Mr. Young

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EUCLID
Mr. Youngs Geometry Classes, Spring 2005
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Outline
1. What is Euclidean Geometry?
2. What is Non-Euclidean Geometry?
3. Spherical Geometry
4. Spherical Geometry A Real World Application
5. Euclidean vs. Spherical Geometry
6. Other Geometries
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1. What is Euclidean Geometry?

• Euclidean Geometry is the Geometry that is taught
  in High School Geometry Classes

• It is based primarily on a book called The
  Elements written by a Greek Mathematician named
  Euclid who lived from about 325-265 B.C.

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1. What is Euclidean Geometry?

• Euclidean Geometry deals with points, lines and
  planes and how they interact to make more complex
  figures.

• Euclids Postulates define how the points, lines,
  and planes interact with each other.

Remember A Postulate is statement that is
assumed to be true.
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1. What is Euclidean Geometry?

• Euclids First Four Postulates are as follows

1. Through any two points there is exactly one
line
2. Through any three points not on the same line
there is exactly one plane
3. A line contains at least two points
4. All right angles are congruent
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1. What is Euclidean Geometry?

• Euclids Fifth Postulate, called the Parallel
  Postulate seems obvious, but is the source of
  much debate.

5. Given a line and a point not on that line,
there is exactly one line through the point that
is parallel to the line

• You can say that Euclidean Geometry is Geometry
  in which the parallel postulate holds.

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2. What is Non-Euclidean Geometry?

• Non-Euclidean Geometry is any Geometry that uses
  a different set of postulates than Euclid used.

• Most of the time Non-Euclidean Geometry is
  Geometry in which the Parallel Postulate does not
  hold to be true.

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2. What is Non-Euclidean Geometry?

• If the parallel postulate is not true that means
  that given a line and a point not on the line
  there is NOT exactly one line through the point
  which is parallel to the line.

• How is this possible?

Remember that points, lines, and planes are
undefined terms. Their meaning comes only from
postulates. So if you change the postulates you
can change the meaning of points, lines, and
planes, and how they interact with each other.

• This is most easily seen by example

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3. Spherical Geometry

• The main difference between Spherical Geometry
  and Euclidean Geometry is that instead of
  describing a plane as a flat surface a plane is a
  sphere.

• A line is a great circle on the sphere. A great
  circle is any circle on a sphere that has the
  same center as the sphere.

• Points are exactly the same, just on a sphere.

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3. Spherical Geometry

• Are Euclids Postulates true in Spherical
  Geometry?

1. Through any two points there is exactly one
line
TRUE
2. Through any three points not on the same line
there is exactly one plane
TRUE
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3. Spherical Geometry

• Are Euclids Postulates true in Spherical
  Geometry?

3. A line contains at least two points
TRUE
4. All right angles are congruent
TRUE
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3. Spherical Geometry

• Is the Parallel Postulate true in Spherical
  Geometry?

5. Given a line and a point not on that line how
many lines can be drawn through the point that
are parallel to the line?
NONE, Therefore the Parallel Postulate is FALSE
in Spherical Geometry
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3. Spherical Geometry

• Is the Parallel Postulate true in Spherical
  Geometry?

5. Given a line and a point not on that line how
many lines can be drawn through the point that
are parallel to the line?
NONE, Therefore the Parallel Postulate is FALSE
in Spherical Geometry
Common Mistake Except for the circle in the
middle, these horizontal circles do not share a
center with the sphere and are therefore can not
be considered parallel lines, even though they
appear to be parallel.
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3. Spherical Geometry

• Other strange things happen in Spherical Geometry

• Lines always intersect at 2 points, not one.

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3. Spherical Geometry

• Other strange things happen in Spherical Geometry

• In the diagram below B is between A and C, but...

A is between B and C, and...
C is between A and B.
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3. Spherical Geometry

• Other strange things happen in Spherical Geometry

• The angles in a triangle dont have to add to 180º

• In the diagram below ?ABC has 3 right angles,
  which add to 270.

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4. Spherical Geometry A Real World Application

• If Spherical Geometry is so strange why do we
  even bother studying it?

Because the Earth is a Sphere.

• Euclidean geometry can not be used to model the
  Earth because it is a sphere.

• Instead of the Cartesian coordinates used in
  Euclidean Geometry Longitude and Latitude are
  used as to define position of points on the
  Earth.

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4. Spherical Geometry A Real World Application

• Lines of Longitude are great Circles running
  between the North and South Poles.

• The Center Longitude is called the Prime
  Meridian

Degrees West
Degrees East

• Longitude is measured in degrees East or West
  from the prime meridian.

Prime Meridian, 0
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4. Spherical Geometry A Real World Application

• Lines of Latitude are parallel horizontal
  circles, but not great Circles

• The Center Latitude is called the equator

Degrees North
Equator, 0

• Latitude is measured in degrees North or South
  from the equator

Degrees South
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4. Spherical Geometry A Real World Application

• Any Location on the Earth can be found with its,
  a latitude and longitude.

Newberry, FL Lat. 29.6 N Long. 82.6 W
Newberry, FL
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4. Spherical Geometry A Real World Application

• Latitude

• Longitude

90 N
60 N
30 N
180 E, 0
150 W
0
60 E
90 W
60 W
30 W
90 E
30 E
120 W
120 E
150 E
30 S
60 S
90 S
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4. Spherical Geometry A Real World Application

• This picture shows the angles that define the
  degrees for longitude and latitude

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4. Spherical Geometry A Real World Application

• Astronomers use a similar concept to define the
  position of stars and other objects in the sky.

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4. Euclidean vs. Spherical Geometry
Euclid vs. The Sphere
Which Geometry is right?
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4. Euclidean vs. Spherical Geometry

• Spherical Geometry must be used in some cases
• Finding long distances for flights, driving or
  sailing.
• Predicting paths of weather
• Map making

• But Euclidean Geometry works well in most cases
• Finding most distances or lengths
• Most everyday activities that require geometry
  like construction, drawing, etc.

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4. Euclidean vs. Spherical Geometry
Euclid vs. The Sphere
Which Geometry is right?

• Neither Geometry is the right Geometry, but
  since Euclidean Geometry works in most cases and
  is simplest, it is taught in schools.

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5. Other Geometries

• Spherical Geometry is just one Example of
  Non-Euclidean Geometry

• Any Geometry that starts with a different set of
  postulates is Non-Euclidean.

• Some other Geometries have practical applications
  and some are just theoretical

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5. Other Geometries

• Hyperbolic Geometry is used to model space since
  Einsteins theories imply that space is curved.

• Here is a model of a three dimensional hyperbolic
  curve, that Hyperbolic Geometry would be based
  on.

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Summary

• Euclidean Geometry is not the only type of
  Geometry.

• Spherical Geometry is one example of
  Non-Euclidean Geometry that has definite
  practical applications.

• Euclidean Geometry sufficiently describes the
  world that most of us deal with day to day, so it
  is the primary Geometry studied in School.

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The End
Want to learn more about Non-Euclidean geometry?
Do a web search. Or better yet major in Math and
take a course on it in college.