triangle centres Hirst Inverse pairs of pivotal isocubics

Here, we take the case of Hirst inversion in a conic with respect to its centre.
The more general case is discusssed in Hirst Inversion and Cubics

The theory for inversion in the Circumcircle is given in detail in §4.1.1 of the e-book
Special Isocubics in the Triangle Plane by Ehrmann and Gibert.
Some related material appears in Gibert's
CL035 - Circular pK.
CL032 - Hirst Cubics

Notation

T(Q) is the tripolar of Q.
C(Q) is the circumconic with perspector Q.
Q2 is the centre of C(Q), so Q2 is the G-Ceva conjugate of Q.
Q3 is the perspector of the inconic with centre Q, so Q3 is the isotomic conjugate of
the anticomplement of Q. It is also the Q-isoconjugate of Q2.
L is the Line at Infinity.
t(X) is the isotomic conjugate of a point (or curve) X.
hQ(X) denotes the Q-Hirst inverse of X (the inverse in C(Q) with respect to Q2).

Q-conical cubics

We say that a cubic is Q-conical if it contains the infinite points of C(Q).
The case of Q-conical cubics of type nK is discussed in CTC - Glossary

Note that, if Q is on the Steiner Inellipse, then C(Q) is a parabola, so is tangent to the
Line at Infinity. The theory then takes on a different flavour, so we may omit such Q.

In the first reference above, we observe that Hirst inversion in C(Q) with respect to
the centre Q2 maps a cubic K to a sextic.
If K is Q-conical, then the sextic degenerates int a quadric and the tangents to C(Q)
at its infinite points.
If K contains Q2, the centre of inversion, then the sextic degenerates into a quintic
and the Line at Infinity.
Thus, we have

Theorem 1
The cubic K is the Q-Hirst inverse of another cubic if and only if
K is Q-conical and contains Q2.
If K is of type pK, then the second cubic is also of type pK.

The Q-conical pK can be identified as follows.
Let C(Q) have equation f(x,y,z) = 0. Then the intersections with L are given by
F(y,z) = f(-y-z,y,z) = 0.
Let pK(W,P) have equation g(x,y,z) = 0. Then the intersections with L are given by
G(y,z) = g(-y-z,y,z) = 0.
Hence pK(W,P) is Q-conical if and only if F(y,z) divides G(y,z), i.e. G(y,z) = (ky+lz)F(y,z).
The coefficients of y3 and z3 in G(y,z), and those of y2 and z2 in F(y,z) determine
the constants k and l. Then the condition for divisibility takes the form K,L = 0, where
G(y,z)-(ky+lz)F(y,z) = yz(Ky+Lz).
Let W = u:v:w, P = p:q:r. Calculation shows that K and L are each linear in p,q,r and also
in u,v,w.
Of course, we could eliminate y or z instead of x to obtain different looking conditions.

Thus, in general, either of P, W determines the other uniquely. For example we have

Theorem 2
Suppose that P ≠ Q3. Then pK(W,P) is Q-conical if and only if W is the barycentric product
of P and the Q-Hirst inverse of the Q-isoconjugate of P.

Note that the Q-Hirst inverse above is the secondary pivot (isopivot) of the cubic.
The restriction on P is needed to avoid the case where the Q-isoconjugate of P is Q2, since
the Q-Hirst inverse of Q2 is undefined. We shall meet the isocubics with pivot Q3 shortly.

Although the equations K = 0, L = 0 usually determine W from P or P from W, there are
some exceptions. Looking at the conditions, we can solve uniquely unless we have

case 1 W = Q. Then P must be on the Line at Infinity, or
case 2 P = Q3. Then W must be on T(Q3), the tripolar of Q3.

Thus we have two pencils of Q-conical pivotal isocubics, one with common pole Q, and one
with common pivot Q3. In other cases, each of P, W determines the other. See Theorem 2.
Summing up, we have

Theorem 3
Suppose that P = p:q:r, W = u:v:w, Q = U:V:W.
The isocubic pK(W,P) is Q-conical if and only if
(1) W = Q and P is on the Line at Infinity, or
(2) P = Q3 and W is on T(Q3), or
(3) P ≠ Q3 and W is given by barycentrics U(p2(-U+V+W)-(pq+rp+qr)U+p(Vq+Wr)): : , or
(4) W ≠ Q and P is given by barycentrics VWu(v+w-u)-(V2wu+W2uv-U2vw) : : .

Note that (3) gives an alternative description of the pole for pivot P ≠ Q3 to that in Theorem 2.

Pairs of Hirst inverse pivotal isocubics.

