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). Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it That is, they are the binary relations describing how subsets meet. In Mathematics, a binary relation (or a dyadic or 2-place relation) is an arbitrary association of elements within a set or with elements of 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. See Real projective plane and Complex projective plane, for the cases met as manifolds of respective dimension 2 and 4 In Mathematics

Historically, projective geometry was introduced in order to make the propositions of incidence true (without exceptions such as are caused by parallels). Projective geometry is a non- metrical form of Geometry, notable for its principle of duality. From the point of view of synthetic geometry it was considered that projective geometry should be developed using such propositions as axioms. Synthetic geometry is the branch of Geometry which makes use of Theorems and synthetic observations to draw conclusions as opposed to Analytic geometry In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject This turns out to make a major difference only for the projective plane (for reasons to do with Desargues' theorem). In Projective geometry, Desargues' theorem, named in honor of Gérard Desargues, states In a Projective space, two Triangles

The modern approach is to define projective space starting from linear algebra and homogeneous co-ordinates. In Mathematics a projective space is a set of elements constructed from a vector space such that a distinct element of the projective space consists of all non-zero vectors which Linear algebra is the branch of Mathematics concerned with In Mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, allow Affine transformations 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. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added 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. The concept of a linear subspace (or vector subspace) is important in Linear algebra and related fields of Mathematics.

1 Intersection of a pair of lines
2 Determining the line passing through a pair of points
3 Checking for incidence of a line on a point
4 Concurrence
5 Collinearity
6 See also

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. Slope is used to describe the steepness incline gradient or grade of a straight line. In Coordinate geometry, the y -intercept is the y-value of the point where the Graph of a function or relation intercepts the y -axis Moreover let g be the duality mapping

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

which maps lines onto their dual points. In the Geometry of the Projective plane, duality refers to geometric transformations that replace points by lines and lines by points while preserving 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. ↔ In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R

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. In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication This article is about mathematics See Lawson criterion for the use of the term triple product in relation to Nuclear fusion. if and only if

$<L_1,L_2,L_3> = L_1 \cdot L_2 \times L_3 = 0.$

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. 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 In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which r. t. the dot product, like so:

$A \cdot B = gA \cdot gB$

so that equation (3) simplifies to

$(L_1 \times L_2) \cdot L_3 = <L_1,L_2,L_3> = 0.$

Q.E.D.

Collinearity

The dual of concurrency is collinearity. QED is an abbreviation of the Latin phrase "la '''quod erat demonstrandum'''" which means literally "that which was to be demonstrated" 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. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n

Contents

Intersection of a pair of lines

Determining the line passing through a pair of points

Checking for incidence of a line on a point

Concurrence

Collinearity

See also