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On the wikipedia for the trident curve, $xy+ax^3+bx^2+cx=d$, two graphs are shown:

enter image description here enter image description here

Both are for the case where $a=b=c=d=1$, with the first matching what I find in desmos, but the latter being the 'trident curve at $y=\infty$'. What exactly does this notation mean? The plot has domains $x,y\in[-1.0,1.0]$, which I imagine corresponds to $(-\infty,\infty)$ in some sense, but I'm not gleaning much beyond that. Nor am I sure how to interpret the loop which comes about in the second plot.

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It is explained to some extent in the link. Work in the projective plane, so introduce a variable $z$ and make every term in the polynomial homogeneous of degree 3:

$$F(x,y,z) = xyz + ax^3 + bx^2z + cxz^2 - dz^3 = 0.$$

The original curve is obtained by setting $z=1$. Points at infinity are given by setting $z=0$; we see that we have the unique point $[0,1,0]$. To look nearby, we set $y=1$ and use the variables $x$ and $z$, obtaining

$$xz+ax^3+bx^2z+cxz^2-dz^3=0.$$

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  • $\begingroup$ In the second plot, the vertical axis probably should have been labeled $z$ to be consistent with the text. $\endgroup$
    – arkeet
    Commented Jul 3 at 21:38
  • $\begingroup$ Agreed. Especially since $y=\infty$ becomes $y=0$ … $\endgroup$
    – Ted Shifrin
    Commented Jul 3 at 22:26

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