Re: Determining continuity of regions/curves from inequalities

• To: mathgroup at smc.vnet.net
• Subject: [mg67271] Re: [mg67216] Determining continuity of regions/curves from inequalities
• From: "Carl K. Woll" <carlw at wolfram.com>
• Date: Thu, 15 Jun 2006 03:25:54 -0400 (EDT)
• References: <200606130506.BAA23751@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Bonny Banerjee wrote:
> Is there an easy way in Mathematica to determine whether the region or curve
> formed by a system of inequalities is continuous or not?
>
> For example, the output of some function (e.g. Reduce) might be as follows:
>
> x>2 && y>0
>
> which forms a continuous region. Again, the following output
>
> (x<2 && y<0) || (x>2 && y>0)
>
> is not continuous. Similarly, for curves.
>
> Given such a system of inequalities, how to determine whether the
> region/curve it forms is continuous or not? Or in other words, if I pick any
> two random points, say P1 and P2, lying on the output curve/region, does
> there exist a continuous path lying entirely within the output curve/region
> from P1 to P2?
>
> Any help will be appreciated.
>
> Thanks,
> Bonny.
>

The function SemialgebraicComponents in the package
Algebra`AlgebraicInequalities may help. From the help browser:

"The package provides a function for solving systems of strong
polynomial inequalities in one or more unknowns. To be precise,
SemialgebraicComponents[ineqs, vars] gives a finite set of solutions of
the system of inequalities. That is, within the set of solutions, any
solution can be connected by a continuous path to a solution in the
finite set. The variable ineqs is a list of strong inequalities, where
both sides of each inequality are polynomials in variables vars with
rational coefficients. In other words, SemialgebraicComponents[ineqs,
vars] gives at least one point in each connected component of the open
semialgebraic set defined by inequalities ineqs. "

Needs["Algebra`AlgebraicInequalities`"]

First example: x>2 && y>0

SemialgebraicComponents accepts a list of inequalities:

In[6]:=
SemialgebraicComponents[{x>2,y>0},{x,y}]

Out[6]=
{{3,1}}

Only one point is returned, so the region is connected.

Second example: (x<2 && y<0) || (x>2 && y>0)

This example is not a list of inequalities due to the Or. However, in
this case it is easy to construct an equivalent inequality that
encompasses the same region:

In[5]:=
SemialgebraicComponents[(x-2)y>0,{x,y}]

Out[5]=
{{0,-1},{3,1}}

There are two components, so the region is not connected.

Carl Woll
Wolfram Research

```

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