CGAL users discussion list

In CG there are these notions of width:
1. The width of a set of points P ⊆ R^d is the minimum distance  
between parallel hyperplanes supporting the convex hull of P.
2. Given a normalized vector v, the directional width is the distance  
between parallel hyperplanes supporting the convex hull of P and  
orthogonal to v.

So the width (resp. the directional width) of P is the penetration  
depth (resp. the directional penetration depth) of P and a copy of P.

width(P) = penetration-depth(P,P) = inf{|| t|| | t /∈ (P ⊕ −P), t ∈ R^d}

In simple words the width of a polygon is the smallest distance of the  
origin from the boundary of the Minkowski sum of P and -P (P reflected  
about the origin). The directional width in the direction of a given  
vector v is the distance of the origin from the boundary of the  
Minkowski sum of P and -P in the direction of v.

Quoting "Ben Supnik" 
<>:

> Hi Y'all,
>
> I'm looking for the right term for an algorithm to determine the  
> largest "width" of a polygon.  Colloquially, I want to identify how  
> "thin" a polygon is...in other words, I want to identify slivers and  
> long thin winding polygons vs. large round areas.
>
> Naively I think I could do an inset using a minkowski sum and if my  
> resulting set is empty, I know that no point was "wider" than the 2x  
> the diamater of the shape I used for convolution.
>
> Are there metrics or algorithms that compute this kind of value more  
> directly?  If anyone has a search term for the type of computational  
> geometry problem I'm looking at, a hint would be much appreciated.
>
> cheers
> Ben
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