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A line in **R***P*^{2} has equation *ax* + *by* + *cz* = 0 and so it is determined by the triple (*a*, *b*, *c*). Note that any non-zero multiple of this triple will determine the same line.

Hence the set of all lines: { [*a*, *b*, *c*] | *a*, *b*, *c* are homogeneous coordinates } is another copy of **R***P*^{2}. This is called the *dual space*.

The so-called *duality* comes about because of the symmetry between the homogeneous coordinates of a point and of a line in the above equation.

Any theorem in projective geometry then gives a theorem in this dual space which can be translated into a new theorem by using the correspondence:

Ordinary space | Dual space |

Line | Point |

Point | Line |

Meet of lines | Join of points |

Join | Meet |

We can state Desargues' Theorem as:

*If two triangles have the joins of corresponding vertices concurrent then the meets of corresponding sides are collinear.*

and so its dual is:

*If two triangles have the meets of corresponding sides collinear then the joins of corresponding vertices are concurrent.*

(This is the *converse* of the theorem.)

**Remark**

Let *V* be a vector space over a field *F*. A *linear functional* on *V* is a linear map from *V* to the field *F*. The space of all linear functionals on *V* is called the *dual space* *V** of *V*.

Then the dual of the projective space *P*(*V*) is *P*(*V**).

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