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Historically, Projective Geometry was introduced in order to make the propositions of incidence true (without exceptions such as are caused by parallels). From the point of view of Synthetic Geometry it was considered that projective geometry ''should be'' developed using such propositions as Axiom s. This turns out to make a major difference only for the projective plane (for reasons to do with Desargues' Theorem ).

The modern approach is to define s L and '''M''' of projective space '''P''' meet provided dim L + dim '''M''' is at least dim '''P'''.


INTERSECTION OF A PAIR OF LINES

Let ''L''1 and ''L''2 be a pair of lines, both in a projective plane and expressed in homogeneous coordinates:
: L_1 : : b_1 : 1 _L

: L_2 : : b_2 : 1 _L
where ''m''1 and ''m''2 are Slope s and ''b''1 and ''b''2 are Y-intercept s. Moreover let ''g'' be the Duality Mapping
: g : : y : z \mapsto : -z : y
which maps lines onto their dual points. Then the intersection of lines ''L''1 and ''L''2 is point ''P''3 where
: P_3 = g(L_1 imes L_2).


DETERMINING THE LINE PASSING THROUGH A PAIR OF POINTS

Let ''P''1 and ''P''2 be a pair of points, both in a projective plane and expressed in homogeneous coordinates:
: P_1 : : y_1 : z_1 ,

: P_2 : : y_2 : z_2 .
Let ''g''−1 be the inverse duality mapping:
: g^{-1} : : y : z \mapsto : z : -y
which maps points onto their dual lines. Then the unique line passing through points ''P''1 and ''P''2 is ''L''3 where
: L_3 = g^{-1}(P_1 imes P_2).


CHECKING FOR INCIDENCE OF A LINE ON A POINT

Given line ''L'' and point ''P'' in a projective plane, and both expressed in homogeneous coordinates, then ''P''⊂''L'' If And Only If the dual of the line is Perpendicular to the point (so that their Dot Product is zero); that is, if
: gL \cdot P = 0
where ''g'' is the duality mapping.

An equivalent way of checking for this same incidence is to see whether
: L \cdot g^{-1} P = 0
is true.


CONCURRENCE

Three lines in a projective plane are concurrent if all three of them intersect at one point. That is, given lines ''L''1, ''L''2, and ''L''3; these are concurrent if and only if
: L_1 \cap L_2 = L_2 \cap L_3 = L_3 \cap L_1.
If the lines are represented using homogeneous coordinates in the form {Link without Title} ''L'' with ''m'' being slope and ''b'' being the y-intercept, then concurrency can be restated as
: L_1 imes L_2 \equiv L_2 imes L_3 \equiv L_3 imes L_1.

''Theorem.'' Three lines ''L''1, ''L''2, and ''L''3 in a projective plane and expressed in homogeneous coordinates are concurrent if and only if their Scalar Triple Product is zero, viz. if and only if
: = L_1 \cdot L_2 imes L_3 = 0.
''Proof.'' Letting ''g'' denote the duality mapping, then
: L_1 \cap L_2 = gL_1 imes gL_2. \qquad \qquad (1)
The three lines are concurrent if and only if
: (L_1 \cap L_2) \subset L_3.
According to the Previous Section , the intersection of the first two lines is a subset of the third line if and only if
: gL_3 \cdot (L_1 \cap L_2) = 0 \qquad \qquad (2)
Substituting equation (1) into equation (2) yields
: (gL_1 imes gL_2) \cdot gL_3 = 0 \qquad \qquad (3)
but ''g'' distributes with respect to the Cross Product , so that
: g(L_1 imes L_2) \cdot gL_3 = 0,
and ''g'' can be shown to be isomorphic w.r.t. the dot product, like so:
: A \cdot B = gA \cdot gB
so that equation (3) simplifies to
: (L_1 imes L_2) \cdot L_3 = = 0.
'' Q.E.D. ''


COLLINEARITY

The dual of concurrency is collinearity. Three points ''P''1, ''P''2, and ''P''3 in the projective plane are collinear if they all lie on the same line. This is true If And Only If
: P_1.P_2 \equiv P_2.P_3 \equiv P_3.P_1,
but if the points are expressed in homogeneous coordinates then these three different equations can be collapsed into one equation:
: = P_1 \cdot P_2 imes P_3 = 0
which is more symmetrical and whose computation is straightforward.

If ''P''1 : (''x''1 : ''y''1 : ''z''1), ''P''2 : (''x''2 : ''y''2 : ''z''2), and ''P''3 : (''x''3 : ''y''3 : ''z''3), then ''P''1, ''P''2, and ''P''3 are collinear if and only if