
Duality and Polarity 

The Axioms of Incidence^{(*)(**)} constrain the members of the threefold Projective Geometric Community  Point, Line and Plane  to engage with each other in definite ways, and only in those ways. But within that constraint, "uncountably many" configurations are possible. 
"For every configuration 

Any given point (such as black A, on the right) may be made the dual of any given line, (such as dashed black line a, on the right), by drawing two more lines, (such as the feint blue lines, b and c, on right), in the given point to cut the given line in two more points (blue points B and C). These extra lines must obviously be in the plane common to the given line and the given point. Then the given point is defined by the two new lines, while, dually, the given line is defined by the two new points corresponding to those two new lines. There need be no other manner of association between such lines and points, so they are generally pairable, but not uniquely paired. 

Therefore, if for some reason we wish to constrain duality in such a way that particular element instances are dual uniquely and only with each other—that is to say on a strict, onetoone basis, so that a particular point (say just P_{3} from points P_{0} to P_{∞}), is the dual of a particular line (say just p_{9} from lines p_{0} to p_{∞})—we go beyond simple duality, and enter the special province of  and we will need a special method of bringing polarity about.The interactive animation below shows
Drag point D, or drag line f (by dragging E, or f round E). Note especially thatwhen a Pole lies in the conic, so does its Polar. This illustrates that if a line is to lie in a curve at a point, it does so by taking up the direction of the curve at that point. That is, the line (the Polar) is tangent to the curve at the Pole. 

The method requires Curvature.
But there is no provision for Curvature in the axioms of Projective Geometry. The consequences of this are worked through in detail here, where it is shown that the special curve we need is a plane cut of a cone—which is a conic section, or conic, for short. 

The key here is that 
We see that point D is defined by two tangents at two points on the conic, and that line d joining those points is a chord of the conic . It follows that Polarity propagates from its parent conic! 

DevelopmentLet points A and B, respectively, be defined (in the way indicated above right) as the poles of chords a and b, respectively, of a conic. Let
Then, because they are uniquely associated—
And because M and m are thus shown, indirectly, to be Pole and Polar, we know that the two tangents to the conic through point M touch the conic at the two points, K and L, of intersection of line m with the conic. This is the direct expression of the polarity of M and m. Then—
Thus—
Also—

The statements made on the left in respect of the above construction underlie the proof, supplied on the Conics(3) page, that the four lines projecting a constant crossratio from one line to another always envelope a conic, whether or not the conic is degenerate—as it is, for example, in the case where the conic is just a point. 

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