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- See real projective plane and complex projective plane, for the cases met as manifolds of respective dimension 2 and 4

In mathematics, a projective plane has two possible definitions, one of them coming from linear algebra, and another (which is more general) coming from axiomatic and finite geometry. The first definition quickly produces planes that are homogeneous spaces for some of the classical groups, including the real projective plane $mathbb\{P\}^2$. The second is suitable for an exhaustive study of the simple incidence properties of plane geometry.

In the projective plane P^{2}, a point x is represented by the homogeneous coordinate (x_{1}, x_{2}, x_{3}). If we think of (x_{1}, x_{2}, x_{3}) as a point in real space R^{3} with the third value of the homogeneous coordinate as a value in the z direction, then P^{2} can be visualized as R^{3}.

A line in P^{2} can be represented by the equation ax + by + c = 0. If we treat a, b, and c as the column vector l and x, y, 1 as the column vector x then the equation above can be written in matrix form as:

- x
^{T}l = 0 or l^{T}x = 0.

Using vector notation we may instead write

- x ⋅ l = 0 or l ⋅ x = 0.

The equation k(x^{T}l) = 0 sweeps out a plane that goes through zero in R^{3} and k(x) sweeps out a ray, again going through zero. The plane and ray are subspaces in R^{3}, which always go through zero.

In P^{2} the equation of a line is ax + by + c = 0 and this equation can represent a line on any plane parallel to the x, y plane by multiplying the equation by k.

If z = 1 we have a normalized homogeneous coordinate. All points that have z = 1 create a plane. Let's pretend we are looking at that plane and there are two parallel lines drawn on the plane. From where we are standing we can see only so much of the plane (the area outlined in red). If we walk away from the plane along the z axis we can see more of the plane. In our field of view original points have moved. We can reflect this movement by dividing the homogeneous coordinate by a constant. In the image to the right we have divided by 2 so the z value now becomes 0.5. If we walk far enough away what we are looking at becomes a point in the distance. As we walk away we see more and more of the parallel lines. The lines will meet at a line at infinity (a line that goes through zero on the plane at z = 0). Lines on the plane when z = 0 are ideal points. The plane at z = 0 is the line at infinity.

The homogeneous point (0, 0, 0) is where all the real points go when you're looking at the plane from an infinite distance, a line on the z = 0 plane is where parallel lines intersect.

In the equation x^{T}l = 0 there are two column vectors. You can keep either constant and vary the other. If we keep the point constant x and vary the coefficients l we create new lines that go through the point. If we keep the coefficients constant and vary the points that satisfy the equation we create a line. We look upon x as a point because the axes we are using are x, y, and z. If we instead plotted the coefficients using axis marked a, b, c points would become lines and lines would become points. If you prove something with the data plotted on axis marked x, y, and z the same argument can be used for the data plotted on axis marked a, b, and c. That is duality.

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Last updated on Monday October 06, 2008 at 08:07:29 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Monday October 06, 2008 at 08:07:29 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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