>From Theorem 1, we know that such a pair arises when the cubics are Q-conical and contain Q2.
Obviously, if a cubic has these properties, then so does its Q-Hirst invese. Thereom 3 shows when
a cubic is Q-conical. Then we need only ensure that it contains Q2. The calculations are different
for the pencils (1),(2) in Theorem 3, and in the generic cases (3),(4).

Theorem 4a
Suppose that Q is not on the Steiner Inellipse, and is not the centroid G.
(1) The pencil {pK(W,Q3) : W on T(Q3) } contains one cubic through Q2.
This has W the intersection of T(Q3) and the line Q-Q2.
(2) The pencil { pK(Q,P) : P at infinity } contains just one cubic through Q2.
This has P the infinite point of the line Q2-Q3.
The Q-Hirst inverse of this cubic is pK(W,Q2), with W as in Theorem 2.

The proofs are easy. In the case of Q on the Steiner Inellipse, Q2 is always on the cubics. In the
case Q = G, we have Q = Q2 = Q3 = G. We deal with this later.

Theorem 4b
Suppose that Q is not on the Steiner Inellipse, and is not the centroid G.
(1) Suppose that Q ≠ W. Then the Q-conical cubic pK(W,P) contains Q2 if and only if W is on
(a) the circumconic C(Q4), Q4 the barycentric product of Q and Q2, or
(b) the line Q-Q2.
(2) Suppose that P ≠ Q3. Then the Q-conical cubic pK(W,P) contains Q2 if and only if P is on
(a) the circumconic C(Q), or
(b) the line Q2-Q3.
In either case (a), the cubic is self-inverse under Q-Hirst inversion.
In such cases, W is the barycentric product of P and Q2.
In either case (b), the Q-Hirst inverse of pK(W,P) is pK(W*,P*), where
P* is the Q-Hirst inverse of P, and
W* is the intersection of the polar of W in C(Q4) with Q-Q2.
Also W is the barycentric product of Q2 and P1, W* the barycentric product of Q2 and P1*,
with P1, P1* on Q2-Q3 (as W, W* on Q-Q2), and P1, P1* are Q-Hirst inverse.

There are two non-circular examples in CTC :
K237 = pK(X115,X2)
K238 = pK(X115,X4)

We can also describe the relation between P and W in geometric terms. This uses a generalisation
of perpendicularity. We say that lines L, M are C(Q)-conjugate if the infinite point of M lies on the
polar in C(Q) of the infinite point of L. Observe that, given a line L and a point X, there is a unique
line through X which is C(Q)-conjugate with L.

Let S be the tripole of the line Q-Q2.

Case 1 P from W.
Let L be the tripolar of the isotomic conjugate of the Q-isoconjugate of W.
Let R be the intersection of L and its C(Q)-conjugate through S.
Then P is the reflection of S in R.

Case 2 W from P.
Let R be the mid-point of P and S.
Let L be the line through R C(Q)-conjugate with P-S.
Then W is the Q-isoconjugate of the isotomic conjugate of the tripole of L.

These can be proved by Maple. The conditions that the mid-point of P and S is on the line L,
and that P-S and L are C(Q)-conjugate are both satisfied for W on Q-Q2, and pK(W,P) Q-conical.

G-Hirst inversion

Here, the theory simplifies. The cubics already appear in CL041 - Grassmann Cubics.
We will examine this approach below.

Theorem 5
The cubic pK(W,P) is G-conical if and only if
(1) P = G, W at Infinity, or
(2) W = G, P at Infinity, or
(3) P,W (≠ G) are G-Hirst inverse.
In any case, G is on pK(W,P) and the G-Hirst inverse cubic is pK(P,W).

Corollary 5.1
The cubic pK(W,P) is invariant under G-Hirst inversion if and only if
W is on the Steiner Ellipse and P = W. (part of CL007).

Observe that, if pK(W,P) is G-conical, then it contains G. It follows that its isotomic conjugate
also has these properties, so will also be the G-Hirst inverse of a similar cubic. Now note that
t(pK(W,P)) = pK(t(W),P*), where P* is the barycentric product of P and t(W). We see that t(W)
and P* are G-Hirst inverse when W, P are. This is easy to verify directly. In the first two cases
t(pK(G,P)) = pK(G,P), and t(pK(W,G) = pK(t(W),t(W)) - and this is self-inverse.

Also, if X is at infinity, then pK(X,G) is the complement and the G-Hirst inverse of pK(G,X).
If we write c(Y), a(Y) for the complement and anticomplement of Y, then we have a pair of
involutions hG°c on pK(G,X) and hG°a on pK(X,G).

examples

G-Hirst invariant cubics mentioned in CTC
pK(X290,X290) see CL007

G-Hirst Inverse pairs in CTC
K322, K354 : X694, X1916 are G-Hirst inverse (not in ETC).
K355, K357.

