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[edit]

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[edit]

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[edit]

Given line L and point P in a projective plane, and both expressed in homogeneous coordinates, then PL 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[edit]

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[edit]

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.

See also[edit]

References[edit]

  1. ^ 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.