Home page | Catalogue | Classes | Tables | Glossary | Notations | Links | Bibliography | Thanks | Downloads | Related Curves

When four tangents are drawn from a point M on the Thomson cubic to the Thomson cubic itself, their anharmonic ratio is constant (Salmon) and, up to a permutation of these tangents, is equal to :

which is obtained when the tangents are those drawn from K to the cubic namely AK, BK, CK, GK in this order.

Now, given another pivotal cubic pK with pole Ω=p:q:r and pivot P=u:v:w, the tangents at A, B, C, P concur at P* and their anharmonic ratio is evaluated similarly.

pK is said to be equivalent to the Thomson cubic if and only if these ratios are equal. This gives the condition :

For a given pivot P, the pole Ω must lie on the circum-conic with perspector X512 x P^2. For example, any pK with pivot G must have its pole on the circum-conic through G and K.

For a given pole Ω, the pivot P must lie on the diagonal conic passing through the square roots of Ω and whose center is Ω÷X512. For example, any isogonal pK must have its pivot on the Steiner hyperbola i.e. the diagonal conic through the in/excenters and G. This is the polar conic of G in the Thomson cubic.

Any projection or linear transformation, any isoconjugation with pole Q transform the Thomson cubic into an equivalent cubic. For example, the barycentric product of the Thomson cubic by a point Q is the cubic pK(X6 x Q^2, Q), a cubic equivalent to the Thomson cubic for any point Q. Any such cubic is called a multiple of the Thomson cubic (yellow lines in the table).

The following table gives a selection of such cubics equivalent to the Thomson cubic.

Ω

P

points on the cubic Xi for i =

cubic

6

2

1, 2, 3, 4, 6, 9, 57, 223, 282, 1073, 1249

K002 Thomson

6

20

1, 3, 4, 20, 40, 64, 84, 1490, 1498, 2130, 2131

K004 Darboux

2

69

2, 4, 7, 8, 20, 69, 189, 253, 329, 1032, 1034

K007 Lucas

37

8

1, 4, 8, 10, 40, 65, 72

K033 Spieker central

2

75

1, 2, 7, 8, 63, 75, 92, 280, 347, 1895

K034 Spieker perspector

394

69

2, 3, 20, 63, 69, 77, 78, 271, 394

K099 Darboux perspector

32

3

3, 6, 25, 55, 56, 64, 154, 198, 1033, 1035, 1436

K172

32

1

1, 6, 19, 31, 48, 55, 56, 204, 221, 2192

K175

76

76

2, 69, 75, 76, 85, 264, 312

K184

81

86

1, 2, 7, 21, 29, 77, 81, 86

K317

1333

21

1, 3, 21, 28, 56, 58, 84, 1394, 2360

K318

1333

81

1, 6, 57, 58, 81, 222, 284, 1172, 1433

K319

6

63

1, 9, 19, 40, 57, 63, 84, 610, 1712, 2184

K343

81

86 x 329

7, 20, 21, 27, 63, 84

K344

37

2

1, 2, 9, 10, 37, 226, 281, 1214

K345

1501

6

6, 25, 31, 32, 41, 184, 604, 2199

K346

213

1

1, 6, 33, 37, 42, 55, 65, 73, 2331

K362

321

75

2, 8, 10, 75, 307, 318, 321, 1441

K366

2207

4

4, 6, 19, 25, 33, 34, 64, 208, 393

K445

6 x 577

3

3, 6, 48, 154, 184, 212, 577, 603, 2188

K576

2052

264

2, 4, 92, 253, 264, 273, 318, 342, 2052

K647

213

40

19, 40, 55, 64, 65, 71, 2357

K750

393

2

2, 4, 278, 281, 393, 1249

K879

15

298

1, 2, 6, 298, 616

16

299

1, 2, 6, 299, 617

32

610

6, 19, 48, 198, 610, 1436, 2155

32

1498

6, 64, 154, 221, 1498, 2192

37

329

4, 9, 72, 226, 329, 1490, 1903, 2184

76

304

75, 85, 92, 304, 309, 312, 322

81

333

2, 27, 57, 63, 81, 189, 333, 1817

213

9

6, 9, 19, 37, 71, 198, 1400, 1903

213

1490

33, 64, 73, 198, 1490, 1903

220

8

1, 8, 9, 40, 55, 200, 219, 281

249

99

99, 110, 643, 648, 662, 1414

279

85

2, 7, 57, 77, 85, 189, 273, 279

279

348

2, 7, 278, 279, 347, 348, 1440

393

92

1, 4, 19, 92, 158, 196, 278, 281, 2184

393

1895

1, 4, 158, 1712, 1895

394

326

1, 63, 77, 78, 326

577

63

1, 3, 48, 63, 219, 222, 255, 268, 610

577

394

3, 6, 219, 222, 394, 1073, 1433, 1498

593

86

21, 27, 58, 81, 86, 285, 1014, 1790

762

10

10, 12, 37, 201, 210, 594, 756

1333

1817

3, 28, 57, 284, 610, 1436, 1817

1407

7

1, 7, 56, 57, 84, 222, 269, 278

1427

2

2, 57, 223, 226, 278, 1214, 1427, 2184

1427

7

1, 4, 7, 57, 65, 196, 226, 1439

1427

347

1, 278, 347, 1214

1446

75

7, 75, 253, 273, 307, 347, 1441

1446

85

2, 7, 85, 92, 226, 342, 1441, 1446

1500

10

10, 37, 42, 65, 71, 210, 227, 1826

2052

75

75, 92, 158, 273, 318, 1895

2205

6

6, 31, 41, 42, 213, 607, 1400, 1409

2205

55

25, 31, 42, 55, 228, 1402, 2187, 2357

2205

198

25, 41, 198, 228, 1400, 2155

2207

1

1, 19, 33, 34, 204, 207, 1096, 2331

2207

1249

6, 19, 393, 1033, 1249, 2331

2207

1712

19, 204, 208, 1712

2287

333

2, 8, 9, 21, 63, 271, 333, 2287, 2322

2287

1043

1, 8, 20, 21, 29, 78, 280, 1043

6 x 220

9

6, 9, 33, 41, 55, 198, 212, 220