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pK60+

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)

X(14844)

K097

 

X(11070)

X(1138)

X(3258)

Fermat K515

 

X(11080)

X(13)

X(13)

see below

 

X(11085)

X(14)

X(14)

see below

 

X(1990)

X(5667)

X(14847)

K543

 

Remark 1 : when P is one of the Fermat points X(13) or X(14), the pK60+ decomposes into the cevian lines of P. The corresponding poles are the barycentric squares X(11080), X(11085) of X(13), X(14) respectively.

Remark 2 : the hessian cubic of a pK60+ is always a focal cubic with singular focus the common point X of the three asymptotes.

 

Other stelloids

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

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.

stelloid

X = X(i)

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

13364

o

 

n

spK

4, 5, 143

 

K071

5891

o

 

 

psK, spK

4, 5, 20, 76, 5562, 15318

 

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, 15318

K405

K094

599

o

 

 

nK

 

 

K097

14844

o

 

 

pK

1, 79

 

K100

3

 

c

 

 

1, 3, 40, 1670, 1671

 

K115

3060

o

 

 

spK0

4, 6243

 

K139

5627

o

c

 

nK

30, 5627

 

K205

14846

o

c

 

nK

 

 

K213

2

o

c

 

nK

2, 11058

 

K230

14629

o

 

n

cK

80, 2222

 

K258

549

 

 

n

 

1, 3, 5, 39, 2140

 

K268

3819

o

 

 

spK0

4, 20, 140, 15318

 

K309

5054

o

 

 

spK

3, 4, 376, 1340, 1341

 

K358

3545

o

 

 

spK

3, 4, 381

 

K412

14845

o

 

 

spK

2, 4, 5, 51, 262

 

K513

14846

o

 

 

 

6, 15, 16, 74, 265, 3016

 

K514

6781

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

14847

o

 

 

pK

5667

 

K580

568

o

 

 

spK

4, 847, 5889

 

K581

5055

o

 

 

spK

2, 3, 4, 262

 

K582

14848

o

 

 

spK

2, 4, 6, 262, 4846

 

K594

5603

o

 

n

spK

1, 4, 1482

 

K595

14849

o

 

n

 

74, 98, 265, 290, 671, 9140

 

K596

14850

o

 

n

 

74, 99, 265, 290

 

K597

14851

o

 

n

 

30, 74, 265, 477

 

K598

6

 

 

 

 

 

 

K613

14643

o

 

 

nK, spK

4, 110, 1113, 1114

 

K643

14561

o

 

 

spK0

4, 6, 4846, 8743

 

K665

549

o

 

 

spK

3, 4, 39, 550

K664

K669

14644

o

 

 

nK, spK

4, 74, 265

 

K670

14852

o

 

 

psK, spK

4, 26, 64, 68, 847

 

K708

14853

o

 

 

spK0

4, 1344, 1345, 1351

 

K714

14640

o

 

 

spK

4

 

K724

14854

o

 

n

psK

74, 265, 5961, 6344, 11060

 

K827

14855

o

 

 

spK

4, 20, 550, 15318

 

K833

2

 

c

 

 

2, 3, 381

 

K852

376

o

 

 

spK

3, 4, 1657

 

K910

140

 

 

 

 

1, 3, 546, 1385

 

K911

5054

 

 

 

 

1, 3, 381, 1340, 1341, 3576

 

K928

10606

o

 

 

spK

4, 20, 64, 68, 12084, 15318

 

K929

15061

o

 

 

 

74, 265, 1113, 1114

 

K930

15362

o

 

 

spK

2, 4, 23, 74, 262, 265

 

Additional remarks :

• See the related Table 54 for green cells. These are the spK(X3, Q on the Euler line).

• the circum-cubics highlighted in yellow are those of Table 51 : they are spK(X3, Q on the Brocard axis). They form a pencil of K0s passing through the infinite points of K003, X(4) and the imaginary foci of the Brocard ellipse i.e. common points of the Brocard axis and the Kiepert hyperbola. The asymptotes concur at X on the line X(2), X(51), X(262), X(263), X(373), X(511), X(2979), X(3060), etc.