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Topic: an obvious geometric property (but proof required)
Replies: 21   Last Post: Jun 13, 2014 5:22 PM

 Messages: [ Previous | Next ]
 Mike Terry Posts: 767 Registered: 12/6/04
Re: an obvious geometric property (but proof required)
Posted: Jun 12, 2014 9:21 PM

"quasi" <quasi@null.set> wrote in message
news:g1gkp9letl24vp97fo3nuhvqimfpgu72ne@4ax.com...
> David Hartley wrote:
> >quasi wrote:
> >>
> >>(2) If S is a closed (but not necessarily simple) polygonal
> >>curve, then V = R^2.
> >>
> >>James Waldby's very simple counterexample disproves (1).
> >>
> >>But (2) is still alive, and I'm convinced it's true. It's sort
> >>of "obviously true", but I've been fooled before by flawed
> >>visualizations, so even if it really is obviously true, that's
> >>not good enough -- proof is required.

> >
> >Just take scattered's example, with the lines slightly
> >separated, and then join each extended line back to the end of
> >the line it was originally joined too. As long as the
> >extensions are long enough the new joining pieces are not
> >"visible" from the centre.

>
> Nice.
>
> Ok, then I'll retreat to this:
>
> (2') If S is a simple closed polygonal curve (with finitely many
> edges), then V contains all points of R^2 which are outside of S.
>

This still doesn't work - we can make a saw-tooth shape, "surrounding" a
central point in the sense that all the point can "see" is saw teeth, each
of which is partly obscured by another tooth. The teeth are mostly directly
joined up to their adjacent teeth, but the last two of them join by a
polygonal curve going right around the outside of all the teeth. This puts
our central point on the outside of the polygon...

Regards,
Mike.

Date Subject Author
6/12/14 quasi
6/12/14 scattered
6/12/14 scattered
6/12/14 quasi
6/12/14 quasi
6/12/14 Rick Decker
6/12/14 quasi
6/12/14 quasi
6/12/14 quasi
6/12/14 quasi
6/12/14 David Hartley
6/12/14 quasi
6/12/14 Mike Terry
6/12/14 quasi
6/13/14 David Hartley
6/13/14 quasi
6/13/14 Mike Terry
6/13/14 quasi
6/12/14 David Hartley
6/12/14 James Waldby
6/12/14 quasi
6/12/14 quasi