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A more complete description can be found in the FG paper "Orthocorrespondence and orthopivotal cubics". See the Downloads page. Here is a short summary with several additional informations. See also the related Droz-Farny cubics in CL039 and Table 61.

First recall that the orthocorrespondent of P is the point, here denoted by oc(P), defined as follows : the perpendiculars drawn through P to the lines AP, BP, CP meet BC, CA, AB respectively at three collinear points called the orthotraces of P. These three points lie on a line called orthotransversal of P. The trilinear pole of this line is oc(P). Now, in general, there are two points P1 and P2 sharing the same orthocorrespondent P. These points (although not always real) are said to be orthoassociates and they are the antiorthocorrespondents of P.

An orthopivotal cubic is denoted by O(P) where P is its orthopivot. It is equivalently :

• the locus of M such that the points P, M, oc(M) are collinear.
• the locus of the intersections of a line through P with the circle which is its antiorthocorrespondent
• the locus of the perspectors of the equilateral triangles centered at P which are perspective with ABC. (Jean-Pierre Ehrmann)

O(P) is a circular circum-cubic and passes through the Fermat points X(13) and X(14). It is a K0 for every point P.

With P = (u : v : w), the equation of O(P) is :

∑ x [(c^2 u – 2 SB w) y^2 – (b^2 u – 2 SC v) z^2] = 0.

The involution Psi

Denote by F1, F2 the real foci of the Steiner inellipse and by F1', F2' the other foci. The singular focus of O(P) is F = Psi(P), the Psi-transform of P. Psi is a quadratic transformation and Psi(P) is equivalently :

• the reflection in the line F1F2 of the inverse of P in the circle with diameter F1F2. Psi is clearly an involution hence F = Psi(P) <=> P = Psi(F).

• the center of the polar conic of P (a rectangular hyperbola) in the cubics K003 (McCay) or K024 (Kjp) hence in any cubic of the pencil generated by these two latter cubics. See Table 22.

• the isogonal conjugate of P in the non proper triangle with vertices X(2) and the circular points at infinity (these are the singular points of Psi). This triangle is the diagonal triangle of the quadrilateral whose vertices are the four foci the Steiner inellipse (these are the fixed points of Psi).

• the pole of P in the pencil of concentric rectangular hyperbolas with center X(2) passing through the four foci the Steiner inellipse. Let (Ha), (Hb), (Hc) be the members of the pencil that pass through A, B, C respectively. The construction of (Ha) is easy since it contains A, A' = reflection of A in X(2), Ab = AB /\ reflection of AC in X(2), Ac = AC /\ reflection of AB in X(2), the infinite points of the bisectors at A in ABC. Moreover the tangent at A to (Ha) passes through X(6). (Ha) is in fact the polar conic of the infinite point of the altitude AH (resp. the sideline BC) in the McCay cubic K003 (resp. the Kjp cubic K024).

***

Here is a list of pairs {P, Psi(P)} by Peter Moses (updated 2022-06-05).

