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A general study of spK cubics is to be found in CL055.

In this page, we consider cubics spK(P, Q) with P, Q on the Euler line such that P ≠ X(30). These cubics form a net and all the cubics pass through A, B, C, X(3), X(4).

Naturally, when P = Q, we find the pKs of the Euler pencil of Table 27. This net contains other pencils of remarkable cubics as detailed below. Each pencil is characterized by four other points, not necessarily distinct from the previous five points in which case the tangents at the "repeated" points are the same for all cubics of the pencil.

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Each line of the following table corresponds to a given point Q and all the cubics in this line must pass through the foci of the inconic with center Q. In other words, the cubics in each line are in a same pencil which can be generated by K187 and a cubic pK(X6, Q) of the Euler pencil. Recall that K187 is the locus of foci of inconics with center on the Euler line.

If Q is the point with abscissa t in (X3, X2), any cubic of the pencil related to Q is K(k) = pK(X6, Q) + k K187 where k is a real number or infinity (giving K187).

K(k) is spK(Pk, Qk) where Pk, Qk are the points with abscissa t – k, t + k respectively.

Since pK(X6, Q) and K187 are both self-isogonal cubics (K187 is a nK), the pencil K(k) is invariant under isogonal conjugation and K(k)* = K(– k).

See the comments below to examine various special cubics of a pencil associated with special values of k.

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On the other hand, each column contains cubics related to a certain point P either fixed on the Euler line or simply related to Q. Each column also contains the cubics of a same pencil. These cubics pass through four additional points mentioned at the bottom of the table.

Notations :

• aQ, cQ, gQ, tQ are the anticomplement, complement, isogonal, isotomic of Q respectively,

• [Xn] is the reflection of an ETC center Xn about Q. The cubic spK([Xn], Q) is the isogonal transform of spK(Xn, Q) : this is shown by columns of same colour in the table.

• D denotes a decomposed cubic into the line at infinity and the Jerabek hyperbola. D* denotes a decomposed cubic into the circumcircle and the Euler line.

• When a cubic is not listed in CTC, the corresponding point P is given when listed in ETC.

Comments :

• psK1 = spK(aQ, Q) = psK(gaQ, taQ, X3) obtained with k = 3(t – 1).

• psK2 = spK(aaQ, Q) = psK(gtaQ, X2, X3) obtained with k = – 3(t – 1). See Table 50.

• spK(X3, Q) is a stelloid with asymptotes parallel to those of the McCay cubic K003 obtained with k = t. The radial center X is the homothetic of Q under h(X4, 2/3).

• spK([X3], Q) is a CircumNormal cubic obtained with k = – t. See Table 25.

• spK(X20, Q) is a cubic with three real asymptotes parallel to the altitudes of ABC obtained with k = t + 3. See Table 58 for further details and complements.

• spK([X20], Q) is a central cubic with center X3 obtained with k = – (t + 3).

• spK(X2, Q) is a cubic with three real asymptotes parallel to those of the Thomson cubic K002 obtained with k = t – 1.

• spK([X2], Q) is a cubic passing through the vertices of the Thomson triangle obtained with k = 1 – t.

• spK(S, Q) is a cubic passing through the four X3-OAP points. S is the homothetic of Q under h(X4, 4/3). It is obtained with k = (3 – t) / 7. See also Table 53.

• spK(S', Q) is the isogonal transform of spK(S, Q), hence a cubic passing through the isogonal conjugates of the four X3-OAP points. It is obtained with k = – (3 – t) / 7. S' is the homothetic of Q under h(X4, 2/3) hence it is the radial center X of the corresponding stelloid spK(X3, Q).

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Notes :

• note 1 : these cubics are circular cubics passing through X(30), X(74) but they are not proper spKs. They form a pencil generated by two decomposed cubics namely the union of the line at infinity and the Jerabek hyperbola, the union of the circumcircle of ABC and the Euler line. This pencil also contains the focal cubic K187 and the axial cubic K448.

• note 2 : any spK(P, X2) passes through the foci of the Steiner inellipse.

• note 3 : any spK(P, X5) passes through the four foci of the MacBeath inconic : X(3), X(4) and two imaginary. The tangents at X(3), X(4) pass through X(74) when they are defined i.e. when the cubic is not a nodal cubic such as K009 and K028. See Table 80 for further details.

• note 4 : any spK(P, X3) passes through the foci of the inellipse with center X3.

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Other pencils :

See Table 16 for two other pencils of spKs passing through X(3), X(4) and the points Ua, Ub, Uc for the former, Ua*, Ub*, Uc* for the latter as mentioned in K001.