A planar point set S is an (i, t) set of ghost chimneys if there exist lines H0,H1, ..., Ht-1 such that the orthogonal projection of S onto Hj consists of exactly i + j distinct points. We give upper and lower bounds on the maximum value of t in an (i, t) set of ghost chimneys, showing that it is linear in i

Charlton, D. (David)

Demaine, E.D. (Erik D.)

Demaine, M.L. (Martin L.)

Dujmović, V. (Vida)

Morin, P. (Pat)

Uehara, R. (Ryuhei)

Carleton University's Institutional Repository

Ghost chimneys

URL to paper listed on conference site: http://www.cs.umanitoba.ca/~cccg2010/accepted.htmlA planar point set S is an (i, t) set of ghost chimneys if
there exist lines H[subscript 0],H[subscript 1],...,H[subscript t-1] such that the orthogonal projection of S onto H[subscript j] consists of exactly i + j distinct points. We give upper and lower bounds on the maximum value of t in an (i, t) set of ghost chimneys, showing that it is linear in i

Charlton, David

Demaine, Erik D.

Demaine, Martin L.

Dujmovic, Vida

Morin, Pat

Uehara, Ryuhei

DSpace@MIT

CCCG 2010, Winnipeg MB, August 9–11, 2010Ghost ChimneysDavid Charlton∗† Erik D. Demaine∗ Martin L. Demaine∗Vida Dujmovic´‡ Pat Morin‡ Ryuhei Uehara§AbstractA planar point set S is an (i, t) set of ghost chimneys ifthere exist lines H0, H1, . . . ,Ht−1 such that the orthog-onal projection of S onto Hj consists of exactly i + jdistinct points. We give upper and lower bounds on themaximum value of t in an (i, t) set of ghost chimneys,showing that it is linear in i.1 IntroductionOnce upon a time in Japan, there was a power plantwith four chimneys called “ghost chimneys” (obake en-totsu, or ?????); see Figure 1. Although thesechimneys were dismantled in 1964, they are still famousin Japan, with toys, books, manga, and movies refer-encing them (Figure 2). They are considered a kind ofsymbol of industrialized Japan in the old, good age ofthe Showa era [Ada09].One of the reasons why they are famous and are called“ghost chimneys” is that they could be seen as two chim-neys, three chimneys, or four chimneys depending onthe point of view. This phenomenon itself was an ac-cident, but it raises several natural questions. Whatinterval of integers can be realized by such chimneys?How many chimneys do we need to realize the inter-val? How can we arrange the chimneys to realize theinterval?More precisely, we consider the following problem:given an integer i, what is the maximum value t(i)such that there exists a set of points S ⊂ R2 anda set H0, H1, . . . ,Ht(i)−1 of lines where, for each j ∈{0, 1, . . . , t(i)− 1}, the orthogonal projection of S ontoHj consists of exactly i + j distinct points? We provethe following result:Theorem 1 For any integer i ≥ 1, 2i ≤ t(i) ≤ 123.33i.∗Computer Science and Artificial Intelligence Laboratory,Massachusetts Institute of Technology, Cambridge, USA. {dchar,edemaine,mdemaine}@mit.edu.†Supported in part by Akamai Presidential Fellowship.‡School of Computer Science, Carleton University, Ottawa,Canada, {morin,vida}@scs.carleton.ca. Research supported inpart by NSERC.§School of Information Science, Japan Advanced Institute ofScience and Technology, Ishikawa 923-1292, Japan. uehara@jaist.ac.jpFigure 1: The ghost chimneys in Japan, part of theSenju Thermal Power Station (1926–1964) maintainedby the Tokyo Electric Power Company. [Used with per-mission from Adachi City.]Figure 2: Hello GhostyIn addition to Theorem 1, we show that t(1) = 2,t(2) = 5, t(3) = 9, and t(4) ≥ 12. These results showthat neither the lower bound nor the upper bound ofTheorem 1 is tight for all values of i. Theorem 1 isan immediate consequence of Lemma 1 and Lemma 4,which we prove in the next two sections, respectively.22nd Canadian Conference on Computational