The Visibility Arrangement and Line Shelling Arrangement of a

89 Pages·2012·0.56 MB·English

The Visibility Arrangement and
Line Shelling Arrangement of a
Convex Polytope
Richard P. Stanley
M.I.T.
TheVisibilityArrangementandLineShellingArrangementofaConvexPolytope–p.
Visible facets
P Rd
: a d-dimensional convex polytope in
Rd
Certain facets of P are visible from points v ∈
TheVisibilityArrangementandLineShellingArrangementofaConvexPolytope–p.
Visible facets
P Rd
: a d-dimensional convex polytope in
Rd
Certain facets of P are visible from points v ∈
no facets are
visible
green facets are visible
TheVisibilityArrangementandLineShellingArrangementofaConvexPolytope–p.
The visibility arrangement
Rd
aff(S): the affine span of a subset S ⊂
visibility arrangement:
vis(P) = {aff(F ) : F is a facet of P}
TheVisibilityArrangementandLineShellingArrangementofaConvexPolytope–p.
The visibility arrangement
Rd
aff(S): the affine span of a subset S ⊂
visibility arrangement:
vis(P) = {aff(F ) : F is a facet of P}
Regions of vis(P) correspond to sets of facets
Rd
that are visible from some point v ∈ .
TheVisibilityArrangementandLineShellingArrangementofaConvexPolytope–p.
An example
1
12
14
1
φ
P
2
4 2
4
3
3
34 23
234
TheVisibilityArrangementandLineShellingArrangementofaConvexPolytope–p.
Number of regions
v(P): number of regions of vis(P), i.e., the
number of visibility sets of P
χ (q): characteristic polynomial of the
A
arrangement A
Zaslavsky’s theorem. Number of regions of A is
d
(−1) χ (−1).
A
TheVisibilityArrangementandLineShellingArrangementofaConvexPolytope–p.
Number of regions
v(P): number of regions of vis(P), i.e., the
number of visibility sets of P
χ (q): characteristic polynomial of the
A
arrangement A
Zaslavsky’s theorem. Number of regions of A is
d
(−1) χ (−1).
A
In general, v(P) and χ (q) are hard to
vis(P)
compute.
TheVisibilityArrangementandLineShellingArrangementofaConvexPolytope–p.
A simple example
P = n-cube
n
n
χ (q) = (q − 2)
vis(P )
n
n
v(P ) = 3
n
TheVisibilityArrangementandLineShellingArrangementofaConvexPolytope–p.
A simple example
P = n-cube
n
n
χ (q) = (q − 2)
vis(P )
n
n
v(P ) = 3
n
For any facet F , can see either F , −F , or neither.
TheVisibilityArrangementandLineShellingArrangementofaConvexPolytope–p.