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The pK with pole W = (p : q : r) and pivot P = (u : v : w) is a pK60+ if and only if P lies on the Neuberg cubic= K001. Its pole W lies on the cubic Co = K095. The locus of the common point of the asymptotes is the bicircular quartic Q004.

The correspondences between W and P are given by the formulas :

The table shows some remarkable examples of pK60+ with the intersection X of the asymptotes on Q004.

pole W

pivot P

point X

cubic

hessian or centers on the hessian

X(6)

X(3)

X(2)

McCay K003

McCay hessian cubic K048

X(53)

X(4)

X(51)

McCay orthic K049

 

X(395)

X(617)

X(619)

Fermat K046b

X(533), X(619)

X(396)

X(616)

X(618)

Fermat K046a

X(532), X(618)

X(1989)

X(30)

X(476)

Tixier K037

X(13), X(14), X(476), X(542)

X(2160)

X(1)

(2SA + bc + ca + ab)/(2SA+ bc) : :

K097

 

X(30)xX(1138)

X(1138)

X(3258)

Fermat K515

 

X(13)^2

X(13)

X(13)

see below

 

X(14)^2

X(14)

X(14)

see below

 

X(1990)

X(5667)

?

K543

 

Note : when P is one of the Fermat points X(13) or X(14), the pK60+ decomposes into the cevian lines of P.

The hessian cubic of a pK60+ is always a focal cubic with singular focus the common point X of the three asymptotes.

 

Stelloids

The following table gathers together all listed stelloids with their respective types and their listed isogonal transforms (if any) in the last column.

See the related Table 54 for green cells. These are the spK(X3, Q on the Euler line). The orange (resp. light blue) cells correspond to stelloids having the same asymptotic directions as K003 (resp. K024). Their isogonal transforms are CircumNormal (resp. CircumTangential) cubics as in Table 25.

The "orange" cubics are called McCay stelloids when they are circum-cubics and then, they are spK(X3, Q) for some Q. The asymptotes concur at X such that QX = 1/3 QH (vectors). See here for further properties.

Notations :

o, c, n denote a circumscribed, central, nodal cubic respectively.

1/3 (i, j) means that X is the image of X(j) under the homothety with center X(i), ratio 1/3.

stelloid

X = X(i) for i = or ...

o

c

n

type

X(i) on the curve for i =

stelloid*

K003

2

o

 

 

pK

1, 3, 4, 1075, 1745, 3362

K003

K024

2

o

 

 

nK0

 

K024

K026

5

o

c

 

psK, spK

3, 4, 5, 5403, 5404, 8798

K361

K028

381

o

 

n

psK, spK

3, 4, 8, 76, 847, 3557, 3558, 3730, 8743, 10571

K009

K037

476

o

 

 

pK

30, 5627

K374

K046a

618

o

c

 

pK

13, 616, 618

 

K046b

619

o

c

 

pK

14, 617, 619

 

K049

51

o

 

 

pK

4, 5, 52, 847

K373

K054

midpoint of 5, 51

o

 

n

spK

4, 5, 143

 

K071

5891

o

 

 

psK, spK

4, 5, 20, 76, 5562

 

K077

376

 

 

 

 

1, 3, 20, 170, 194, 7991

 

K078

3524

 

 

 

 

1, 2, 3, 165, 5373, 6194

 

K080

3

o

c

 

spK

3, 4, 20, 1670, 1671

K405

K094

599

o

 

 

nK

 

 

K097

79 x 1654

o

 

 

pK

1, 79

 

K100

3

 

c

 

 

1, 3, 40, 1670, 1671

 

K115

2, 511 /\ 6, 22

o

 

 

spK0

4, 6243

 

K139

1989 รท 30

o

c

 

nK

30, 5627

 

K205

2, 249 /\ 111, 265

o

c

 

nK

 

 

K213

2

o

c

 

nK

2, 11058

 

K230

1/3 (80, 2222)

o

 

n

cK

80, 2222

 

K258

549

 

 

n

 

1, 3, 5, 39, 2140

 

K268

2, 511 /\ 3, 64

o

 

 

spK0

4, 20, 140

 

K309

5054

o

 

 

spK

3, 4, 376, 1340, 1341

 

K358

3545

o

 

 

spK

3, 4, 381

 

K412

1/3 (5, 51)

o

 

 

spK

2, 4, 5, 51, 262

 

K513

1/3 (187, 265)

o

 

 

 

6, 15, 16, 74, 265, 3016

 

K514

3, 1506 /\ 20, 32

o

 

 

spK

4, 15, 16, 39, 6781

 

K515

3258

o

 

 

pK

30, 1138

 

K516

262

o

 

 

spK0

4, 3095

 

K525

4

o

c

 

spK

3, 4, 382

 

K543

1/3 (107, 125)

o

 

 

pK

5667

 

K580

568

o

 

 

spK

4, 847, 5889

 

K581

5055

o

 

 

spK

2, 3, 4, 262

 

K582

1/3 (6, 381)

o

 

 

spK

2, 4, 6, 262, 4846

 

K594

1/3 (1, 4)

o

 

n

spK

1, 4, 1482

 

K595

1/3 (98, 265)

o

 

n

 

74, 98, 265, 290, 671, 9140

 

K596

1/3 (99, 265)

o

 

n

 

74, 99, 265, 290

 

K597

1/3 (477, 265)

o

 

n

 

30, 74, 265, 477

 

K598

6

 

 

 

 

 

 

K613

1/3 (5, 110)

o

 

 

nK, spK

4, 110, 1113, 1114

 

K643

centroid of OHK

o

 

 

spK

4, 6, 4846, 8743

 

K665

549

o

 

 

spK

3, 4, 39, 550

K664

K669

1/3 (5, 265)

o

 

 

nK, spK

4, 74, 265

 

K670

1/3 (5, 68)

o

 

 

psK, spK

4, 26, 64, 68, 847

 

K708

1/3 (6, 4)

o

 

 

spK

4, 1344, 1345, 1351

 

K714

3, 107 /\ 5, 53

o

 

 

spK

4

 

K724

1/3 (265, 5961)

o

 

n

psK

74, 265, 5961, 6344, 11060

 

K827

3, 64 /\ 20, 52

o

 

 

spK

4, 20, 550

 

K852

376

o

 

 

spK

3, 4, 1657