{1, 1054}, {3, 110}, {4, 125}, {5, 3448}, {6, 111}, {8, 11814}, {10, 17777}, {11, 18343}, {13, 14}, {15, 16}, {17, 22739}, {18, 22738}, {20, 5972}, {22, 15462}, {23, 182}, {24, 13198}, {25, 5622}, {32, 14660}, {36, 5197}, {39, 9998}, {51, 14651}, {54, 14656}, {61, 14705}, {62, 14704}, {69, 126}, {74, 14685}, {76, 13518}, {83, 13519}, {98, 1316}, {99, 5108}, {100, 1083}, {101, 46407}, {105, 5091}, {107, 18338}, {109, 46410}, {112, 35901}, {114, 36163}, {115, 6792}, {122, 18337}, {136, 18347}, {137, 18335}, {140, 14683}, {141, 14360}, {147, 11007}, {148, 32525}, {154, 38699}, {184, 186}, {187, 353}, {193, 6719}, {194, 48444}, {195, 14706}, {237, 15920}, {262, 22735}, {316, 34512}, {352, 574}, {371, 7599}, {372, 7598}, {373, 12243}, {376, 5642}, {381, 9140}, {382, 15059}, {403, 1899}, {468, 6776}, {476, 18332}, {485, 13521}, {486, 13520}, {487, 32500}, {488, 32501}, {549, 9143}, {599, 10717}, {616, 619}, {617, 618}, {620, 38940}, {621, 624}, {622, 623}, {627, 14081}, {628, 14082}, {631, 24981}, {663, 41157}, {671, 9169}, {675, 24279}, {691, 9129}, {737, 13515}, {739, 11650}, {846, 5529}, {858, 1352}, {868, 11005}, {1055, 41160}, {1113, 13415}, {1114, 13414}, {1151, 7602}, {1152, 7601}, {1302, 6795}, {1325, 37527}, {1340, 5639}, {1341, 5638}, {1379, 6141}, {1380, 6142}, {1560, 35902}, {1627, 14701}, {1637, 6794}, {1649, 11006}, {1976, 37930}, {1992, 9172}, {1993, 15560}, {1994, 15563}, {1995, 11579}, {2070, 5012}, {2071, 9306}, {2072, 11442}, {2482, 14916}, {2549, 5913}, {2715, 8429}, {2937, 27866}, {2979, 15561}, {3047, 37814}, {3060, 38224}, {3094, 32526}, {3098, 7711}, {3111, 9147}, {3117, 11654}, {3120, 6788}, {3124, 14700}, {3146, 6723}, {3148, 11653}, {3153, 21243}, {3165, 3166}, {3258, 18331}, {3410, 37938}, {3543, 45311}, {3569, 34235}, {3573, 46409}, {3589, 20099}, {3642, 5979}, {3643, 5978}, {3734, 5971}, {3821, 5211}, {3923, 5205}, {4414, 45763}, {4427, 6789}, {5000, 32619}, {5001, 32618}, {5026, 6031}, {5027, 32531}, {5077, 9759}, {5085, 9157}, {5092, 9999}, {5112, 9744}, {5118, 9153}, {5121, 24248}, {5159, 5921}, {5189, 24206}, {5463, 5464}, {5465, 5466}, {5613, 36185}, {5617, 36186}, {5640, 11632}, {5651, 7464}, {5652, 9828}, {5653, 5968}, {5655, 9159}, {5912, 35606}, {5967, 36166}, {5980, 22689}, {5981, 22687}, {6032, 11646}, {6036, 36181}, {6054, 36194}, {6108, 10654}, {6109, 10653}, {6298, 14905}, {6299, 14904}, {6302, 33443}, {6303, 33441}, {6306, 33442}, {6307, 33440}, {6322, 9100}, {6566, 33502}, {6567, 33503}, {6770, 32461}, {6771, 44466}, {6773, 32460}, {6774, 44462}, {6779, 16241}, {6780, 16242}, {6787, 9148}, {6800, 14649}, {7418, 46130}, {7426, 11179}, {7468, 47049}, {7473, 35912}, {7575, 11003}, {7620, 37746}, {7697, 33873}, {7698, 12188}, {7709, 47638}, {7998, 8724}, {8289, 11655}, {8371, 9144}, {8594, 13084}, {8595, 13083}, {9138, 46127}, {9158, 40280}, {9178, 46131}, {9185, 50149}, {9191, 50146}, {9544, 15646}, {9832, 12177}, {10151, 23291}, {10160, 14712}, {10168, 37901}, {10418, 48945}, {10989, 11178}, {11002, 49102}, {11162, 11184}, {11171, 11673}, {11177, 34094}, {11180, 47097}, {11186, 15921}, {11402, 15565}, {11422, 15744}, {11628, 34319}, {11647, 11648}, {11656, 46339}, {11752, 11789}, {11799, 18911}, {12827, 18531}, {13492, 22111}, {13510, 33665}, {13636, 31862}, {13722, 31863}, {14826, 47090}, {14899, 35607}, {14932, 14995}, {15915, 36213}, {17596, 47623}, {17974, 36176}, {18403, 23293}, {18581, 33517}, {18582, 33518}, {18906, 48439}, {20021, 36183}, {20344, 24250}, {20481, 45012}, {22112, 37946}, {22489, 36330}, {22490, 35752}, {26913, 31726}, {30508, 47089}, {30509, 47088}, {31959, 39157}, {35608, 35609}, {35749, 48312}, {35903, 36203}, {35904, 36202}, {35910, 35911}, {36327, 48311}, {37925, 43650}, {38064, 47313}, {38227, 39906}, {38941, 45674}, {39162, 39162}, {39163, 39163}, {39164, 39164}, {39165, 39165}, {39206, 39304}, {39207, 39305}, {39208, 39306}, {39209, 39307}, {40330, 46517}, {45510, 47631}, {45511, 47632}, {46053, 47859}, {46054, 47860}, {46465, 46466}, {47361, 47363}, {47362, 47364}, {48443, 49488}.

