Hyperbolic Geometry

1
Hyperbolic Geometry

• Chapter 11

2
Hyperbolic Lines and Segments

• Poincaré disk model
• Line circular arc, meets fundamental circle
  orthogonally
• Note
• Lines closer tocenter of fundamentalcircle are
  closer to Euclidian lines
• Why?

3
Poincaré Disk Model

• Model of geometric world
• Different set of rules apply
• Rules
• Points are interior to fundamental circle
• Lines are circular arcs orthogonal to fundamental
  circle
• Points where line meets fundamental circle are
  ideal points -- this set called ?
• Can be thought of as infinity in this context

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Poincaré Disk Model

• Euclids first four postulates hold
• Given two distinct points, A and B, ? a unique
  line passing through them
• Any line segment can be extended indefinitely
• A segment has end points (closed)
• Given two distinct points, A and B, a circle with
  radius AB can be drawn
• Any two right angles are congruent

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Hyperbolic Triangles

• Recall Activity 2 so how do you find ?
  measure?
• We find sum of angles might not be 180?

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Hyperbolic Triangles

• Lines that do not intersect are parallel lines
• What if a triangle could have 3 vertices on the
  fundamental circle?

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Hyperbolic Triangles

• Note the angle measurements
• What can you concludewhen an angle is 0? ?

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Hyperbolic Triangles

• Generally the sum of the angles of a hyperbolic
  triangle is less than 180?
• The difference between the calculated sum and 180
  is called the defect of the triangle
• Calculatethe defect

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Hyperbolic Polygons

• What does the hyperbolic plane do to the sum of
  the measures of angles of polygons?

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Hyperbolic Circles

• A circle is the locus of points equidistant from
  a fixed point, the center
• Recall Activity 11.2

What seems wrong with these results?
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Hyperbolic Circles

• What happens when the center or a point on the
  circle approaches infinity?
• If center could beon fundamentalcircle
• Infinite radius
• Called a horocycle

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Distance on Poincarè Disk Model

• Rule for measuring distance ? metric
• Euclidian distance
• Metric Axioms
• d(A, B) 0 ? A B
• d(A, B) d(B, A)
• Given A, B, C points, d(A, B) d(B, C) ? d(A,
  C)

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Distance on Poincarè Disk Model

• Formula for distance
• Where AM, AN, BN, BM are Euclidian distances

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Distance on Poincarè Disk Model

• Now work through axioms
• d(A, B) 0 ? A B
• d(A, B) d(B, A)
• Given A, B, C points, d(A, B) d(B, C) ? d(A,
  C)

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Circumcircles, Incircles of Hyperbolic Triangles

• Consider Activity 11.6a
• Concurrency of perpendicular bisectors

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Circumcircles, Incircles of Hyperbolic Triangles

• Consider Activity 11.6b
• Circumcircle

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Circumcircles, Incircles of Hyperbolic Triangles

• Conjecture
• Three perpendicular bisectors of sides of
  Poincarè disk are concurrent at O
• Circle with center O, radius OA also contains
  points B and C

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Circumcircles, Incircles of Hyperbolic Triangles

• Note issue of ? bisectors sometimes not
  intersecting
• More on this later

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Circumcircles, Incircles of Hyperbolic Triangles

• Recall Activity 11.7
• Concurrence of angle bisectors

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Circumcircles, Incircles of Hyperbolic Triangles

• Recall Activity 11.7
• Resulting incenter

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Circumcircles, Incircles of Hyperbolic Triangles

• Conjecture
• Three angle bisectors of sides of Poincarè disk
  are concurrent at O
• Circle with center O, radius tangent to one side
  is tangent to all three sides

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Congruence of Triangles in Hyperbolic Plane

• Visual inspection unreliable
• Must use axioms, theorems of hyperbolic plane
• First four axioms are available
• We will find that AAA is now a valid criterion
  for congruent triangles!!

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Parallel Postulate in Poincaré Disk

• Playfairs Postulate
• Given any line l and any point P not on l, ?
  exactly one line on P that is parallel to l
• Definition 11.4Two lines, l and m are parallel
  if the do not intersect

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Parallel Postulate in Poincaré Disk

• Playfares postulate Says ? exactly one line
  through point P, parallel to line
• What are two possible negations to the postulate?
• No lines through P, parallel
• Many lines through P, parallel
• Restate the first Elliptic Parallel Postulate
• There is a line l and a point P not on l such
  that every line through P intersects l

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Elliptic Parallel Postulate

• Examples of elliptic space
• Spherical geometry
• Great circle
• Straight line on the sphere
• Part of a circle with center atcenter of sphere

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Elliptic Parallel Postulate

• Flat map with great circle will often be a
  distorted straight line

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Elliptic Parallel Postulate

• Elliptic Parallel Theorem
• Given any line l and a point P not on l every
  line through P intersects l
• Let line l be the equator
• All other lines (great circles) through any
  pointmust intersect the equator

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Hyperbolic Parallel Postulate

• Hyperbolic Parallel Postulate
• There is a line l and a point P not on l such
  that more than one line through P is parallel
  to l

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Parallel Lines, Hyperbolic Plane

• Lines outside the limiting rays will beparallel
  to line AB
• Calledultraparallel orsuperparallel
  orhyperparallel
• Note line ED is limiting parallel with D at ?

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Parallel Lines, Hyperbolic Plane

• Consider Activity 11.8
• Note the congruent angles, ?DCE ? ?FCD

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Parallel Lines, Hyperbolic Plane

• Angles ?DCE ?FCD are called the angles of
  parallelism
• The angle betweenone of the limitingrays and CD
• Theorem 11.4The two anglesof parallelismare
  congruent

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Hyperbolic Parallel Postulate

• Result of hyperbolic parallel postulateTheorem
  11.4
• For a given line l and a point P not on l, the
  two angles of parallelism are congruent
• Theorem 11.5
• For a given line l and a point P not on l, the
  two angles of parallelism are acute

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The Exterior Angle Theorem

• Theorem 11.6
• If ABC is a triangle in the hyperbolic plane and
  ?BCD is exterior for this triangle, then ? BCD
  is larger than either ? CAB or ? ABC.

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Parallel Lines, Hyperbolic Plane

• Note results of Activity 11.8
• CD is a commonperpendicular tolines AB, HF
• Can be proved inthis context
• If two lines do not intersect then eitherthey
  are limiting parallelsor have a
  commonperpendicular

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Quadrilaterals, Hyperbolic Plane

• Recall results of Activity 11.9
• 90? angles at B and A

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Quadrilaterals, Hyperbolic Plane

• Recall results of Activity 11.10
• 90? angles at B, A, and D only
• Called a Lambert quadrilateral

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Quadrilaterals, Hyperbolic Plane

• Saccheri quadrilateral
• A pair of congruent sides
• Both perpendicular to a third side

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Quadrilaterals, Hyperbolic Plane

• Angles at A and B are base angles
• Angles at E and F aresummit angles
• Note they are congruent
• Side EF is the summit
• You should have foundnot possible to
  constructrectangle (4 right angles)

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Hyperbolic Geometry

• Chapter 11