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-In [[geometry]], the [[Relation (mathematics)|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 L<sub>1</sub> intersects line L<sub>2</sub>', in three-[[dimension]]al space). That is, they are the [[binary relation]]s describing how [[subset]]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 a [[projective plane]], though not true in [[Euclidean space]] of two dimensions where lines may be [[Parallel (geometry)|parallel]].
+== Its your body Cheap Beats Studio Australia ==
-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 [[axiom]]s.  This turns out to make a major difference only for the projective plane (for reasons to do with [[Desargues' theorem]]).
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-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 space]]s: given subspaces ''U'' and ''V'' of a vector space ''W'', the dimension of their intersection is at least dim ''U'' + dim ''V'' &minus; dim ''W''. Bearing in mind that the dimension of the projective space '''P'''(''W'') associated to ''W'' is dim ''W'' &minus; 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'''.<ref>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 &minus; dim (L ∩ M). Thus dim L + dim M > dim P implies dim (L ∩ M) > 0</ref>
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-Let ''L''<sub>1</sub> and ''L''<sub>2</sub> be a pair of lines, both in a projective plane and expressed in homogeneous coordinates:
-:<math> L_1 : [m_1 : b_1 : 1]_L </math>
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-:<math> L_2 : [m_2 : b_2 : 1]_L </math>
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-where ''m''<sub>1</sub> and ''m''<sub>2</sub> are [[slope]]s and ''b''<sub>1</sub> and ''b''<sub>2</sub> are [[y-intercept]]s.  Moreover let ''g'' be the [[duality (projective geometry)|duality mapping]]
-:<math> g : [x : y : z] \mapsto [x : -z : y] </math>
-which maps lines onto their dual points.  Then the intersection of lines ''L''<sub>1</sub> and ''L''<sub>2</sub> is point ''P''<sub>3</sub> where
-:<math> P_3 = g(L_1 \times L_2). </math>
-==Determining the line passing through a pair of points==
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-Let ''P''<sub>1</sub> and ''P''<sub>2</sub> be a pair of points, both in a projective plane and expressed in homogeneous coordinates:
-:<math> P_1 : [x_1 : y_1 : z_1], </math>
+   <li>[http://www.bbs.fobqingdao.com/forum.php?mod=viewthread&tid=214752 http://www.bbs.fobqingdao.com/forum.php?mod=viewthread&tid=214752]</li>
-:<math> P_2 : [x_2 : y_2 : z_2]. </math>
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-Let ''g''<sup>&minus;1</sup> be the inverse duality mapping:
-:<math> g^{-1} : [x : y : z] \mapsto [x : z : -y] </math>
+   <li>[http://www.snesup-paris13.org/spip.php?article16/ http://www.snesup-paris13.org/spip.php?article16/]</li>
-which maps points onto their dual lines.  Then the unique line passing through points ''P''<sub>1</sub> and ''P''<sub>2</sub> is ''L''<sub>3</sub> where
-:<math> L_3 = g^{-1}(P_1 \times P_2). </math>
+   <li>[http://shijiren1999.w42-e2.ezwebtest.com/forum.php?mod=viewthread&tid=135397 http://shijiren1999.w42-e2.ezwebtest.com/forum.php?mod=viewthread&tid=135397]</li>
-==Checking for incidence of a line on a point==
+ </ul>
-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
-:<math> gL \cdot P = 0 </math>
-where ''g'' is the duality mapping.
-An equivalent way of checking for this same incidence is to see whether
-:<math> L \cdot g^{-1} P = 0 </math>
-is true.
