Non-Euclidean

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Non-Euclidean
Geometry
Part I
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What is Euclidean Geometry?

• It is an axiom system about
• points
• lines
• consisting of five axioms
• two points determine a unique line
• any terminated line may be extended indefinitely
• a circle may be drawn with any given point as
  center and any given radius
• all right angles are equal

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and

• 5. If two lines lying in a plane are met by
  another line, and if the sum of the internal
  angles on one side is less than two right angles,
  then the lines will meet if extended sufficiently
  on the side on which the sum of the angles is
  less than two right angles.

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Parallel Postulate

• Equivalent formulations of Euclids Fifth
• If a line intersects one of two parallels, it
  will intersect the other.
• Lines parallel to the same line are parallel to
  each other.
• Two lines which intersect cannot be parallel to
  the same line.
• and

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Playfair formulation of the parallel postulate

• Given a line L and a point P not on L, there
  exists a unique line parallel to L through the
  point P.

P
L
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If not the parallel postulate, then what?

• Girolamo Saccheri (1667-1733)
• For quadrilateral ABCD, with right angles A,B and
  ADBC, one of these holds
• C and D are both right angles
• C and D are both obtuse angles
• C and D are both acute angles.
• Johann Lambert (1728-1777)
• With ?Playfair, proved existence of an absolute
  unit of length, and considered this a
  contradiction.

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Other Non Euclidean Geometers

• Gauss (1777-1855)
• Bolyai (1802-1860)
• Lobachevsky (1792-1856)
• Riemann (1826-1866)

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• Lobachevskian or Hyperbolic Geometry
• Given a line L and a point P not on L, there
  exists a unique line parallel to L through the
  point P.
• Riemannian or Elliptic Geometry
• Given a line L and a point P not on L, there
  exists a unique line parallel to L through the
  point P.

NO
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Model for Elliptic Geometry

• On the surface of a sphere,
• points are antipodal point pairs
• lines are great circles (the shortest distance
  between two points)
• Any pair of lines must intersect.
• Saccheri Hypothesis is angles C and D are obtuse.
• Angle sum of a triangle is greater than p.
• Triangle area is proportional to excess of angle
  sum.

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Any pair of lines intersect
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Saccheri Hypothesis is angles C and D are obtuse.
C
D
A
B
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Angle sum of a triangle is greater than p.
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Triangle area is proportional to excess of angle
sum.
Surface area of unit sphere 4p
Surface area of hemisphere 2p
Surface area of lune of angle A 2A
When a third line is drawn, a triangle ABC is
formed.
The angles at B and C mark two other lunes of
area 2B and 2C, respectively.
The triangle is part of each lune. Call the
triangles area K
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Lune A is divided into the triangle of area K and
an area I.
Lune B is divided into the triangle of area K and
an area II.
Lune C is divided into the triangle of area K and
an area III.
The areas IIIIIIK 2p
The areas of Lunes A,B and C sum to 3KIIIIII
2 (ABC).
Subtracting the first equation from the second
yields 2K2(ABC)-2p,
or K (ABC)-p.
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Model for Hyperbolic Geometry

• On the surface of a pseudosphere,
• points
• a line is the shortest path between two points on
  the line
• There are multiple lines through a point P not on
  a line L that do not intersect L
• Saccheri Hypothesis is angles C and D are acute
• Angle sum of a triangle is less than p
• Triangle area is proportional to defect of angle
  sum.

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Curvature

• The surface of a sphere has constant positive
  curvature
• The surface of a pseudosphere has constant
  negative curvature