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H. M. Cundy and C. F. Parry have written several papers on cubics (see the bibliography) and, in particular, have studied a transformation we shall call the CundyParry transformation Phi : M > M1 = OM /\ HM*, where M* is the isogonal conjugate of M. Naturally, we can also define the related other transformation Psi : M > M2 = HM /\ OM*. These two transformations yield a large number of interesting facts concerning cubics. They are given by : 



Properties of Phi and Psi




Phi and Psi transforms of a line L and a circumconic L*
In the most general case, L is transformed into a nodal (with node O) circumcubic K1 under Phi and a nodal (with node H) circumcubic K2 under Psi. It is clear that K1 and K2 are isogonal conjugates and both contain O and H. If L* denotes the circumconic which is the isogonal transform of the line L then K2 = Phi(L*) and K1 = Psi(L*). When L passes through a singular point, K1 and K2 decompose. For example :
Special cubics K1
Special cubics K2




Phi and Psi transforms of non circumconics
In general, the Phi or Psi transform of a conic is a sextic but when the conic contains O and H, both transforms are circumcubics with nodes at O and H respectively. See K389 for example. 



Phi and Psi transforms of circumcubics
In the most general case, Phi and Psi transform a circumcubic into a sextic. It is more interesting to consider circumcubics passing through O and H which will transform into other circumcubics passing through O and H.
Phi and Psi transforms of the cubics of the Euler pencil An isogonal pK with pivot P on the Euler line is a member of the Euler pencil. See Table 27. It always contains the in/excenters, O, H, P and its isogonal conjugate P* on the Jerabek hyperbola. Phi and Psi both transform this pK into another pK of the same pencil whose pivot Q is the harmonic conjugate of P with respect to O and H. Psi fixes the McCay cubic and Phi fixes the Orthocubic. Under Psi and Phi, these two cubics are globally invariant. The following table gives examples of such cubics with centers X(i) on the cubics. 



Phitransform of a pK(W,O) passing through H A pK(W,O) passing through H must have its pole W on the line HK. All these cubics form a pencil since they all contain A, B, C, O, H, X(1075) and the vertices of the cevian triangle of O. The Psitransform of such pK(W,O) is another pK with :
Phitransform of a pK(W,H) passing through O A pK(W,H) passing through O must have its pole W on the line OK. All these cubics form a pencil since they all contain A, B, C, O, H, X(155) and the vertices of the orthic triangle. The Psitransform of such pK(W,H) is another pK with same pivot and pole another point on the line OK. The cubic is invariant if and only if W is the orthocenter or the point X(1609).
Phi / Psitransform of a nK(X6, R, X3) passing through O and H Any nK(X6, R, X3) is globally invariant under both transformations. See CL062. 



Phi and Psi transforms of other curves Q150 is a quintic invariant under Phi. Q151 is a quintic invariant under Psi. 
