Duality (projective geometry)

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Duality in the projective plane refers to the interchangeability between points and lines which preserves incidence properties.

Notice that both points and lines can be represented (on a plane) by means of ordered pairs. A point is represented by the ordered pair (x,y), where x is the abscissa and y is the ordinate, which together are coordinates of the point. A line can likewise be represented by an ordered pair (m,b) where m is the slope and b is the y-intercept.

Given three points

these three points are collinear iff their coordinates satisfy the equation

Likewise, given three lines

one can verify that these three lines are concurrent iff their parameters satisfy the equation

Equations (1) and (2) are equivalent to each other up to an exchange of x with m and y with b. Therefore there exists is a way to exchange lines with points in such a way that concurrency is exchanged with collinearity.

It is possible to distinguish lines from points by conjugating ordered pairs. That is, let line (m,b) be represented instead by its conjugate Then it can be verified that the intersection L₁.L₂ of a pair of lines L₁ and L₂ is

where b₁ and b₂ are negative y-intercepts. Also, the common line P₁.P₂ passing through a pair of points P₁ and P₂ is

Equation (4) can be seen to be the same as equation (3), after exchanging m with x and b with y, and applying the following rules of conjugation:

Indeed, if equation (3) is represented as

then applying rule (6) yields

Applying rule (7) then yields

and applying rule (5) finally yields

which is equation (4).

Thus it is possible to imagine a pair of planes S₁ and S₂, and a bijective relation between loci of points in the two planes, such that points in S₂ correspond to lines in S₁, and points in S₁ correspond to lines in S₂.

One way to establish such bijection is to model the projective plane, not as an extended affine plane, but as a "unit sphere modulo antipodes", i.e. a unit sphere in which antipodal points are equivalent. Then through points P₁ and P₂ in S₁ passes a line L₃ which is actually a great circle. But to these two original points correspond a pair of great circles L₁ and L₂ in S₂, such that if S₂ and S₁ are superposed, then L₁ is the unique great circle in a plane perpendicular to the line through the pair of points P₁, and L₂ is the unique great circle in a plane perpendicular to the line through the pair of points P₂. These great circles L₁ and L₂ intersect at a pair of points P₃ in S₂. The line through P₃ is the cross product of the lines through P₁ and P₂. Then the unique great circle perpendicular to point P₃ is line L₃ in S₁.

Therefore to every great circle in S₁ corresponds a unique pair of points (which are actually the same point) in S₂, such that if S₁ and S₂ are superposed, then the line passing through the pair of points is perpendicular to the great circle. The above sentence remains true if S₁ and S₂ are exchanged. This establishes the bijective nature of the duality in the projective plane.

It must be noted that in the "unit sphere modulo antipodes", one "line", i.e. great circle, must be chosen to be the line at infinity. This line may be chosen to be the equator by convention.

There is also a duality in projective 3-space, in which points correspond to planes, and lines correspond to lines. This is analogous to duality of polyhedra in solid geometry, where points are dual to faces, and sides are dual to sides, so that the icosahedron is dual to the dodecahedron, and the cube is dual to the octahedron.