Question

What is a cubic curve called?

Answer 1

An algebraic curve of curve order 3 is called a cubic curve.

What do you mean by cubic curve?

In the event that a cubic curve has a singular point, it can be parametrized in terms of a projective line. In contrast, it is known that a non-singular cubic curve, over an algebraically closed field like the complex numbers, has nine points of inflexion.

Equation f(X,Y)=0 describes an algebraic curve over a field K, where f(X,Y) is a polynomial in X and Y with coefficients in K, and the degree of f is the highest degree that any one of its terms can reach (monomials).

Newton demonstrated that the projection of the five divergent cubic parabolas can produce all cubics. The "Curves" chapter of John Harris' 1710 London-published Lexicon Technicum included Newton's classification of cubic curves. Newton also divided all cubics into 72 types, but six of them were left out. Additionally, he demonstrated that any cubic can be obtained by projecting the elliptic curve in the proper manner.

                        $y^2 = ax^3+bx^2+cx+d$                                      ....................(1)

where the general cubic can alternatively be expressed as, and the projection is a birational transformation.

                               $y^2 = x^3+ax+b$                                           ...................(2)

Equations of the type are the first class in Newton:

                 $xy^2+ey = ax^3+bx^2+cx+d$                                    ...................(3)

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The serpentine curve is one of the subcases of this case, which is the toughest case. The third graders were:

                           $ay^2=x(x^2-2bx+c)$                                     ......................(4)

which are also referred to as Newton's diverging parabolas. The Newton trident was his 66th curve.

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