# The Three Reflections Theorem

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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 / 19

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 / 19

3 The Three Two-dimensional Geometries The Euclidean plane Euclidean The Euclidean plane is with the Euclidean distance d ( (x 1, y 1 ),(x 2, y 2 ) ) = E 2 = {(x, y) x, y R}, (x 1 x 2 ) 2 + (y 1 y 2 ) 2. d (x 2, y 2 ) (x 1, y 1 ) Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

4 Arc length The Three Two-dimensional Geometries Euclidean If γ : [a, b] E 2 is a smooth curve then length(γ) = b a ds, where ds 2 = dx 2 + dy 2 is the infinitesimal metric. γ(b) γ(a) γ (t) = ( dx dt ) 2 ( ) 2 + dy dt 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 / 19

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 / 19

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 / 19

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 / 19

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 / 19

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 / 19

10 The Three Two-dimensional Geometries Spherical geometry Spherical Restrict the 3-dimensional Euclidean metric to the unit sphere S 2 in R 3. ds 2 = dx 2 + dy 2 + dz 2 Arc length on S 2 is given by (3d) Euclidean arc length. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

11 The Three Two-dimensional Geometries Lines in spherical geometry Spherical Lines in spherical geometry are great circles: the intersection of a plane 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 / 19

12 The Three Two-dimensional Geometries Lines in spherical geometry Spherical Lines in spherical geometry are great circles: the intersection of a plane 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 / 19

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 / 19

14 The Three Two-dimensional Geometries Hyperbolic Hyperbolic geometry: the upper half plane model Hyperbolic geometry may be modelled by the upper half plane H 2 = {(x, y) R 2 y > 0}, with metric ds 2 = dx 2 + dy 2 y 2. The vectors shown all have the same hyperbolic length. Hyperbolic angle in H 2 co-incides with Euclidean angle. Other models exist, including the conformal disc model. y x Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

15 The Three Two-dimensional Geometries Hyperbolic Lines in the upper half plane model A line in H 2 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 / 19

16 The Three Two-dimensional Geometries Hyperbolic Lines in the upper half plane model A line in H 2 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 / 19

17 The Three Two-dimensional Geometries Hyperbolic Lines in the upper half plane model A line in H 2 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 / 19

18 The Three Two-dimensional Geometries Hyperbolic Hyperbolic isometries The metric ds 2 = dx 2 + dy 2 y 2 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 i Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

19 The Three Two-dimensional Geometries Hyperbolic Hyperbolic isometries The metric ds 2 = dx 2 + dy 2 y 2 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 i Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

20 The Three Two-dimensional Geometries Hyperbolic Hyperbolic isometries The metric ds 2 = dx 2 + dy 2 y 2 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 i Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

21 The Three Two-dimensional Geometries Hyperbolic Hyperbolic isometries The metric ds 2 = dx 2 + dy 2 y 2 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 i Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

22 The Three Two-dimensional Geometries Hyperbolic Hyperbolic isometries The metric ds 2 = dx 2 + dy 2 y 2 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 i Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

23 The Three Reflections Theorem Statement The Three Reflections Theorem The following hold in each of the three geometries E 2, S 2 and H 2. Theorem (Characterisation of lines) P 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. 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 / 19

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 P C 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 / 19

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. A P Consequently, any isometry is completely determined by the images of any three non-collinear points. Q B C 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 / 19

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. A P Consequently, any isometry is completely determined by the images of any three non-collinear points. Q B C 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 / 19

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 φ(a) φ(c) φ(b) C B The product of these reflections must be φ, because it co-incides on A, B, C. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

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 φ(a) φ(c) φ(b) C B B C The product of these reflections must be φ, because it co-incides on A, B, C. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

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 φ(a) φ(c) φ(b) C C B B C The product of these reflections must be φ, because it co-incides on A, B, C. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

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 If C φ(c), reflect in the line equidistant from C and φ(c). A φ(a) φ(c) φ(b) C C B B C The product of these reflections must be φ, because it co-incides on A, B, C. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

31 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 / 19

32 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 / 19

33 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 / 19

34 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 / 19

35 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 / 19

36 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 / 19

37 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 / 19

38 The sphere Orientation preserving isometries Any two distinct lines in S 2 intersect = every orientation preserving isometry of S 2 is a rotation. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

39 Orientation preserving isometries The hyperbolic plane Three cases: intersecting lines: rotations asymptotic lines: limit rotations (includes horizontal Euclidean translations) ultraparallel lines: hyperbolic translations (includes Euclidean dilations) Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

40 Orientation preserving isometries The hyperbolic plane Three cases: intersecting lines: rotations asymptotic lines: limit rotations (includes horizontal Euclidean translations) ultraparallel lines: hyperbolic translations (includes Euclidean dilations) Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

41 Orientation preserving isometries The hyperbolic plane Three cases: intersecting lines: rotations asymptotic lines: limit rotations (includes horizontal Euclidean translations) ultraparallel lines: hyperbolic translations (includes Euclidean dilations) Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

42 Orientation preserving isometries The hyperbolic plane Three cases: intersecting lines: rotations asymptotic lines: limit rotations (includes horizontal Euclidean translations) ultraparallel lines: hyperbolic translations (includes Euclidean dilations) Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

43 Orientation preserving isometries Orientation preserving isometries, classified by pairs of reflections Spherical intersecting lines rotation disjoint lines Euclidean rotation parallel lines: translation Hyperbolic rotation asymptotic lines: ultraparallel lines: limit rotation translation Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

44 Going further Going further In each geometry, an orientation reversing isometry is a glide reflection. Subgroups of the isometry group lead to quotient surfaces with the given geometry. Euclidean three-space has a Four Reflections Theorem. There are eight model geometries in three dimensions: E 3, S 3, H 3, S 2 E 1, H 2 E 1, Nil, SL 2 R, Solv. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

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