- Incidence (geometry)
In

geometry , therelation s 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 L_{1}intersects line L_{2}', in three-dimension al space). That is, they are thebinary relation s describing howsubset s 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 aprojective plane , though not true inEuclidean 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 ofsynthetic geometry it was considered that projective geometry "should be" developed using such propositions asaxiom s. This turns out to make a major difference only for the projective plane (for reasons to do withDesargues' theorem ).The modern approach is to define

projective space starting fromlinear algebra andhomogeneous co-ordinates . Then the propositions of incidence are derived from the following basic result onvector space s: 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 subspace 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::$L\_1\; :\; [m\_1\; :\; b\_1\; :\; 1]\; \_L$:$L\_2\; :\; [m\_2\; :\; b\_2\; :\; 1]\; \_L$where "m"

_{1}and "m"_{2}areslope s and "b"_{1}and "b"_{2}arey-intercept s. 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 isperpendicular to the point (so that theirdot 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 scalartriple product is zero, viz. if and only if:$,l\_2,l\_3>\; =\; L\_1\; cdot\; L\_2\; imes\; L\_3\; =\; 0.$"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 thecross 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:$(L\_1\; imes\; L\_2)\; cdot\; L\_3\; =,l\_2,l\_3>\; =\; 0.$"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 trueif 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::$,p\_2,p\_3>\; =\; P\_1\; cdot\; P\_2\; imes\; P\_3\; =\; 0$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:$left|\; egin\{matrix\}\; x\_1\; y\_1\; z\_1\; \backslash \; x\_2\; y\_2\; z\_2\; \backslash \; x\_3\; y\_3\; z\_3\; end\{matrix\}\; ight|\; =\; 0,$i.e. if and only if thedeterminant 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)

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