the three reflections theorem
TRANSCRIPT
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The Three Reflections Theorem
Christopher Tuffley
Institute of Fundamental SciencesMassey University, Palmerston North
24th June 2009
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Outline
1 The Three Two-dimensional GeometriesEuclideanSphericalHyperbolic
2 The Three Reflections TheoremStatementProof
3 Orientation preserving isometries
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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
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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}.
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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
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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
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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
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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
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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
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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.
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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.
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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.
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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.
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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.������������������������������������������������������
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x
y
Other models exist, including the conformal disc model.
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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.
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Disjoint lines may be asymptotic or ultraparallel.The x-axis together with ∞ forms the circle at infinity.
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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.
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Disjoint lines may be asymptotic or ultraparallel.The x-axis together with ∞ forms the circle at infinity.
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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.
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Disjoint lines may be asymptotic or ultraparallel.The x-axis together with ∞ forms the circle at infinity.
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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
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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
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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
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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
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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
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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
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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
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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
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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
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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
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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
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Orientation preserving isometries
The sphere
Any two distinct lines in S2 intersect
=⇒ every orientation preserving isometry of S2 is a rotation.
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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)
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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)
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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)
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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)
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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
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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