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Part I
What is Euclidean Geometry?
It is an axiom system about
•
•
points
lines
consisting of five axioms
1. two points determine a unique line
2. any terminated line may be extended
indefinitely
3. a circle may be drawn with any given point
as center and any given radius
4. all right angles are equal
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.
Parallel Postulate
Equivalent formulations of “Euclid’s 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
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
If not the parallel postulate,
then what?
• Girolamo Saccheri (1667-1733)
– For quadrilateral ABCD, with right angles
A,B and AD=BC, 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.
Other Non Euclidean Geometers
• Gauss (1777-1855)
• Bolyai (1802-1860)
• Lobachevsky (17921856)
• Riemann (1826-1866)
• 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
NO
the point P.
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 π.
Triangle area is proportional to excess of angle
sum.
Any pair of lines intersect
Saccheri Hypothesis is angles C and
D are obtuse.
D
A
C
B
Angle sum of a triangle is greater
than π.
Triangle area is proportional to excess of
angle sum.
Surface area of unit sphere = 4π
Surface area of hemisphere = 2π
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 triangle’s area K
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 I+II+III+K = 2π
The areas of Lunes A,B and C sum to
3K+I+II+III = 2 (A+B+C).
Subtracting the first equation from the second yields
2K=2(A+B+C)-2π,
or
K = (A+B+C)-π.
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 π
Triangle area is proportional to defect of angle
sum.
Curvature
• The surface of a sphere has constant
positive curvature
• The surface of a pseudosphere has
constant negative curvature