 Intersecting Plane Curves - Maple Help

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Intersecting plane curves

For Maple17, a new command, intersectcurves, has been added to the algcurves package.

 > $\mathrm{with}\left(\mathrm{algcurves}\right):$

Given two plane curves expressed as polynomials in two variables,

 > $\mathrm{c1}:=-{x}^{3}+2{y}^{2}+x:$$\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{c2}≔{x}^{2}-{y}^{2}-1:$

The intersectcurves command computes the points of intersection along with their multiplicities. Each set of intersection points is expressed as a list containing three values. The first two are irreducible polynomials, the second of which is in only one of the variables. The third value is either 0 or 1. When the third value is 1, the point is affine; when the third value is 0, it is a point in the projective plane (at infinity).

 > $\mathrm{intersectcurves}\left(\mathrm{c1},\mathrm{c2},x,y\right)$
 $\left[\left[{2}{,}\left[{1}{+}{x}{,}{y}{,}{1}\right]\right]{,}\left[{2}{,}\left[{-}{1}{+}{x}{,}{y}{,}{1}\right]\right]{,}\left[{1}{,}\left[{-}{2}{+}{x}{,}{{y}}^{{2}}{-}{3}{,}{1}\right]\right]\right]$ (1)

This is to be interpreted as follows. There are two intersection points of multiplicity 2, namely, (1,0) and (-1,0), indicating that they are points where the curves intersect tangentially. In addition, there are two intersection points of multiplicity 1, namely, $\left(2,\sqrt{3}\right)$ and $\left(2,-\sqrt{3}\right)$. We can see all intersection points graphically with a simple implicit plot.

 > $\mathrm{with}\left(\mathrm{plots}\right):$
 > $\mathrm{implicitplot}\left(\left[\mathrm{c1},\mathrm{c2}\right],x=-3..3,y=-3..3,\mathrm{gridrefine}=3,\mathrm{scaling}=\mathrm{constrained},\mathrm{color}=\left["Red","Blue"\right]\right)$ The following example illustrates the case when there are intersection points at infinity.

 > $\mathrm{d1}:={x}^{2}y-y-1:$$\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{d2}:={x}^{2}-x:$
 > $\mathrm{intersectcurves}\left(\mathrm{d1},\mathrm{d2},x,y\right)$
 $\left[\left[{1}{,}\left[{x}{,}{y}{+}{1}{,}{1}\right]\right]{,}\left[{5}{,}\left[{x}{,}{1}{,}{0}\right]\right]\right]$ (2)
 > $\mathrm{implicitplot}\left(\left[\mathrm{d1},\mathrm{d2}\right],x=-3..3,y=-3..3,\mathrm{gridrefine}=3,\mathrm{scaling}=\mathrm{constrained},\mathrm{color}=\left["Red","Blue"\right]\right)$ There is one affine intersection point (0,-1) of multiplicity 1, plus one intersection point at infinity, of multiplicity 5. The tangent at this point is the line $x=0$. Each of the two blue vertical lines intersects each of the two asymptotes of the red curve at the infinite point $\left(0,1,0\right).$ All but one of these intersections have multiplicity 1, and the exceptional one, between the blue line and the asymptote to the red curve at $x=1$, comes from a tangential intersection, of multiplicity 2.

We can perform a change of coordinates in order to plot this in the affine plane. First, we homogenize the equations, introducing a third variable $z$, and then we substitute $y=1$ in order to move the intersection point that was previously at infinity to the origin.

 >
 ${\mathrm{h1}}{:=}{-}{{z}}^{{3}}{+}{{x}}^{{2}}{-}{{z}}^{{2}}$
 ${\mathrm{h2}}{:=}{{x}}^{{2}}{-}{x}{}{z}$ (3)
 > $\mathrm{implicitplot}\left(\left[\mathrm{h1},\mathrm{h2}\right],x=-1.5..1.5,z=-1.5..1.5,\mathrm{gridrefine}=3,\mathrm{scaling}=\mathrm{constrained},\mathrm{color}=\left["Red","Blue"\right]\right)$ > $\mathrm{intersectcurves}\left(\mathrm{h1},\mathrm{h2},x,z\right)$
 $\left[\left[{5}{,}\left[{x}{,}{z}{,}{1}\right]\right]{,}\left[{1}{,}\left[{x}{,}{z}{+}{1}{,}{1}\right]\right]\right]$ (4)

After the coordinate transformation, all intersection points (of $\mathrm{h1}$ and $\mathrm{h2}$) are visible in the plot and there are no intersection points at infinity. See Also