Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Given a set of n points is it always possible to construct a non self intersecting polygon?

share|improve this question
    
Closed polygon with these as only vertices? We need conditions, take $n$ collinear points. –  André Nicolas Jan 5 '13 at 6:39
    
Polygon that does what? –  Alex Becker Jan 5 '13 at 6:40
    
@AndréNicolas I want to know given any random non collinear points is it always possible to construct a non self-intersecting closed polygon. –  harish.venkat Jan 5 '13 at 6:56
    
An obviously necessary and sufficient condition is that the set of points be "non-reentrant", i.e. no point is inside the convex hull of the other points. –  Ewan Delanoy Jan 5 '13 at 7:01
2  
@EwanDelanoy Why is this necessary? The polygon only needs to be simple, not convex. –  Erick Wong Jan 5 '13 at 7:15
show 1 more comment

1 Answer 1

up vote 5 down vote accepted

Choose a point $x_0$ of your set and order the other points around $x_0$ counter-clockwise. Label them $x_1,x_2,\ldots x_{n-1}$ according to that order. You get a non intersecting polygon and $x_0$ is in its kernel.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.