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MathGroup Archive 2006

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Re: Determining continuity of regions/curves from inequalities

  • To: mathgroup at smc.vnet.net
  • Subject: [mg67277] Re: [mg67216] Determining continuity of regions/curves from inequalities
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Thu, 15 Jun 2006 03:26:16 -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. 
> 

One approach would be to see if any pair of boundary curves intersect, 
and, if so, whether intersection points are in both regions. This 
reduces (if you will) the problem to solving bivariate polynomials over 
the reals.

This will only tell you if it is connected. Actual computation of paths 
would be a separate matter.

Daniel Lichtblau
Wolfram Research


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