Course MT3818 Topics in Geometry

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## Duality

A line in RP2 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 RP2. 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

Example

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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JOC March 2003