Re: Smallest (n+1)-polygon enclosing an n-polygon
- From: Robert Israel <israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx>
- Date: Tue, 22 Jan 2008 10:57:56 -0600
"Philippe 92" <nospam@xxxxxxxxxxxx> writes:
Rod wrote :
Does anybody have a handle on this?
I mean regular polygons of course.
When I drew a nonagon enclosing an octagon it looked like there were
7 points of contact. That's probably not the case but it does mean it
is not very obvious which the points of contact are.
I would be very surprised if there were more than 4 contact points.
I have a puzzle about constructing a regular n-gon, knowing n points
on the sides, with example of a pentagon through 4 points :
http://chephip.free.fr/pbg_en/sol122c.html
As I wrote there :
"with just 4 of the n [points], we can reconstruct the n-gon."
Hence for more than 4 given vertices of the n-gon, the construction
of the circumscribed (n+1)-gon through these touching points "may" be
impossible.
Trying to circumscribe a 'nonagon' (enneagon ?) around an octogon,
the best I got is with vertices 1,2,4,5 of the octogon touching sides
1,2,5,6 of enneagon.
Vertices 6,7,8 don't seem to touch. Neither vertex 3.
http://i29.tinypic.com/2k33ww.gif
Trying to increase side 1-2 of octogon in this construction results
into vertex 5 being outside enneagon.
Exact calculations seem tough...
Breaking the symetry is also to consider...
An interresting question !
[OT] About "enneagon" versus "nonagon" :
http://en.wikipedia.org/wiki/Nonagon
Think about the question this way: given a regular (n+1)-gon, find the largest
regular n-gon inside it. The (n+1)-gon can be specified by n+1 linear
inequalities a_j x + b_j y <= c_j. If the n-gon has centre (x_0, y_0)
and the vector from there to one vertex is (v, w), the constraints on
(x_0,y_0,v,w) are n(n+1) linear inequalities, and we want to maximize
the convex function v^2 + w^2. By convexity, the maximum must occur at
one of the extreme points of the feasible region (a basic feasible solution,
in the terminology of linear programming). In principle, we can
enumerate these and test each one. In a basic feasible solution, at
least four of the constraints will be binding.
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.
- References:
- Smallest (n+1)-polygon enclosing an n-polygon
- From: Rod
- Re: Smallest (n+1)-polygon enclosing an n-polygon
- From: Philippe 92
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