-==Concurrence==
-Three lines in a projective plane are [[Concurrent lines|concurrent]] if all three of them intersect at one point.  That is, given lines ''L''<sub>1</sub>, ''L''<sub>2</sub>, and ''L''<sub>3</sub>; these are concurrent if and only if
-:<math> L_1 \cap L_2 = L_2 \cap L_3 = L_3 \cap L_1. </math>
-If the lines are represented using homogeneous coordinates in the form [''m'':''b'':1]<sub>''L''</sub> with ''m'' being slope and ''b'' being the y-intercept, then concurrency can be restated as
-:<math> L_1 \times L_2 \equiv L_2 \times L_3 \equiv L_3 \times L_1. </math>
-''Theorem.'' Three lines ''L''<sub>1</sub>, ''L''<sub>2</sub>, and ''L''<sub>3</sub> in a projective plane and expressed in homogeneous coordinates are concurrent if and only if their [[scalar (mathematics)|scalar]] [[triple product]] is zero, viz. if and only if
-:<math> \langle L_1,L_2,L_3\rangle = L_1 \cdot L_2 \times L_3 = 0. </math>
-''Proof.'' Letting ''g'' denote the duality mapping, then
-:<math> L_1 \cap L_2 = gL_1 \times gL_2. \qquad \qquad (1)</math>
-The three lines are concurrent if and only if
-:<math> (L_1 \cap L_2) \subset L_3. </math>
-According to the [[Incidence (geometry)#Checking for incidence of a line on a point|previous section]], the intersection of the first two lines is a subset of the third line if and only if
-:<math> gL_3 \cdot (L_1 \cap L_2) = 0 \qquad \qquad (2)</math>
-Substituting equation (1) into equation (2) yields
-:<math> (gL_1 \times gL_2) \cdot gL_3 = 0 \qquad \qquad (3)</math>
-but ''g'' distributes with respect to the [[cross product]], so that
-:<math> g(L_1 \times L_2) \cdot gL_3 = 0, </math>
-and ''g'' can be shown to be isomorphic w.r.t. the dot product, like so:
-:<math> A \cdot B = gA \cdot gB </math>
-so that equation (3) simplifies to
-:<math> (L_1 \times L_2) \cdot L_3 = \langle L_1,L_2,L_3 \rangle = 0. </math>
-''[[Q.E.D.]]''
-==Collinearity==<!-- This section is linked from [[Bézier curve]] -->
-The dual of concurrency is [[collinearity]].  Three points ''P''<sub>1</sub>, ''P''<sub>2</sub>, and ''P''<sub>3</sub> in the projective plane are '''collinear''' if they all lie on the same line.  This is true [[if and only if]]
-:<math> P_1.P_2 \equiv P_2.P_3 \equiv P_3.P_1, </math>
-but if the points are expressed in homogeneous coordinates then these three different equations can be collapsed into one equation:
-:<math> <P_1,P_2,P_3> = P_1 \cdot P_2 \times P_3 = 0 </math>
-which is more symmetrical and whose computation is straightforward.
-If ''P''<sub>1</sub> : (''x''<sub>1</sub> : ''y''<sub>1</sub> : ''z''<sub>1</sub>), ''P''<sub>2</sub> : (''x''<sub>2</sub> : ''y''<sub>2</sub> : ''z''<sub>2</sub>), and ''P''<sub>3</sub> : (''x''<sub>3</sub> : ''y''<sub>3</sub> : ''z''<sub>3</sub>), then ''P''<sub>1</sub>, ''P''<sub>2</sub>, and ''P''<sub>3</sub> are collinear if and only if
-:<math> \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,</math>
-i.e. if and only if the [[determinant]] of the homogeneous coordinates of the points is equal to zero.
-==See also==
-* [[Menelaus theorem]]
-* [[Ceva's theorem]]
-* [[Concyclic]]
-* [[Incidence matrix]]
-* [[Incidence algebra]]
-* [[Angle of incidence]]
-* [[Incidence structure]]
-* [[Incidence geometry]]
-* [[Levi graph]]
-* [[Hilbert's axioms]]
-==References==
-{{Reflist}}
-* Harold L. Dorwart (1966) ''The Geometry of Incidence'', [[Prentice Hall]].
-[[Category:Projective geometry]]
-[[Category:Articles containing proofs]]

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