[PPT] - The Three Reflections Theorem Christopher Tuffley Institute of PowerPoint Presentation

SLIDE 1

The Three Reflections Theorem

Christopher Tuffley

Institute of Fundamental Sciences Massey University, Palmerston North

24th June 2009

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

SLIDE 2

Outline

1

The Three Two-dimensional Geometries Euclidean Spherical Hyperbolic

2

The Three Reflections Theorem Statement Proof

3

Orientation preserving isometries

Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 2 / 19

SLIDE 3

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

SLIDE 4

The Three Two-dimensional Geometries Euclidean

Arc length

If γ : [a, b] → E2 is a smooth curve then length(γ) = b

a

ds, where ds2 = dx2 + dy2 is the infinitesimal metric. γ(a) γ(b) |γ′(t)| = dx

dt

2 +

dy

dt

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

SLIDE 5

The Three Two-dimensional Geometries Euclidean

Euclidean isometries

Definition An 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

SLIDE 6

The Three Two-dimensional Geometries Euclidean

Euclidean isometries

Definition An 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

SLIDE 7

The Three Two-dimensional Geometries Euclidean

Euclidean isometries

Definition An 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

SLIDE 8

The Three Two-dimensional Geometries Euclidean

Euclidean isometries

Definition An 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

SLIDE 9

The Three Two-dimensional Geometries Euclidean

Euclidean isometries

Definition An 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

SLIDE 10

The Three Two-dimensional Geometries Spherical

Spherical geometry

Restrict the 3-dimensional Euclidean metric ds2 = dx2 + dy2 + dz2 to the unit sphere S2 in R3. Arc length on S2 is given by (3d) Euclidean arc length.

Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 6 / 19

SLIDE 11

The Three Two-dimensional Geometries Spherical

Lines in spherical geometry

Lines in spherical geometry are great circles: the intersection of a plane through the origin with S2. 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

SLIDE 12

The Three Two-dimensional Geometries Spherical

Lines in spherical geometry

Lines in spherical geometry are great circles: the intersection of a plane through the origin with S2. 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

SLIDE 13

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 as a reflection in the corresponding great circle, i.e. spherical line.

Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 8 / 19

SLIDE 14

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) ∈ R2|y > 0}, with metric ds2 = dx2 + dy2 y2 . The vectors shown all have the same hyperbolic 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

SLIDE 15

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

SLIDE 16

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

SLIDE 17

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

SLIDE 18

The Three Two-dimensional Geometries Hyperbolic

Hyperbolic isometries

The metric ds2 = dx2 + dy2 y2 is preserved by Horizontal translations z → z + c, c real Euclidean dilations z → ρz, ρ > 0 Reflections in vertical rays e.g. z → −¯ z Inversions in semi-circular lines e.g. z → 1/¯ z

1

i

Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 11 / 19

SLIDE 19

The Three Two-dimensional Geometries Hyperbolic

Hyperbolic isometries

The metric ds2 = dx2 + dy2 y2 is preserved by Horizontal translations z → z + c, c real Euclidean dilations z → ρz, ρ > 0 Reflections in vertical rays e.g. z → −¯ z Inversions in semi-circular lines e.g. z → 1/¯ z

1

i

Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 11 / 19

SLIDE 20

The Three Two-dimensional Geometries Hyperbolic

Hyperbolic isometries

The metric ds2 = dx2 + dy2 y2 is preserved by Horizontal translations z → z + c, c real Euclidean dilations z → ρz, ρ > 0 Reflections in vertical rays e.g. z → −¯ z Inversions in semi-circular lines e.g. z → 1/¯ z

1

i

Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 11 / 19

SLIDE 21

The Three Two-dimensional Geometries Hyperbolic

Hyperbolic isometries

The metric ds2 = dx2 + dy2 y2 is preserved by Horizontal translations z → z + c, c real Euclidean dilations z → ρz, ρ > 0 Reflections in vertical rays e.g. z → −¯ z Inversions in semi-circular lines e.g. z → 1/¯ z

1

i

Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 11 / 19

SLIDE 22

The Three Two-dimensional Geometries Hyperbolic

Hyperbolic isometries

The metric ds2 = dx2 + dy2 y2 is preserved by Horizontal translations z → z + c, c real Euclidean dilations z → ρz, ρ > 0 Reflections in vertical rays e.g. z → −¯ z Inversions in semi-circular lines e.g. z → 1/¯ z

1

i

Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 11 / 19

SLIDE 23

The Three Reflections Theorem Statement

The Three Reflections Theorem

The following hold in each of the three geometries E2, S2 and H2. Theorem (Characterisation of lines) The set of points equidistant from a pair of distinct points P and Q is a line. Reflection in this line exchanges P and Q. P Q Conversely, every line is the set of points equidistant from a suitably chosen 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

SLIDE 24

The Three Reflections Theorem Proof

Step 1: three points determine an isometry

Lemma Any point P is uniquely determined by its distances to three non-collinear points A, B, C. Consequently, any isometry is completely determined by the images of any three non-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

SLIDE 25

The Three Reflections Theorem Proof

Step 1: three points determine an isometry

Lemma Any point P is uniquely determined by its distances to three non-collinear points A, B, C. Consequently, any isometry is completely determined by the images of any three non-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

SLIDE 26

The Three Reflections Theorem Proof

Step 1: three points determine an isometry

Lemma Any point P is uniquely determined by its distances to three non-collinear points A, B, C. Consequently, any isometry is completely determined by the images of any three non-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

SLIDE 27

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 = φ(A), reflect in the line equidistant from A and φ(A).

2

If B′ = φ(B), reflect in the line equidistant from B′ and φ(B).

3

If C′′ = φ(C), reflect in the line equidistant from C′′ and φ(C). A B C φ(A) φ(B) φ(C) The product of these reflections must be φ, because it co-incides

n A, B, C.

Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 14 / 19

SLIDE 28

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 = φ(A), reflect in the line equidistant from A and φ(A).

2

If B′ = φ(B), reflect in the line equidistant from B′ and φ(B).

3

If C′′ = φ(C), reflect in the line equidistant from C′′ and φ(C). A B C B′ C′ φ(A) φ(B) φ(C) The product of these reflections must be φ, because it co-incides

n A, B, C.

Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 14 / 19

SLIDE 29

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 = φ(A), reflect in the line equidistant from A and φ(A).

2

If B′ = φ(B), reflect in the line equidistant from B′ and φ(B).

3

If C′′ = φ(C), reflect in the line equidistant from C′′ and φ(C). A B C B′ C′ C′′ φ(A) φ(B) φ(C) The product of these reflections must be φ, because it co-incides

n A, B, C.

Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 14 / 19

SLIDE 30

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 = φ(A), reflect in the line equidistant from A and φ(A).

2

If B′ = φ(B), reflect in the line equidistant from B′ and φ(B).

3