Download - 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
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 2 / 19
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
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 3 / 19
The Three Two-dimensional Geometries Euclidean
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}.
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 4 / 19
The Three Two-dimensional Geometries Euclidean
Euclidean isometries
DefinitionAn isometry is a distance preserving map.
Euclidean examples
Orientation preserving:
Translations
Rotations
Orientation reversing:
Reflections
Glide reflections
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 5 / 19
The Three Two-dimensional Geometries Euclidean
Euclidean isometries
DefinitionAn isometry is a distance preserving map.
Euclidean examples
Orientation preserving:
Translations
Rotations
Orientation reversing:
Reflections
Glide reflections
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 5 / 19
The Three Two-dimensional Geometries Euclidean
Euclidean isometries
DefinitionAn isometry is a distance preserving map.
Euclidean examples
Orientation preserving:
Translations
Rotations
Orientation reversing:
Reflections
Glide reflections
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 5 / 19
The Three Two-dimensional Geometries Euclidean
Euclidean isometries
DefinitionAn isometry is a distance preserving map.
Euclidean examples
Orientation preserving:
Translations
Rotations
Orientation reversing:
Reflections
Glide reflections
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 5 / 19
The Three Two-dimensional Geometries Euclidean
Euclidean isometries
DefinitionAn isometry is a distance preserving map.
Euclidean examples
Orientation preserving:
Translations
Rotations
Orientation reversing:
Reflections
Glide reflections
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 5 / 19
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.
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 6 / 19
The Three Two-dimensional Geometries Spherical
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.
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 7 / 19
The Three Two-dimensional Geometries Spherical
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.
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 7 / 19
The Three Two-dimensional Geometries Spherical
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.
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 8 / 19
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.
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 9 / 19
The Three Two-dimensional Geometries Hyperbolic
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.
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 10 / 19
The Three Two-dimensional Geometries Hyperbolic
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.
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 10 / 19
The Three Two-dimensional Geometries Hyperbolic
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.
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 10 / 19
The Three Two-dimensional Geometries Hyperbolic
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
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 11 / 19
The Three Two-dimensional Geometries Hyperbolic
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
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 11 / 19
The Three Two-dimensional Geometries Hyperbolic
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
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 11 / 19
The Three Two-dimensional Geometries Hyperbolic
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
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 11 / 19
The Three Two-dimensional Geometries Hyperbolic
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
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 11 / 19
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.
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 12 / 19
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.
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 13 / 19
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
Q
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.
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 13 / 19
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
Q
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.
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 13 / 19
The Three Reflections Theorem 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.
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 14 / 19
The Three Reflections Theorem 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 B′ C′
φ(A)φ(B)
φ(C)
The product of these reflections must be φ, because it co-incideson A, B, C.
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 14 / 19
The Three Reflections Theorem 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 B′ C′
C′′
φ(A)φ(B)
φ(C)
The product of these reflections must be φ, because it co-incideson A, B, C.
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 14 / 19
The Three Reflections Theorem 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 B′ C′
C′′
φ(A)φ(B)
φ(C)
The product of these reflections must be φ, because it co-incideson A, B, C.
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 14 / 19
Orientation preserving isometries
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
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 15 / 19
Orientation preserving isometries
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
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 15 / 19
Orientation preserving isometries
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
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 15 / 19
Orientation preserving isometries
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
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 15 / 19
Orientation preserving isometries
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
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 15 / 19
Orientation preserving isometries
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
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 15 / 19
Orientation preserving isometries
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
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 15 / 19
Orientation preserving isometries
The sphere
Any two distinct lines in S2 intersect
=⇒ every orientation preserving isometry of S2 is a rotation.
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 16 / 19
Orientation preserving isometries
The hyperbolic plane
Three cases:intersecting lines: rotationsasymptotic lines: “limit rotations”
(includes horizontal Euclidean translations)ultraparallel lines: hyperbolic translations
(includes Euclidean dilations)
����������������������������������������������������������������������������������������������������������������
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 17 / 19
Orientation preserving isometries
The hyperbolic plane
Three cases:intersecting lines: rotationsasymptotic lines: “limit rotations”
(includes horizontal Euclidean translations)ultraparallel lines: hyperbolic translations
(includes Euclidean dilations)
����������������������������������������������������������������������������������������������������������������
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 17 / 19
Orientation preserving isometries
The hyperbolic plane
Three cases:intersecting lines: rotationsasymptotic lines: “limit rotations”
(includes horizontal Euclidean translations)ultraparallel lines: hyperbolic translations
(includes Euclidean dilations)
����������������������������������������������������������������������������������������������������������������
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 17 / 19
Orientation preserving isometries
The hyperbolic plane
Three cases:intersecting lines: rotationsasymptotic lines: “limit rotations”
(includes horizontal Euclidean translations)ultraparallel lines: hyperbolic translations
(includes Euclidean dilations)
����������������������������������������������������������������������������������������������������������������
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 17 / 19
Orientation preserving isometries
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
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 18 / 19
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.
Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 19 / 19