Incidence (geometry)

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.

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::$L\_1\; :\; [m\_1\; :\; b\_1\; :\; 1]\; \_L$

:$L\_2\; :\; [m\_2\; :\; b\_2\; :\; 1]\; \_L$where "m"₁ and "m"₂ are slopes and "b"₁ and "b"₂ 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 "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::$P\_1\; :\; [x\_1\; :\; y\_1\; :\; z\_1]\; ,$

:$P\_2\; :\; [x\_2\; :\; y\_2\; :\; z\_2]\; .$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 "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 ["m":"b":1] _"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:"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:"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::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:i.e. if and only if the determinant of the homogeneous coordinates of the points is equal to zero.

ee also

* Menelaus theorem
* Ceva's theorem
* concyclic
* Incidence matrix
* incidence algebra
* angle of incidence
* incidence structure
* incidence geometry
* Levi graph
* Hilbert's axioms
* Incidence (descriptive geometry)