From:baloglou@panix.com (George Baloglou)Date:31 Jul 1998 01:27:00 -0400Newsgroups:sci.math.researchSubject:Polygons as projections of polytopes

The recent thread on "how to pixelize a sphere" (and related short tour of polytopes) made me think of a question I posted on sci.math a couple of years ago but remained unnoticed at worst and unanswered at best: Is every convex polygon the projection of a polytope in such a way that the polytope's vertices become the polygon's vertices and intersections of diagonals, the polytope's edges become the polygon's edges and "diagonal segments", and the polytope's faces become the polygon's "between-the- diagonals" regions (save for the polytope's "base" that is the whole polygon)? For example, "lifting" the intersection point of a quadrilateral's two diagonals vertically and "dragging" the diagonals along, we do get the desired polytope (a pyramid); likewise, a "lifted diagonals' pentagon" helps us solve the problem for n = 5, but it should by now be clear that it is hard to generate the desired polytope for large n preserving both the sought polytope's planarity of faces and the polygon's linearity of diagonals (after projection). I arrived at this question when I became aware of a slick way of counting a polygon's "between-the-diagonals" regions using "Euler's polytope formula", r = e-v+2. [See for example Alan Tucker's "Applied Combinatorics" (2nd ed), p.184] George Baloglou baloglou@oswego.edu http://www.oswego.edu/~baloglou "The Mathematics of our brain is not our Mathematics"

Date:Mon, 16 Aug 1999 13:42:23 +1000To:baloglou@oswego.eduFrom:Andrew Kepert <andrew@frey.newcastle.edu.au>Subject:geometry junkyardFrom:Andrew Kepert <andrew@frey.newcastle.edu.au>To:baloglou@oswego.eduSubject:geometry junkyardDate:Mon, 16 Aug 1999 13:42:23 +1000

Hi George (& Cc: to David Eppstein) Just browsing through the Geometry Junkyard, and found your posting on polygons+diagonals being able to be lifted into a 3-d surface : http://www.ics.uci.edu/%7Eeppstein/junkyard/diagonal-projection.html >From: baloglou@panix.com (George Baloglou) >Date: 31 Jul 1998 01:27:00 -0400 >Newsgroups: sci.math.research >Subject: Polygons as projections of polytopes > . . . > >Is every convex polygon the projection of a polytope in such a way that >the polytope's vertices become the polygon's vertices and intersections of >diagonals, the polytope's edges become the polygon's edges and "diagonal >segments", and the polytope's faces become the polygon's "between-the- >diagonals" regions (save for the polytope's "base" that is the whole >polygon)? > . . . it is listed as "new" on the Junkyard, but it seems over a year old. If you haven't already found a solution, here is mine, firstly inductively, then non-inductively. Suppose you have, for an existing polygonal region P, a function f giving the height at each point, being equal to 0 on dP, the boundary of P and continuous and affine on the polygonal subregions in question. Add another vertex v, making a polygon Q. Define g on Q to be that whose graph is a pyramid with base Q and apex v. [e.g. if x in Q\{V}, there is a unique lambda in [0,1] and y in (dP intersect dQ) such that x=(lambda)v+(1-lambda)y. Define g(x)=lambda.] Then g+f is as required except on Q\P. But this is easy to fix, as the diagonals passing through Q\P divide it into triangles with a vertex at v. [e.g. define h to have graph equal to the top surface of the convex hull of graph( (g+f)|P ) union Q x {0} , OR define it as some lambda in a convex combination, as above] Come to think of it, this construction can be done in closed form. Read a_b as a-subscript-b here: Let P be the polygon and V the finite vertex set. For each v in V, define g_v on P by g_v(x) = min{ lambda : exists y in P such that x=lambda v + (1-lambda) y } and define g = ( sum_v g_v ) - 1 . Cheers, Andrew [ Dr Andrew Kepert e-mail: andrew@frey.newcastle.edu.au ] [ http://frey.newcastle.edu.au/~andrew ] [ Central Coast Campus, Univ. of Newc., Ourimbah NSW 2258, AUSTRALIA ] [ Phone: 02 4348 4116 Fax: 02 4348 4145 ] [ Mathematics, University of Newcastle, Callaghan NSW 2308 AUSTRALIA ] [ Phone: 02 4921 5190 Fax: 02 4921 5548 ]