This question is about three specific types of irreducible cubic curves in the plane with genus zero. Additionally, any curve which is not the zero set of a polynomial in two variables with real coefficients is not a curve this question is concerned with.
- The first type is the type which $y^2=x^2(x-1)$ is an example of
- The second type is the cusp curve which $y^2=x^3$ is an example of
- The third type is the alpha curve, which $y^2=x(x-1)^2$ is an example of
After some appropriate translation and some appropriate rotation, any arbitrary cubic curve which falls into one of the above three categories can be written in the form
$$y^{2}=ax^{3}-3a\sqrt[3]{\frac{b^{2}}{4}}x+ab$$
where $a\ne0$ and $a,b$ $\in$ $ℝ$
- This is a cusp curve if and only if $b=0$
- This is an alpha curve if and only if $ab>0$
- This is neither a cusp curve nor an alpha curve if and only if $ab<0$
Let $t$ $\in$ $[0,2\pi)$. Let $h,k$ $\in$ $ℝ$.
Replacing $x$ with $x\cos(t)+y\sin(t)-h$ and replacing $y$ with $-x\sin(t)+y\cos(t)-k$ gives
$$0=a\left(\cos\left(t\right)\right)^{3}x^{3}+3a\cos^{2}\left(t\right)\sin\left(t\right)x^{2}y+3a\sin^{2}\left(t\right)\cos\left(t\right)xy^{2}+a\left(\sin\left(t\right)\right)^{3}y^{3}+\left(-3ah\cos^{2}\left(t\right)-\sin^{2}\left(t\right)\right)x^{2}+2\sin\left(t\right)\cos\left(t\right)\left(1-3ah\right)xy+\left(-3ah\sin^{2}\left(t\right)-\cos^{2}\left(t\right)\right)y^{2}+\left(3ah^{2}\cos\left(t\right)-2k\sin\left(t\right)-3a\sqrt[3]{\frac{b^{2}}{4}}\cos\left(t\right)\right)x+\left(3ah^{2}\sin\left(t\right)+2k\cos\left(t\right)-3a\sqrt[3]{\frac{b^{2}}{4}}\sin\left(t\right)\right)y+ab+3a\sqrt[3]{\frac{b^{2}}{4}}h-ah^{3}-k^{2}$$ $\tag1$
Using (1) one can make a number of observations about what conditions must be satisfied by $A, B, C, D, E, F, G, H, I$ and $J$ in order for the zero set of $Ax^3+Bx^2y+Cxy^2+Dy^3+Ex^2+Fxy+Gy^2+Hx+Iy+J$ to be one of the three kinds of curves which this question is concerned with.
My observations so far are the following:
- $8AD(E+G+2)^3=F^3(\sqrt[3]{A^2}+\sqrt[3]{D^2})^3$
- $BC=9AD$
- $AC^3=B^3D$
- $A$ and $D$ are not both zero
- Either $ABCDF\ne0$ or $AD=0=B=C=F$ (exactly one of these two statements must be true)
Of course, conditions 1,2 and 3 are arbitrary, since one could always use the three equations to derive a new set of three equations which together carry the same information.
Clearly, more conditions - regarding $H,I$ and $J$ - are needed in order to complete the list of conditions that must be satisfied by $A, B, C, D, E, F, G, H, I$ and $J$ in order for the zero set of $Ax^3+Bx^2y+Cxy^2+Dy^3+Ex^2+Fxy+Gy^2+Hx+Iy+J$ to be one of the three kinds of curves which this question is concerned with.
Which conditions are missing from the list I have so far?
Another idea I had was to figure out conditions that must be satisfied by $A, B, C, D, E, F, G, H, I$ and $J$ in order for the zero set of $Ax^3+Bx^2y+Cxy^2+Dy^3+Ex^2+Fxy+Gy^2+Hx+Iy+J$ to be irreducible and have a rational parametrization. However, I'm not sure how to do this.