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 . The second is suitable for an exhaustive study of the simple incidence properties of plane geometry.
In the projective pl...
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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 . The second is suitable for an exhaustive study of the simple incidence properties of plane geometry.
In the projective plane P, a point x is represented by the homogeneous coordinate (x1, x2, x3). If we think of (x1, x2, x3) as a point in real space R with the third value of the homogeneous coordinate as a value in the z direction, then P can be visualized as R.
A line in P 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:
Using vector notation we may instead write
The equation k(xℓ) = 0 sweeps out a plane that goes...
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