserving isometries


Orientation preserving isometries are products of two reflections.In Euclidean geometry, there are two cases:

intersecting mirror lines: rotations

parallel mirror lines: translations







































The sphere

Any two distinct lines in S2 intersect

=⇒ every orientation preserving isometry of S2 is a rotation.



The hyperbolic plane

Three cases:intersecting lines: rotationsasymptotic lines: “limit rotations”

(includes horizontal Euclidean translations)ultraparallel lines: hyperbolic translations

(includes Euclidean dilations)

��







��







��







��



Orientation preserving isometries, classified bypairs of reflections

intersecting lines disjoint lines

Spherical rotation

Euclidean rotation parallel lines: translation

Hyperbolic rotation asymptotic lines: ultraparallel lines:limit rotation translation


Going further

Going further

In each geometry, an orientation reversing isometry isa glide reflection.

Subgroups of the isometry group lead to quotient surfaces withthe given geometry.

Euclidean three-space has a “Four Reflections Theorem”.

There are eight “model geometries” in three dimensions:

E3, S

3, H3, S

2 × E1, H

2 × E1, Nil, S̃L2R, Solv.


Documents

The Three Reflections Theorem