Quadric, cubic and quartic cones

Anatoly B. Korchagin, David A. Weinberg

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


There are 2 irreducible quadric cones (real and imaginary) required for obtaining the affine classification of the 4 irreducible conic sections. According to Newton there are 5 irreducible cubic cones required for obtaining his classification of 59 irreducible cubic sections. In this historical survey paper we show how it follows from Gudkov's classification of forms of real projactive quartic curves that 1037 quartic cones are required for obtaining a similar classification of irreducible quartic sections. We also present the singular-isotopy classification of the unions of irreducible affine cubic curves with their asymptotes, which consists of 99 classes. This classification sheds a new light on Newton's famous classification consisting of 78 species.

Original languageEnglish
Pages (from-to)1627-1656
Number of pages30
JournalRocky Mountain Journal of Mathematics
Issue number5
StatePublished - 2005


  • Topological and isotopy classifications of real algebraic curves


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