COLLAPSING WALLS THEOREM

Transcription

1 COLLAPSING WALLS THEOREM IGOR PAK AND ROM PINCHASI Abstact. Let P R 3 be a pyamid with the base a convex polygon Q. We show that when othe faces ae collapsed (otated aound the edges onto the plane spanned by Q) they cove the whole Q.. Intoduction Let P be a convex pyamid in R 3 ove the base Q which is a convex polygon in a hoizontal plane. Think of the othe faces F of P as the walls of a wooden box that each wall F is hinged to the base Q along the edge. Suppose now that the walls ae collapsed i.e. otated aound the edges towads the base onto the hoizontal plane. The question is: do they cove the whole base Q? Figue. An impossible configuation of fou collapsing walls of a pyamid leaving a hole in the base. At fist this may seem obvious but in fact the poblem is aleady non-tivial even in the case of fou-sided pyamids which can possibly have some obtuse dihedal angles (see Figue ). Fomally we have the following esult: Collapsing Walls Theoem. Let P R 3 be a pyamid ove a convex polygon Q. Fo a face F of P denote by e F the edge between F the base: e F = F Q let A F denotes the esult of otation of F aound e F in the diection of P onto the plane which contains Q. Then Q F A F whee the union is ove all faces F of P diffeent fom Q. Fo example suppose pyamid P in the theoem has a vey lage height so that all walls ae nealy vetical. The theoem then implies that evey point z Q has an othogonal pojection into the inteio of some edge e of Q. This is a classical esult Date: 8 Mach 009. School of Mathematics Univesity of Minnesota Minneapolis MN; pak@umn.edu. Depatment of Mathematics Technion Haifa Isael; oom@math.technion.ac.il.

2 IGOR PAK AND ROM PINCHASI with a numbe of fa-eaching genealizations (see [Pak 9]). Thus the collapsing walls theoem can be viewed as yet anothe genealization of this esult (cf. Section 3).. Poof of the theoem Conside R 3 endowed with the stad Catesian coodinates (x x x 3 ). Without loss of geneality assume that the plane H spanned by Q is hoizontal i.e. given by x 3 = 0 that P is contained in the half-space x 3 0. Denote by F...F m the faces of P diffeent fom Q by H i the planes spanned by F i by e i = F i Q the edges of Q fo all i m. Denote by Φ i the otation about e i of H i onto H (the otation is pefomed in the diection dictated by P so that thoughout the otation H i intesects the inteio of P). Similaly let A i = Φ i (F i ) is the otation of the face F of P onto Q i m. We need to show that evey point in Q lies in m i= A i. Without loss of geneality we can take this point to be the oigin O. Futhe denote by L i = H i H the line though e i. Let i be the distance fom the oigin to L i let α i be the dihedal angle of P at e i i.e the angle between H H i which contains P. Suppose now F is a face such that τ i = i tan α i is minimized at τ. We will show that the oigin O is contained in A. In othe wods we pove that if O / A then τ i < τ fo some i >. Let z H such that the otation of z onto Q is the oigin: Φ (z) = O. It suffices to show that z F. Let v = (v v 0) be the unit vecto that is a nomal to L in the hoizontal plane. It is easy to see that Oz = ( ) ( cosα )v ( cosα )v sin α. To pove the theoem assume to the contay that z / F. Then thee exists a face of P say F such that H sepaates z fom the oigin. Denote by y the closest point to z on L by α the angle between the line (zy) the hoizontal plane H whee the angle is taken with the half-plane of H which contains Q ( thus the oigin). In this notation the above condition implies that α > α. Without loss of geneality we may assume that line L is given by equations x = x 3 = 0. Then y = ( ( cosα )v 0 ) cosα = cosôyz = ( cosα )v sin α + ( ( cos α )v ). Note that the function x/ a + x is monotone inceasing as a function of x that v. We get cosα ( cos α ) sin α + ( ( cosα )).

