The Real Projective Plane
The specification real simply means that we are considering a
projective plane whose coordinates, if we were to calculate them,
would all be real numbers. So now, what is the projective plane?
Projective geometry is based on the Euclidean geometry that we all
know and love, but it is more generalized. The projective plane can
be thought of as a Euclidean plane with the addition of a line at
infinity. The addition of the line at infinity closes the projective
plane, giving it special properties that the Euclidean plane does not share.
- On the projective plane, any (every) two coplanar lines
intersect at a point. (Parallel lines intersect at the point at
- Similarly, any two planes in projective geometry intersect at a
line. Parallel planes intersect at the line at
- A line on a Euclidean plane can be divided into two segments by
"cutting" at one point. However, a line on a projective plane
needs two distinct points to divide it into two segments. This
is because a line on a projective plane meets itself at the
point at infinity, and is therefore closed. It may be helpful
to visualize the line as a circle -- a circle would need to be
"cut" at two distinct points to form two segments.
- On a line on the projective plane, the order of points is
cyclic, whereas for a line on a Euclidean plane, the order of
points is serial. This is because a line on the
projective plane is closed (as defined above), while a
line on the Euclidean plane is open.
A page about the
projective plane with helpful diagrams.
The Geometry Center's page about the
real projective plane.
Source:"The Real Projective Plane" by
H. S. M. Coxeter (Cambridge University Press, 1960)
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