Incidence (geometry)

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In geometry, the relations of incidence are those such as 'lies on' between points and lines (as in 'point P lies on line L'), and 'intersects' (as in 'line L₁ intersects line L₂', in three-dimensional space). That is, they are the binary relations describing how subsets meet. The propositions of incidence stated in terms of them are statements such as 'any two lines in a plane meet'. This is true in a projective plane, though not true in Euclidean space of two dimensions where lines may be parallel.

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 axioms. 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 projective space starting from linear algebra and homogeneous co-ordinates. Then the propositions of incidence are derived from the following basic result on vector spaces: given subspaces U and V of a vector space W, the dimension of their intersection is at least dim U + dim V − dim W. Bearing in mind that the dimension of the projective space P(W) associated to W is dim W − 1, but that we require an intersection of subspaces of dimension at least 1 to register in projective space (the subspace {0} being common to all subspaces of W), we get the basic proposition of incidence in this form: linear subspaces L and M of projective space P meet provided dim L + dim M is at least dim P.^[1]

Intersection of a pair of lines

Let L₁ and L₂ be a pair of lines, both in a projective plane and expressed in homogeneous coordinates:

$L_{1}:[m_{1}:b_{1}:1]_{L}$

$L_{2}:[m_{2}:b_{2}:1]_{L}$

where m₁ and m₂ are slopes and b₁ and b₂ are y-intercepts. Moreover let g be the duality mapping

$g:[x:y:z]\mapsto [x:-z:y]$

which maps lines onto their dual points. Then the intersection of lines L₁ and L₂ is point P₃ where

$P_{3}=g(L_{1}\times L_{2}).$

Determining the line passing through a pair of points

Let P₁ and P₂ be a pair of points, both in a projective plane and expressed in homogeneous coordinates:

$P_{1}:[x_{1}:y_{1}:z_{1}],$

$P_{2}:[x_{2}:y_{2}:z_{2}].$

Let g⁻¹ be the inverse duality mapping:

$g^{{-1}}:[x:y:z]\mapsto [x:z:-y]$

which maps points onto their dual lines. Then the unique line passing through points P₁ and P₂ is L₃ where

$L_{3}=g^{{-1}}(P_{1}\times P_{2}).$

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 L₁, L₂, and L₃; 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 [m:b:1]_L with m being slope and b being the y-intercept, then concurrency can be restated as

$L_{1}\times L_{2}\equiv L_{2}\times L_{3}\equiv L_{3}\times L_{1}.$

Theorem. Three lines L₁, L₂, and L₃ 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

$\langle L_{1},L_{2},L_{3}\rangle =L_{1}\cdot L_{2}\times L_{3}=0.$

Proof. Letting g denote the duality mapping, then

$L_{1}\cap L_{2}=gL_{1}\times gL_{2}.\qquad \qquad (1)$

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

$(gL_{1}\times gL_{2})\cdot gL_{3}=0\qquad \qquad (3)$

but g distributes with respect to the cross product, so that

$g(L_{1}\times L_{2})\cdot gL_{3}=0,$

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

$(L_{1}\times L_{2})\cdot L_{3}=\langle L_{1},L_{2},L_{3}\rangle =0.$

Q.E.D.

Collinearity

The dual of concurrency is collinearity. Three points P₁, P₂, and P₃ 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:

$<P_{1},P_{2},P_{3}>=P_{1}\cdot P_{2}\times P_{3}=0$

which is more symmetrical and whose computation is straightforward.

If P₁ : (x₁ : y₁ : z₁), P₂ : (x₂ : y₂ : z₂), and P₃ : (x₃ : y₃ : z₃), then P₁, P₂, and P₃ are collinear if and only if

$\left|{\begin{matrix}x_{1}&y_{1}&z_{1}\\x_{2}&y_{2}&z_{2}\\x_{3}&y_{3}&z_{3}\end{matrix}}\right|=0,$

i.e. if and only if the determinant of the homogeneous coordinates of the points is equal to zero.

References

↑ Joel G. Broida & S. Gill Williamson (1998) A Comprehensive Introduction to Linear Algebra, Theorem 2.11, p 86, Addison-Wesley ISBN 0-201-150065-5 . The theorem says dim (L + M) = dim L + dim M − dim (L ∩ M). Thus dim L + dim M > dim P implies dim (L ∩ M) > 0

Harold L. Dorwart (1966) The Geometry of Incidence, Prentice Hall.

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