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## The projective plane

The projective plane is the projective space . A point of is represented by a 3-vector . A line is also represented by a 3-vector. A point is located on a line if and only if

 (B1)

This equation can however also be interpreted as expressing that the line passes through the point . This symmetry in the equation shows that there is no formal difference between points and lines in the projective plane. This is known as the principle of duality. A line passing through two points and is given by their vector product . This can also be written as
 (B2)

The dual formulation gives the intersection of two lines. All the lines passing through a specific point form a pencil of lines. If two lines and are distinct elements of the pencil, all the other lines can be obtained through the following equation:
 (B3)

for some scalars and . Note that only the ratio is important.

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Marc Pollefeys 2002-11-22