3 Applying cosα < cos α we conclude: () COLLAPSING WALLS THEOREM 3 ( cosα ) sin α + ( ( cosα )) < cosα. Recall the assumption that τ τ. This gives tan α tan α o () tan α tan α. The est of this section is dedicated to showing that both () () ae impossible. This gives a contadiction with ou assumptions poves the claim. We split the poof into two cases depending on whethe the dihedal angle α is acute o obtuse. In each case we epeatedly ewite () () eventually leading to a contadiction. Case (obtuse angles). Suppose π < α < π. In this case cosα < 0 () is equivalent to (3) + (4) sin α ( ( cos α )) < cos α sin α ( cosα ) > tan α. This can be futhe ewitten as: (5) < cosα + sin α. tanα Now (5) () togethe imply tan α tan α < cosα + sin α tan α which is impossible. Indeed suppose fo some 0 < a b < π we have (6) tan a tan b < cosa + sin a tanb. Dividing both sides by (tan a ) afte some easy manipulations we conclude that (6) is equivalent to (7) < sin a + + cosa tan b tanb which in tun is equivalent to ( (8) tan b ) sin b < cos(a b). tan b Since the left h side of (8) is equal to we get a contadiction complete the poof in Case.

4 4 IGOR PAK AND ROM PINCHASI Case (ight acute angles). Suppose now that 0 < α π. Then cosα 0 0 < tan α. Let us fist show that the numeato of () is nonnegative i.e. that ( cosα ). Fom the contay assumption we have / < ( cosα ). Togethe with () this implies: cosα > α tan tan α tan α which is impossible fo all 0 < α < π. Fom above we can exclude the ight angle case α = π fo else the l.h.s. of () is nonnegative while.h.s. is equal to zeo. Thus cos α > 0. Theefoe the inequality () in this case can be ewitten as (9) + (0) sin α ( ( cosα )) > cos α sin α ( cosα ) > tanα. Note now that (0) coincides with (4). Since (6) holds fo all 0 < a b < π we obtain the contadiction vebatim the poof in Case. This completes the analysis of Case finishes the poof of the theoem. 3. Final emaks 3.. The collapsing walls theoem extends vebatim to highe dimensions. Moeove it also extends to evey polytope P R d as follows. Fix one facet Q of P assume all othe facets F of P ae otated aound the affine subspace H F H onto the hypeplane H containing Q then they cove the whole facet Q. Hee H F denotes the hypeplane that contains the facet F. We efe to [PP] whee this esult is poved in full geneality is used to show that a smalle polyhedon can always be sequentially cut out of a bigge polyhedon in any dimension. 3.. Let us note that when the walls of a pyamid ae collapsed outside athe than onto the base they ae paiwise non-intesecting (see Figue ). We leave this easy execise to the eade Continuing with the example of vetical walls as given in the intoduction ight afte the theoem ecall that fo the cente of mass z = cm(q) thee ae at least two such edges onto which othogonal pojection of z lies in the inteio (see e.g. [Pak 9]). It would be inteesting to see if this esult extends to the setting of the theoem (of couse the notion of the cente of mass would have to be modified appopiately). Let us note hee that the cente of mass esult is closely elated to the fou vetex theoem [Tab] fails in highe dimension [CGG]. One can give a constuction with thee is only one such edge if the cente of mass is eplaced by a geneal point in Q (see [CGG] [Pak 9]).

5 COLLAPSING WALLS THEOREM 5 Figue. Walls of a pyamid collapsing outside the base do not intesect The poof of the theoem is based on an implicit subdivision of Q given by the smallest of the linea functions τ i at evey point z Q. Recall that τ i is a weighted distance to the edge e i. Thus this subdivision is in fact a weighted analogue of the dual Voonoi subdivision in the plane (see [Au Fo]). As a consequence computing this subdivision can be done efficiently both theoetically pactically. Acknowledgments. The authos ae thankful to Yui Rabinovich fo the inteest in the poblem. The fist autho was patially suppoted by the National Secuity Agency the National Science Foundation. The second autho was suppoted by the Isaeli Science Foundation (gant No. 938/06). Refeences [Au] F. Auenhamme Voonoi diagams a suvey of a fundamental geometic data stuctue ACM Comput. Suv. 3 (99) [CGG] J. H. Conway M. Goldbeg R. K. Guy Poblem 66- SIAM Review (969) [Fo] S. Fotune Voonoi diagams Delaunay tiangulations in Computing in Euclidean [Pak] geomety (F. Hwang D. Z. Du eds.) Wold Scientific Singapoe I. Pak Lectues on Discete Polyhedal Geomety monogaph to appea 009; available at [PP] I. Pak R. Pinchasi How to cut out a convex polyhedon pepint (009). [Tab] S. Tabachnikov Aound fou vetices Russian Math. Suveys 45 (990) 9 30.