The CellLocation command returns a non-negative integer, the index of the open cell in which the point lies, or $0$ if the point does not lie in any of the open cells of $m$.

The CellLocation command determines the cell of $m$ in which the given point lies.

The point can be specified in two different formats:

as a list s of equations of the form parameter=rational number, or

as a list p of rational numbers, in which case the $i$th parameter in m:-Parameters gets replaced by $p_i$, for all $i$.

This command is part of the RootFinding[Parametric] package, so it can be used in the form CellLocation(..) only after executing the command with(RootFinding[Parametric]). However, it can always be accessed through the long form of the command by using RootFinding[Parametric][CellLocation](..).

$\mathrm{with}\left(\mathrm{RootFinding}\left[\mathrm{Parametric}\right]\right)\:$

$m≔\mathrm{CellDecomposition}\left(\left[x^2+y^2=a\&comma;x-y=b\&comma;0<a\right]\&comma;\left[x\&comma;y\right]\right)$

$m≔\left[\begin{array}{lll}\mathrm{Equations}& \&equals;& \left[x^2+y^2-a\&comma;x-y-b\right]\\ \mathrm{Inequalities}& \&equals;& \left[a\right]\\ \mathrm{Filter}& \&equals;& 0\ne 1\\ \mathrm{Variables}& \&equals;& \left[x\&comma;y\right]\\ \mathrm{Parameters}& \&equals;& \left[a\&comma;b\right]\\ \mathrm{DiscriminantVariety}& \&equals;& \left[\left[a\right]\&comma;\left[-b^2+2a\right]\right]\\ \mathrm{ProjectionPolynomials}& \&equals;& \left[\left[b\right]\&comma;\left[a\&comma;-b^2+2a\right]\right]\\ \mathrm{SamplePoints}& \&equals;& \left[\left[a=\frac{302231454903657293676531}{1208925819614629174706176}\&comma;b=\mathrm{-1}\right]\&comma;\left[a=1\&comma;b=\mathrm{-1}\right]\&comma;\left[a=\frac{302231454903657293676531}{1208925819614629174706176}\&comma;b=1\right]\&comma;\left[a=1\&comma;b=1\right]\right]\end{array}\right.$

$\mathrm{CellPlot}\left(m\&comma;'\mathrm{samplepoints}'\right)$

$\mathrm{CellLocation}\left(m\&comma;\left[a=\frac{1}{2}\&comma;b=3\right]\right)$

$3$

$\mathrm{CellLocation}\left(m\&comma;\left[1\&comma;-1\right]\right)$

$2$

$\mathrm{CellLocation}\left(m\&comma;\left[a=\frac{1}{2}\&comma;b=1\right]\right)$

$0$

$\mathrm{CellLocation}\left(m\&comma;\left[-1\&comma;1\right]\right)$

$0$