Duality Mapping Defined
Given a line L in the projective plane, what is its dual point? Draw a line L′ passing through the 2-D origin and perpendicular to line L. Then pick a point P on line L′ on the other side of the origin from line L, such that the distance of point P to the origin is the reciprocal of the distance of line L to the origin.
Expressed algebraically, let g be a one-to-one mapping from the projective plane onto itself:
such that
and
where the L subscript is used to semantically distinguish line coordinates from point coordinates. In words, affine line (m, b) with slope m and y-intercept b is the dual of point (m/b, −1/b). If b=0 then the line passes through the 2-D origin and its dual is the ideal point .
The affine point with Cartesian coordinates (x,y) has as its dual the line whose slope is −x/y and whose y-intercept is −1/y. If the point is the 2-D origin, then its dual is L which is the line at infinity. If the point is, on the x-axis, then its dual is line L which shall be interpreted as a line whose slope is vertical and whose x-intercept is −1/x.
If a point or a line's homogeneous coordinates are represented as a vector in 3x1 matrix form, then the duality mapping g can be represented by a 3x3 matrix
whose inverse is
Matrix G has one real eigenvalue: one, whose eigenvector is . The line L is the y-axis, whose dual is the ideal point which is the intersection of the ideal line with the x-axis.
Notice that L is the y-axis, L is the line at infinity, and L is the x-axis. In 3-space, matrix G is a 90° rotation about the x-axis which turns the y-axis into the z-axis. In projective 2-space, matrix G is a projective transformation which maps points to points, lines to lines, conic sections to conic sections: it exchanges the line at infinity with the x-axis and maps the y-axis onto itself through a Möbius transformation. As a duality, matrix G pairs up each projective line with its dual projective point.
Read more about this topic: Duality (projective Geometry)
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