Generalization of Pick's theorem for surface of polyhedra
Authors: Agfalvi Mihaly , Kadar Istvan , Papp Erik 
Published: January 2007

The Pick's theorem is one of the rare gems of elementary
mathematics because this is a very innocent sounding hypothesis
imply a very surprising conclusion (Bogomolny 1997). Yet the
statement of the theorem can be understood by a fifth grader. Call
a polygon a lattice polygon if the co-ordinates of its vertices are
integers. Pick's theorem asserts that the area of a lattice polygon
P
is given by
A(P)
=
I(P)
+
B(P) / 2 -
1
=
V(P) - B(P) / 2 - 1
where
I(P), B(P)
and
V(P)
are the number of
interior
lattice points, the
number of
boundary
lattice points and the
total
number of lattice points of
P
respectively. It is worth to
mention that the
I(P)
(understand like
digital area
)
is
digital mapping standard
in USA since decade (Morrison,
J. L. 1988 and 1989). Because the Pick's theorem was first
published in 1899 therefore our planned presentation had timing its
100 anniversary. Currently it has greater importance than realized
heretofore because of the Pick's theorem forms a connection between
the old
Euclidean
and the new
digital (discrete)
geometry.
During this long period lots of proof had been made
of Pick's theorem and many trial of its generalization from simple
polygons towards complex polygon networks, moreover tried to extend
it to the direction of 3D geometrical objects as well. It is also
turned out that nowadays the
inverse Pick's formulas
comes
to the front instead of the original ones, consequently of powerful
spreading the digital geometry and mapping. Today the question is
not the old one: how can we produce traditional area without
co-ordinates, using only inside points and boundary points. Just on
the contrary:
how is it possible to simply determine digital
boundary and digital area (namely the number of boundary points and
inside points) using known co-ordinates of vertices.
The
inverse formulas are:
B(P)=ΣGCD (ÄX, ÄY,
ÄZ)
(1D Pick's theorem) and
I(P)=A(P)-B(P)/2+1
(2D
Pick's theorem) where GCD is the Great Common Divisor of the
co-ordinate differences of two-two neighboring vertices. The our
main object is not these formulas to present, but we desire to show
that the
Pick's theorem (after adequate redrafting) indeed valid
for every spatial triangle which are determined by three arbitrary
points of a 3D lattice.
The original planar theorem is only a
special case of it. However if it is true then its valid not only
for triangles but all irregular polygons also which are lying in
space and have its vertices in spatial lattice points. Finally if
the extended Pick's theorem is true for all face of a lattice
polyhedron then it is true for total surface as well. Consequently
we developed so simple and effective algorithms which solve
enumeration tasks without the time- and memory-wasting immediate
computing. These algorithms make possible that using the
vertex-co-ordinate list and the topological description of a convex
or non-convex polyhedron (cube, prism, tetrahedron etc.) getting
answer many elementary questions. For example,
how many vaxels
can be found on the complex surface of a polyhedron, how many on
its edges or on its individual faces.
We succeeded to extend
our results also to the surface of non-cornered geometric objects
(circle, sphere, cylinder, cone, ellipsoid etc.), but anyway, this
have to be object of another presentation.