Geometry, 2010These ghost-chimney problems relate more generallyto understanding what orthogonal projections a single2D or 3D shape can have. In 2D, some closely re-lated problems have been considered in [Ski90, MPv08].Past explorations into structures in 3D, known vari-ously as 3D ambigrams, trip-lets, and shadow sculp-tures, have focused on precise, usually connected projec-tions [KvWW09, MP09]. Our work was originally mo-tivated by considering what happens with disconnectedprojections of unspecified relative position.2 The Lower BoundLemma 1 For each integer i ≥ 1, there exists a setS = S(i) of 3i − 1 points and a set H0, H1, . . . ,H2i−1of lines such that, for each j ∈ {0, 1, . . . , 2i − 1}, theorthogonal projection of S onto Hj has exactly i + jdistinct values.Proof. The point set S consists of the points of ani × 3 grid with the bottom-right corner removed; seeFigure 3. For even j, Hj is a line of slope j/2. Forodd j, Hj is a line of slope −(j + 1)/2. 3 The Upper BoundOur upper-bound proof is closely related to Sze´kely’sproof of the Szeme´redi–Trotter Theorem [Sze´97]. Wemake use of the following version of the CrossingLemma, which was proved by Pach, Radoicic´, Tardos,and To´th [PRTT04]:Lemma 2 (Crossing Lemma) Let β = 103/6, γ =1024/31827, and let G be a graph with no self loops, noparallel edges, v vertices, and e > βv edges. Thencr(G) ≥ γ · e3v2.Lemma 3 Let t = αi, let S be a set of r points, and letH0, H1, . . . ,Ht−1 be a set of lines such that the orthog-onal projection of S onto Hj gives exactly i+ j distinctvalues. Then, t ≤ 34 or r ≤ max{4, 2/α+ 2 + α/2}i/γ.Proof. Each projection direction Hj defines a set Lj ofi + j parallel lines, each of which contains at least onepoint of S. Let G be the geometric graph that containsthe points in S plus t additional points p0, p1, . . . , pt−1.Two vertices in S are connected by an edge in G if andonly if they occur consecutively on some line in⋃t−1j=0 Lj .Additionally, each vertex pj is connected to each of thei + j lexically largest points on each of the lines in Lj .See Figure 4.The graph G has t + r vertices and tr edges. Ob-serve that we have a drawing of G so that the onlycrossings between edges occur where lines in L intersectp0p1 p2Figure 4: The graph G for a set of points with i = 9and t = 3.each other. The total number of intersecting pairs oflines in L isX =t−1∑j=1(i+ j) ·j−1∑k=0(i+ k)≤t−1∑j=1(i+ j)(ij + j2/2)≤t−1∑j=1(i2j + 3ij2/2 + j3/2)≤ i2t2/2 + it3/2 + t4/8 .Applying Lemma 2, we learn that eithertr ≤ β(t+ r) , (1)orX ≥ cr(G) ≥ γ (tr)3(t+ r)2. (2)In the former case, we rewrite (1) to obtaint ≤ β(t/r + 1) ≤ 2β ≤ 34 + 1/3 ,so t ≤ 34 (since t is an integer).In the latter case, we expand (2) to obtaini2t2/2 + it3/2 + t4/8 ≥ γ (tr)3(t+ r)2.Substituting t = αi givesi(12α+12+α8)≥ γ r3(t+ r)2≥ γr/4 ,where the second inequality follows from the fact thatt ≤ i+ t− 1 ≤ r. Rewriting to isolate r finally givesr ≤(2α+ 2 +α2)i/γ .We finish the proof by observing that, for α > 2, theinequality r ≤ 4i/γ obtained by setting α = 2 is strongerand still applies. CCCG 2010, Winnipeg MB, August 9–11, 2010j = 0j = 1 j = 2j = 3 j = 4Figure 3: The set S(i) for i = 9 and the projection directions that yield i, i+ 1, . . . , i+ 4 distinct points.Lemma 4 For all integers i ≥ 1, t(i) ≤ 123.33i.Proof. Observe that i+t−1 ≤ r. Therefore, for i ≥ 18,the lemma follows by applying Lemma 3 with α = 2.For i ∈ {1, 2, . . . , 17}, the lemma follows by setting α =35/i. 4 Small Values of iIn this section we give some tighter bounds on t(i) fori ∈ {1, 2, 3, 4}.Lemma 5 t(1) = 2, and t(2) = 5.Proof. Point sets achieving these bounds are the 1× 2and the 2 × 3 grid, respectively; see Figure 5. Thatthese point sets are optimal follows from the fact thatthe