Generalization of Pick's theorem for surface of polyhedra

• Authors: Mihaly Agfalvi;Istvan Kadar;Erik Papp
• Venue: Proceedings of the conference on APL '99 : On track to the 21st century: On track to the 21st century
• Year: 1998

The Pick's theorem is one of the rare gems of elementarymathematics because this is a very innocent sounding hypothesisimply a very surprising conclusion (Bogomolny 1997). Yet thestatement of the theorem can be understood by a fifth grader. Calla polygon a lattice polygon if the co-ordinates of its vertices areintegers. Pick's theorem asserts that the area of a lattice polygonP is given by A(P) = I(P) + B(P) / 2 -1 = V(P) - B(P) / 2 - 1 where I(P), B(P) andV(P) are the number of interior lattice points, thenumber of boundary lattice points and the totalnumber of lattice points of P respectively. It is worth tomention 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 firstpublished in 1899 therefore our planned presentation had timing its100 anniversary. Currently it has greater importance than realizedheretofore because of the Pick's theorem forms a connection betweenthe old Euclidean and the new digital (discrete)geometry. During this long period lots of proof had been madeof Pick's theorem and many trial of its generalization from simplepolygons towards complex polygon networks, moreover tried to extendit to the direction of 3D geometrical objects as well. It is alsoturned out that nowadays the inverse Pick's formulas comesto the front instead of the original ones, consequently of powerfulspreading the digital geometry and mapping. Today the question isnot the old one: how can we produce traditional area withoutco-ordinates, using only inside points and boundary points. Just onthe contrary: how is it possible to simply determine digitalboundary and digital area (namely the number of boundary points andinside points) using known co-ordinates of vertices. Theinverse formulas are: B(P)=ΣGCD (ÄX, ÄY,ÄZ) (1D Pick's theorem) and I(P)=A(P)-B(P)/2+1 (2DPick's theorem) where GCD is the Great Common Divisor of theco-ordinate differences of two-two neighboring vertices. The ourmain object is not these formulas to present, but we desire to showthat the Pick's theorem (after adequate redrafting) indeed validfor every spatial triangle which are determined by three arbitrarypoints of a 3D lattice. The original planar theorem is only aspecial case of it. However if it is true then its valid not onlyfor triangles but all irregular polygons also which are lying inspace and have its vertices in spatial lattice points. Finally ifthe extended Pick's theorem is true for all face of a latticepolyhedron then it is true for total surface as well. Consequentlywe developed so simple and effective algorithms which solveenumeration tasks without the time- and memory-wasting immediatecomputing. These algorithms make possible that using thevertex-co-ordinate list and the topological description of a convexor non-convex polyhedron (cube, prism, tetrahedron etc.) gettinganswer many elementary questions. For example, how many vaxelscan be found on the complex surface of a polyhedron, how many onits edges or on its individual faces. We succeeded to extendour results also to the surface of non-cornered geometric objects(circle, sphere, cylinder, cone, ellipsoid etc.), but anyway, thishave to be object of another presentation.