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Subject: "Projective Geometry"     Previous Topic | Next Topic
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Conferences The CTK Exchange College math Topic #676
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gd88
Member since Mar-2-08
Mar-29-08, 11:28 AM (EST)
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"Projective Geometry"
 
   The Pojective Geometry is a "Non Euclidean Geometry"?
When I say "Non Euclidean Geometry" I think Hiperbolic Geometry or Elliptic Geometry: Projective Geometry belongs to this set of geometry or not?

Thanks


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alexb
Charter Member
2212 posts
Mar-29-08, 11:38 AM (EST)
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1. "RE: Projective Geometry"
In response to message #0
 
   Absolutely! Projective geometry in which any two lines intersect is non-Euclidean.

Hyperbolic and Elliptic geometries came from different interpretations of the Fifth Postulate and this is why the are commonly labeled non-Euclidean. Projective geometry was an independent development but is as much non-Euclidean as the other two.

In addition, any non-discrete geometry is also non-Euclidean. So geometries may be non-Euclidean but for different reasons.

Quite often the Taxicab geometry is presented as one of the variety. I believe it's a misnomer. In fact, I do not think it's a geometry at all. It's a metric space, yes, but not a geometry. It violates the most common requirement of having a single line through any two points.


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gd88
Member since Mar-2-08
Mar-29-08, 12:20 PM (EST)
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2. "RE: Projective Geometry"
In response to message #1
 
   Ok.
Projective Geometry, Hyperbolic Geometry and Elliptic Gemetry are non-Euclidean Geometry.

And Affine Geometry? Is Affine Geometry a non-Euclidean Geometry?


P.S.
I'm sorry for the quality of my questions, but I can't speak english very well.


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alexb
Charter Member
2212 posts
Mar-29-08, 12:33 PM (EST)
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3. "RE: Projective Geometry"
In response to message #2
 
   Affine geometry is also non-Euclidean. E.g., in affine geometry the is no notion of angle measure. There are no right angles.

Euclidean geometry is based on five postulates and some common notions. There are a few axioms Euclid assumed implicitly without mentioning, e.g. that a straight line cuts the plane into two parts such that is a point from one is joint to a point on the other, the segment will cross the line. Or that two circles under certain conditions intersect in two points.

You should look whether a geometry violates any of the five postulates, although, as I said, I would not call "a geometry" a theory that allows more than one line through two points.


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gd88
Member since Mar-2-08
Mar-29-08, 12:50 PM (EST)
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4. "RE: Projective Geometry"
In response to message #3
 
   Ok.
Thanks very much.


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