Lines and points in the projective plane

  • #1
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im reading a paper about projective geometry and i encountered to a section which describes a duality between points and line:
"Looking once again at figure 2, the similarities between points and lines are striking. Their representations, for example, are identical, and the formula for the intersection of two lines is the same as the formula for the connecting line between two points. These observations are not the result of coincidence but are rather a result of the duality that exists between points and lines in the projective plane. In other words, any theorem or statement that is true for the projective plane can be reworded by substituting points for lines and lines for points, and the resulting statement will be true as well."
if you can substitute the definition for a line to a point than what according to the projective geometry distinguish a line from a point?

here's the link for the quote:
http://robotics.stanford.edu/~birch/projective/node8.html
 

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  • #2
Hurkyl
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Same way I tell clockwise from counterclockwise; you make a definition and stick to it.

One way we might represent the projective plane is by the tuple (P, L, ...) where P is the set of points, L is the set of lines, and ... is where we specify what the operations (like incidence) are. Dualism just says that the tuple (L, P, ...) is also a projective plane. While all projective planes are isometric, they aren't the same.

Because of the isometry, any theorem in (P, L, ...) must also be a theorem in (L, P, ...), which is why the duality works.
 
  • #3
HallsofIvy
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if you can substitute the definition for a line to a point than what according to the projective geometry distinguish a line from a point?
And what, exactly, are the definitions of "line" and "point" in projective geometry? My experience has always been that they are left as undefined terms. Yes, it true that if you swap the words "line" and "point" in a theorem in projective geometry, you get another (true) theorem.

For example, it is certainly true that "two points determine a line" (given any two points, there exist exactly one line containing both). In projective geometry it is also true that "two lines determine a point"- given any two lines, there exist exactly one point lying on both lines. That's not true in Euclidean geometry where two lines may not intersect.
 
  • #4
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so hallsofivy in pp lines always intersect?
 
  • #5
HallsofIvy
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Yes, all lines in projective geometry intersect.

Sometimes, "points at infinity" are added to Euclidean geometry to get a projective geometry. The point of intersection of two (Euclidean) parallel lines is a "point at infinity".
 
  • #6
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yes this point at infinity is coverde in the text ive given.
 
  • #7
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another question popped into my mind in the website im reading it says that to transform a point in the projective plane into euclidean you need to divide by the third coordinate for example in the projective plane the point is represented by the coordinates (x,y,w) so in the euclidean plane this point's coordinates are (x/w,y/w).

i want to know why is this?
is this a definition or there is proof to this arguement?

strangely, in the webpage they dont say why.
 

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