The EnclosingBox function computes a rectangular box containing a point defined by point or by variables and values. The main property of the returned box is that its interior has an empty intersection with the real roots of the polynomials given in constraints.

The box returned is maximal in the sense that the borders of the box contain at least one real root of the constraint polynomials.

constraints is a polynomial $p$ (or a list of polynomials $p_1,\mathrm{...},p_k$) with rational coefficients.  EnclosingBox computes a box that contains no real root of $p$ (or, for a list, no real root of any of the polynomials $p_1,\mathrm{...},p_k$) in its interior.

In the EnclosingBox(constraints, variables, values) calling sequence, variables is a list of variable names $\left[v_1\,\mathrm{...}\,v_n\right]$ and values is a list of numerical values $\left[r_1\,\mathrm{...}\,r_n\right]$.  These are used to specify a point, described by the coordinates $v_1=r_1,\mathrm{...},v_n=r_n$.

pointlist and valueslist can be used to specify a list of points. In this case, an enclosing box will be computed for each point.

In the last calling sequence, decomposition is a data structure returned by the command CellDecomposition. The constraints are taken from the field decomposition:-DiscriminantVariety and the points coordinates are taken from decomposition:-SamplePoints

You can specify the output format.  If output = list, the output is a box described by a set of intervals of the form $\mathrm{name}=\left[\mathrm{rational}\,\mathrm{rational}\right]$ where $\mathrm{name}$ is a variable and the two numerics are the bound of the interval.  This is the default.

If output=plot and the number of variable is less than or equal to 3, then the output is a plot that displays the point, the enclosing box, and the constraints.

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

$p≔x^2+y^2-1$

$p≔x^2+y^2-1$

$\mathrm{inbox}≔\mathrm{EnclosingBox}\left(p\,\left[x=0\,y=0\right]\right)\:$

$\mathrm{evalf}\left(\mathrm{inbox}\right)$

$\left[\left[x=\left[\mathrm{-0.7071067811}\,0.7071067811\right]\,y=\left[\mathrm{-0.7071067812}\,0.7071067812\right]\right]\right]$

$\mathrm{outbox}≔\mathrm{EnclosingBox}\left(p\,\left[x=1\,y=1\right]\right)\:$

$\mathrm{evalf}\left(\mathrm{outbox}\right)$

$\left[\left[x=\left[0.7071067812\,Float\left(\mathrm{\infty }\right)\right]\,y=\left[0.7071067812\,Float\left(\mathrm{\infty }\right)\right]\right]\right]$

$\mathrm{EnclosingBox}\left(p\,\left[\left[x=0\,y=0\right]\,\left[x=1\,y=1\right]\right]\,\mathrm{output}='\mathrm{plot}'\right)$

$\mathrm{evalf}\left(\mathrm{EnclosingBox}\left(p\,\left[x\,y\right]\,\left[\left[0\,0\right]\,\left[1\,1\right]\right]\,\mathrm{output}='\mathrm{list}'\right)\right)$

$\left[\left[x=\left[\mathrm{-0.7071067811}\,0.7071067811\right]\,y=\left[\mathrm{-0.7071067812}\,0.7071067812\right]\right]\,\left[x=\left[0.7071067812\,Float\left(\mathrm{\infty }\right)\right]\,y=\left[0.7071067812\,Float\left(\mathrm{\infty }\right)\right]\right]\right]$

$\mathrm{pts}≔\mathrm{WitnessPoints}\left(p\ne 0\right)$

$\mathrm{pts}≔\left[\left[x=\mathrm{-2}\,y=0\right]\,\left[x=0\,y=\mathrm{-2}\right]\,\left[x=0\,y=0\right]\,\left[x=0\,y=2\right]\,\left[x=2\,y=0\right]\right]$

$\mathrm{EnclosingBox}\left(p\,\mathrm{pts}\,\mathrm{output}='\mathrm{plot}'\right)$

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

$\mathrm{dec}≔\mathrm{CellDecomposition}\left(\left[x^6+a{y}^2-a=0\,a^3{x}^6+a{y}^2-b=0\right]\,\left[x\,y\right]\right)\:$

$\mathrm{EnclosingBox}\left(\mathrm{dec}\,\mathrm{output}='\mathrm{plot}'\right)$