[alogo] Reciprocal polar conics

[1] Given two conics (c) and (c1), the polars of points X on c1 with respect to (c) envelope a third conic (c2).
[2] The polars of points Y on (c2) with respect to (c) are tangent to (c1).
[3] Previous properties justify the term: c1, c2 are reciprocal polar conics with respect to (c). This means that the polars of one of the conics with respect to (c) are tangent to the other and vice-versa. The relation is a symmetric (or involutive) one.
[4] Given X on c1 let X' on c2 be the contact point of the polar pX with respect to X (and c). Then X is the contact point of the polar pX' of X' with respect to c. The map X to X' is the restriction on c1 (mapping it onto c2) of a projectivity.

[0_0] [0_1] [0_2]

Let the two conics (c) and (c1) be defined through quadratic equations xtAx=0 and xtBx=0. For a point z on the second conic the polar pz with respect to the first is given by ztAu=0 (variable u). These lines, for variable z satisfy:
(ztA)(A-1BA-1)(Az) = 0,
which is the dual conic of the one defined through the inverse matrix of (A-1BA-1), which is matrix:
C = AB-1A.
This shows [1] by determining the matrix C representing conic (c2).
The relation implies also that B = AC-1A and this shows the symmetry property of the conjugation.
To see last property interpret the polar pz(u) = ztAu=0 as a tangent of c2:
ztA = z't(AB-1A) => zt = z'tAB-1, which is equivalent to z' = A-1Bz. Last equation represents a projectivity mapping c1 onto c2.

Remark-1 One could define c2 by considering the tangents of c1 and getting their poles with respect to (c).
Remark-2 The duality of pole-polars results by applying this construction to c1=c2=c. An important application of this duality offers the deduction of Brianchon's theorem from Pascal's and vice versa. This is examined in PascalBrianchonDuality.html .
Remark-3 This duality is to be distinguished from the usual duality between projective spaces handled in Duality.html.
Remark-4 There is no need to restrict the curve (c1) to be a conic. One could define (c2) as the envelope of polars of points X of (c1) for general curves c1. By this reciprocity properties between points of c1 correspond to properties between tangents of c2 and vice versa.

See Also



Salmon, G. A treatise on conic sections London 1855, Longmans, Chapter XV p. 253

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