***

Psi transforms any curve (C) of degree n into a curve (C') of degree 2n – m, where m is the number of singular points on (C), counting multiplicity. Obviously, (C') must pass through X(2) and the circular points at infinity.

A good number of such transforms are given in the FG paper "Orthocorrespondence and orthopivotal cubics". See also the related Psi-cubics in Table 60.

See a generalization of Psi here.

Orthopivotal cubics O(P)

Some special cases of O(P) :

• O(P) is a pK if and only if P lies on the Napoleon cubic K005 and, in this case, the pivot of O(P) lies on the cubic Kn = K060 = O(X5) and its pole on the cubic Co = K095. The isoconjugate of the pivot lies on the Neuberg cubic K001. The singular focus F lies on Q041. See the blue cells in the table.

• O(P) is a nK (in fact a nK0) if and only if P lies on the cubic K397. F lies on a complicated bicircular quintic.

• O(P) is a focal if and only if P lies on the Brocard (second) cubic K018. In this case, F also lies on K018 which is therefore invariant under Psi. See Table 60 for other Psi-invariant cubics.

• O(P) is a K0+ if and only if P lies at infinity (in which case F = G) or on an axis of the Steiner ellipse (in which case F also lies on this same axis).

• O(P) is singular if and only if P lies on Q015 giving either decomposed cubics or nodal cubics. See the orange cells in the table.

The table gives a selection of interesting O(P) with singular focus F and its isogonal transform O*(P) with singular focus F', the inverse of F in (O).

Notes

• X(3413) and X(3414) are the points at infinity of the asymptotes of the Kiepert hyperbola and also those of the axes of the Steiner ellipses.

• P450 is the reflection of X(98) in the perpendicular bisector of the Fermat points. SEARCH = -3.1454529314. P450 is now X(11005) in ETC (2016-11-23).

• P451 is the intersection of the parallel at X(99) and the perpendicular at G to the Fermat line. SEARCH = 5.1605352720. P451 is now X(11006) in ETC (2016-11-23).

***

Special pencils of orthopivotal cubics

• when P lies on the line at infinity, O(P) has its singular focus at G. It is the locus of M such that the Euler line of the antipedal triangle of M passes through P. On the other hand, O*(P) has its singular focus at X(23). It is the locus of M such that the Euler line of the pedal triangle of M passes through P. See the green lines in the table above and also Q002, Q003 for related quartics.

• when P lies on the Euler line, O(P) contains X(4), X(30) and the singular focus lies on the line X(2), X(98), X(110), etc. The pencil contains K001, K023, K059, K060, K313, K479, K808. K952, K1251.

• when P lies on the Brocard axis, O(P) contains X(15), X(16) and its singular focus lies on the Parry circle. See the yellow cells in the table.The cubic O*(P) is also an orthopivotal cubic with orthopivot the inverse Q of P in the Brocard circle and singular focus also on the Parry circle. It follows that O(P) and O*(P) belong to a same pencil of circular circum-cubics passing through X(13), X(14), X(15), X(16) generated by K001 and K018. See CL034.

• when P lies on the line X(3), X(54), X(97), etc, O(P) contains X(3), X(1157) and the singular focus lies on the circle passing through X(2), X(110), X(2070), etc, which is the Psi-image of the line. These are the isogonal transforms of the cubics of the Neuberg pencil. See pink cells in the table and also Neuberg cubics. The circumcircle inverse of O(P) is a circular circum-cubic passing through X(3), X(54), X(6104), X(6105). See K073 (the inverse of K001), K112 (an inversible cubic), K947 (the inverse of K937) for example.

• when P lies on the line X(2), X(98), X(110), X(114), X(125), etc, O(P) contains X(98), X(542) and the singular focus lies on the Euler line. O*(P) is a K0+ passing through X(15), X(16), X(511), X(842) with singular focus also on the Euler line since it is the inverse in the circumcircle of the focus of O(P).