G-Hirst Inverse pairs with one member in CTC
K128, pK(X385,X6).
K356, pK(X76,t(X694)).
CTC does not mention that K356 contains the infinite points of the Steiner ellipses.

K357 is the isotomic conjugate of pK(X290,X290) mentioned above.
K356 is the isotomic conjugate of K354.
pK(X76,t(X694)) is the isotomic conjugate of K128.
pK(X385,X6) is the isotomic conjugate of X322.

notes on the class CL041

summary
Suppose that Q is a point other than G or a vertex of the antimedial triangle. The points t(Q)
and Q2, the barycentric square of Q, define a triply perspective desmic structure.With such a
structure, we associate two Grassmann cubics :
GnK(Q) = nK(Q,R) : R is the G-Hirst inverse of Q,
GpK(Q) = pK(Q,R+) : (t(Q),Q2,R,R+) harmonic.
The triangles associated with the latter have perspectors on a further cubic :
GpK'(Q) = pK(t(R),R*) : R* the Q-isoconjugate of R.

new facts
(1) t(R) and R* are G-Hirst inverses : this shows that the cubic is G-conical.
(2) pK(t(R),R*) = t(pK(R,Q)) = t(hG(pK(Q,R)).

further notes
(3) The families {GnK(Q)}, {GpK(Q)}, {GpK'(Q)} are all closed under isotomic conjugation.
Indeed, t(GnK(Q)) = GnK(t(Q)), and t(GpK(Q)) = GpK(t(Q)).
Also, t(GpK'(Q)) = pK(hG(Q),Q) = GpK'(hG°t°hG(Q)).
(4) The families {GnK(Q)} and {GpK'(Q)} are closed under G-Hirst inversion.
Here, hG(GnK(Q)) = GnK(hG(Q)), and hG(GpK'(Q)) = GpK'(hG°t°hG°t°hG(Q)).
(5) The maps ht = hG°t and th = t°hG have order 3.
As a corollary, from (4) and (5), hG(GpK'(Q)) = GpK'(t(Q)).
This is a simple matter of algebraic verification.
(6) ht(GnK(Q)) = GnK(ht(Q)), ht(GpK'(Q)) = GpK'(ht(Q)), and similarly for th.
We get this by combining (3), (4) and (5).

We now see that, for a general point Q, the maps t and hG produce just six images.
For example, with Q = X6, we get {X6,X76,X694,X1916,X385,t(X694)}.
Of the related GnK(Q), just one appears in CTC - GnK(X6) = K017 = nK0(X6,X385).
On the other hand, four of the GpK'(Q) appear :
GpK'(X6) = K322 = pK(X1916,X694) = hG(K354),
GpK'(X76) = K354 = pK(X694,X1916) = hG(K322) = t(K356),
GpK'(t(X694)) = K128 = pK(X6,X385),
GpK'(X1916) = K356 = pK(t(X694),X76) = t(K354).
The missing ones are :
GpK'(X385) = pK(X76,t(X694)) = t(K128) = hG(K356) ,
GpK'(X694) = pK(X385,X6) = t(K322) = hG(K128).

There are two examples where all six points are in ETC :
{X1, X75, X239, X291, X335, X350},
(X4, X69, X98, X287, X297, X325}.

The GpK'(Q) tend to contain many centres. They always include G and four of the six points
associated with Q. in fact, Q and ht(Q) are missing. See CL041 for a lot more!

When we start with a point at infinity or on the Steiner Ellipse, there are just three points,
and hence three cubics. For example, we can start with X290 on the Steiner Ellipse. Then
the set of points is {X2, X290, X511}. The associated pK are
K355 = pK(X2,X511) = hG(K357), closed under t
K357 = pK(X511,X2) = hG(K355),
pK(X290,X290) = t(K357), closed under hG.

notes on the class CL007

This is the class of pivotal isocubics pK(W,W). First, we note some general points :

Theorem
The cubic pK(W,W) is invariant under any general Hirst inversion in a circumconic through
the point W, and having pole G.
The tangent to pK(W,W) at G (the isopivot) is that at G to the circumconic through W and G.
The tangent meets the curve again at the anticomplement of the isotomic conjugate of W.

Proof : see Hirst Inversion and Cubics

We are particularly interested in the case where W is on the Steiner Ellipse, so pK(W,W)
is self-inverse under hG. Then the tangent at G meets the cubic again at t(W), the third
infinte point. Thus this tangent is parallel to the asymptote.

The case of W = X290 has been discussed above.