Hirst Inversion - the Kimberling definition

The general case of Hirst inversion is considered briefly in Hirst Inversion and Cubics.
The case where the pole is the centre of the conic is considered in Hirst Inversion - Isocubics.
Here, we look at the case where the pole is the perspector of the circumconic. This is the version
which appears in ETC - Glossary. We look at cubics of type pK(W,P) and nK0(W,P) since the two
theories have much in common.

C(Q) is the circumconic with perspector Q.
T(X) is the tripolar of X.
X&Y is the barycentric product of X and Y.
t(X) is the isotomic conjugate of X.
i[X] denotes the isoconjugation fixing X, i.e. with pole X&X.
h[Q] denotes Hirst Inversion in C(Q) with pole Q.

If Q = u:v:w and X is x:y:z, then h[Q](X) = vwx2 - u2yz : : .

As in the general case, the image of a cubic K under h[Q] will contain a cubic precisely when K
contains Q and the contacts of the tangents from Q to C(Q). The latter are the intersections of
C(Q) and T(Q) as the latter is the polar of Q in C(Q).

Our first result is proved as in Hirst Inversion - Isocubics. It turns out that, once we know
that the cubic contains the intersections of C(Q) and T(Q), it also contains Q.

Theorem 1
Suppose that K is one of the cubics pK(W,P) or nK0(W,P).
Then K contains the intersections of C(Q) and T(Q) if and only if
(1) P = Q and W is on T(Q&Q), or
(2) W = Q2 and P is on T(Q), or
(3) P ≠ Q, W ≠ Q2, and W = Q&h[Q](P).
     (equivalently, P = t(Q)&h[Q&Q](W)).
When any of the conditions hold, Q lies on the cubic.

Theorem 2 - cases (1) and (2).
Suppose that W is on T(Q&Q).
(1) nK0(W,Q) is the union of T(Q) and C(W&t(Q)).
(2) h[Q](nK0(W,Q)) = nK(Q&Q,W&t(Q))
     W&t(Q) is on T(Q).
     nK(Q&Q,W&t(Q)) is the image of C(W&t(Q)) under h[Q].
     nK(Q&Q,W&t(Q)) is cK(#Q,W&t(Q)) - see notes under CL030.
(3) pK(W,Q) and pK(Q&Q,W&t(Q)) are interchanged by h[Q].

Theorem 3 - case (3).
Suppose that P ≠ Q. Let K denote nK0 or pK.
Then h[Q](K(Q&h[Q](P),P)) = K(Q&P,h[Q](P)).

Corollary 4
The cubics nK0(W,P), pK(W,P) are invariant under h[Q] if and only if
P is on C(Q) and W = Q&P (i.e. W is on C(Q&Q)).

One way to obtain suitable pairs {W,P} is to begin a point X on the line T(t(Q)).
Then take P as the isotomic conjugate of X, and W as the Q-isoconjugate of X.

The families of cubics in Theorem 1 are also closed under another mapping, i[Q].
We begin with a result about the effect of i[Q] on cubics of type nK0 and pK.

Lemma A
Let K denote nK0 or pK. Then i[Q](K(W,P)) = K(i[Q&Q](W),P&i[Q](W)).

Observe that C(Q) and T(Q) are interchanged by i[Q]. It follows that, if K contains the intersections
of these - so is as in Theorem 1 - then so does i[Q](K), so that this is also of the type in Theorem 1.

Theorem 5
(1) nK0(Q&Q,P) and pK(Q&Q,P) are invariant under i[Q].
(2) If W is on T(Q&Q), i[Q](nK0(W,Q)) and i[Q](pK(W,Q)) are invariant under h[Q].
(3) If W = Q&h[Q](P), then i[Q](nK0(W,P)) and i[Q](pK(W,P)) are of the type in Theorem 1.

By algebraic computation, we can also establish a result which shows that the operations i[Q] and h[Q]
applied to one of our isocubics produce sets of just six cubics in general, and just three when we start
with a cubic of type (1) or (2) in Theorem 1, or of the type in Corollary 4.

Lemma B
The (inverse) maps i[Q]°h[Q] and h[Q]°i[Q] are of order 3.

To illustrate the set of six cubics, we note that, if P is the pivot of a pK (or root of an nK0), then the pole
W is determined as Q&h[Q](P). Thus it sufficient to list the values of P.

cubic   pivot/root
K   P  
h[Q](K)   h[Q](P)  
i[Q](K)   i[q]°h[Q]°i[Q](P)  
h[Q]°i[Q](K)   i[q]°h[Q](P)  
i[q]°h[Q](K)   h[Q]°i[Q](P)  
i[q]°h[Q]°i[Q](K)   i[Q](P)  

There appear to be few examples of the above types in the current CTC, apart from those for Q = G
discussed in Hirst Inversion - Isocubics.

The main source is mentioned under CL030. The cubics cK(#F,R) with R on T(F) are of type cK0. In
the current context, these are related to Hirst inversion in C(F).

Q = X1
Then i[Q] is isogonal conjugation. The conic C(Q) has perspector X1, and centre X9.
Those of type cK0(#X1,R) have R on the Antiorthic Axis.
Examples     K040    K137    K221.

Q = X6
The conic C(Q) is the Circumcircle
Those of type cK0(#X6,R) have R on the Lemoine Axis.
Examples     K222    K223    K224    K225.

One sporadic example is K367 = pK(X669,X6), with Q = X6, as X669 is on T(X32) (as well as T(X6)).
The image under i(X6) is pK(X1576, X110), and this must be invariant under h[X6]. Of course, we now
know that nK0(X1576, X110) must also be invariant under h[X6]. There is a general pattern for pairs
(W,P) with pK(W,P), nK0(W,P) h[X6]-invariant. If X is on the de Longchamps Axis, then we have the
pair W the isogonal conjugate of X, P the isotomic conjugate of X. Examples culled from ETC are
(X110, X99), (X692, X100), (X1576, X110), (X1492, X789).

Other Q
There are three further cases :
K147 = cK0(#X110,X6). T(Q) the Brocard Axis.
K217 = cK0(#X523,X1640). This and next have T(Q) = X115-X125.
K218 = cK0(#X523,X1648).