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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 Outline The

  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

  2. Outline The Three Two-dimensional Geometries 1 Euclidean Spherical Hyperbolic The Three Reflections Theorem 2 Statement Proof Orientation preserving isometries 3 Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 2 / 19

  3. The Three Two-dimensional Geometries Euclidean The Euclidean plane The Euclidean plane is E 2 = { ( x , y ) | x , y ∈ R } , with the Euclidean distance � � � ( x 1 − x 2 ) 2 + ( y 1 − y 2 ) 2 . d ( x 1 , y 1 ) , ( x 2 , y 2 ) = ( x 2 , y 2 ) d ( x 1 , y 1 ) Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 3 / 19

  4. The Three Two-dimensional Geometries Euclidean Arc length If γ : [ a , b ] → E 2 is a smooth curve then � b length ( γ ) = ds , a where ds 2 = dx 2 + dy 2 is the infinitesimal metric . γ ( b ) �� dx � � 2 � 2 + dy γ ( a ) | γ ′ ( t ) | = dt 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 2009 4 / 19

  5. The Three Two-dimensional Geometries Euclidean Euclidean isometries Definition An isometry is a distance preserving map. Euclidean examples Orientation preserving: Orientation reversing: Translations Reflections Rotations Glide reflections Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 5 / 19

  6. The Three Two-dimensional Geometries Euclidean Euclidean isometries Definition An isometry is a distance preserving map. Euclidean examples Orientation preserving: Orientation reversing: Translations Reflections Rotations Glide reflections Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 5 / 19

  7. The Three Two-dimensional Geometries Euclidean Euclidean isometries Definition An isometry is a distance preserving map. Euclidean examples Orientation preserving: Orientation reversing: Translations Reflections Rotations Glide reflections Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 5 / 19

  8. The Three Two-dimensional Geometries Euclidean Euclidean isometries Definition An isometry is a distance preserving map. Euclidean examples Orientation preserving: Orientation reversing: Translations Reflections Rotations Glide reflections Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 5 / 19

  9. The Three Two-dimensional Geometries Euclidean Euclidean isometries Definition An isometry is a distance preserving map. Euclidean examples Orientation preserving: Orientation reversing: Translations Reflections Rotations Glide reflections Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 5 / 19

  10. The Three Two-dimensional Geometries Spherical Spherical geometry Restrict the 3-dimensional Euclidean metric ds 2 = dx 2 + dy 2 + dz 2 to the unit sphere S 2 in R 3 . Arc length on S 2 is given by (3d) Euclidean arc length. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 6 / 19

  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 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

  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 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

  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

  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 y The vectors shown all have the same hyperbolic length. Hyperbolic angle in H 2 co-incides x ������������������ ������������������ with Euclidean angle. ������������������ ������������������ ������������������ ������������������ Other models exist, including the conformal disc model . Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 9 / 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 2009 10 / 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 2009 10 / 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 2009 10 / 19

  18. The Three Two-dimensional Geometries Hyperbolic Hyperbolic isometries The metric ds 2 = dx 2 + dy 2 is preserved by y 2 Horizontal translations Reflections in vertical rays e.g. z �→ − ¯ z �→ z + c , c real z Euclidean dilations Inversions in semi-circular lines e.g. z �→ 1 / ¯ z �→ ρ z , ρ > 0 z i 1 ������������������������������� ������������������������������� ������������������������������� ������������������������������� ������������������������������� ������������������������������� Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June 2009 11 / 19

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