existence of H0 and H1 implies that the points ofS lie on the intersection of i parallel lines with anotherset of i+ 1 parallel lines. Thus, |S| ≤ i(i+ 1), so t(i) ≤|S| − i+ 1 ≤ i2 + 1. Notice that the proof of Lemma 5 implies that, forany i, t(i) ≤ i2 + 1. The following lemma shows that,for i ≥ 3, t(i) ≤ i2. Of course, this upper bound istighter than Lemma 4 for i ≤ 123.Lemma 6 t(3) = 9.Proof. The point set S(4) described in the proof ofLemma 1 results in 3 distinct points when projectedonto a vertical line, therefore t(3) ≥ 9.H0H1H0H1 H2H3H4(a) (b)Figure 5: Point sets showing that (a) t(1) ≥ 2 and(b) t(2) ≥ 5.For the upper bound, refer to Figure 6. By an affinetransformation, we may assume that H0 is vertical andH1 is horizontal. Thus, the points of S are containedin the intersection of 3 horizontal lines with 4 verticalH0H1 `Figure 6: The proof of Lemma 6.22nd Canadian Conference on Computational Geometry, 2010H1H0H2H3H4H5H6H9H7 H10H8Figure 7: A (4, 12) set of ghost chimneys.lines. This establishes that |S| ≤ 12, so t(3) ≤ 10.To see that |S| < 12, assume otherwise and considerany line ` that is neither horizontal nor vertical. By areflection through a horizontal line, we may assume that` has positive slope, so that every point on the bottomrow and right column of S has a distinct projection onto`, so S projects onto at least 6 distinct points on `. Inparticular, this implies that there is no line H2 such thatS projects onto 5 distinct points on H2. Lemma 7 12 ≤ t(4) ≤ 15.Proof. The point set and lines H0, H1, . . . ,H10 thatshow t(4) ≥ 12 are shown in Figure 7. (H11 is omittedsince any sufficiently general line will do.)To see that t(4) ≤ 15, we argue as in the proof of thesecond half of Lemma 6. This establishes that |S| ≤ 20.If |S| ∈ {19, 20} then the number of distinct projectionsof S onto ` is at least 7, but this contradicts the existenceof H2. Thus, we must have |S| ≤ 18, to t(4) ≤ 15. 5 ConclusionsWe have given upper and lower bound on the largestpossible value of t, as a function of i, in an (i, t) set ofghost chimneys. These bounds differ by only an (ad-mittedly large) constant factor. Reducing this factorremains an open problem. For small values of i, wehave shown that t(1) = 2, t(2) = 5, t(3) = 9, and12 ≤ t(4) ≤ 18.Another open problem is the generalization of theseresults to three, or higher, dimensions. Given an integeri, what is the maximum value t(i) such that there existsa set of points S ⊂ Rd and a set H0, H1, . . . ,Ht(i)−1 ofhyperplanes where, for each j ∈ {0, 1, . . . , t(i)− 1}, theorthogonal projection of S onto Hj consists of exactlyi+ j distinct points?AcknowledgmentsThis work was initiated at the 25th Bellairs WinterWorkshop on Computational Geometry, co-organizedby Erik Demaine and Godfried Toussaint, held onFebruary 6–12, 2010, in Holetown, Barbados. We thankthe other participants of that workshop for providing astimulating research environment.References[Ada09] Adachi City. ????? (Ghost chimneys,in Japanese). http://www.city.adachi.tokyo.jp/003/d10100040.html, 2009.[KvWW09] J. Keiren, F. van Walderveen, and A. Wolff.Constructability of trip-lets. In Abstractsfrom the 25th European Workshop onComp. Geom., Brussels, Belgium, 2009.[MP09] N. J. Mitra and M. Pauly. Shadow art.ACM Transactions on Graphics, 28(5),2009. Article 157.[MPv08] J. Matousˇek, A. Prˇ´ıveˇtivy´, and P. Sˇkovronˇ.How many points can be reconstructedfrom k projections? SIAM J. on DiscreteMathematics, 22(4):1605–1623, 2008.[PRTT04] J. Pach, R. Radoicic´, G. Tardos, andG. To´th. Improving the crossing lemmaby finding more crossings in sparse graphs.In Proc. of the 20th Annual Symposium onComp. Geom., pages 68–75, 2004.[Ski90] S. Skiena. Counting k-projections of a pointset. J. of Combinatorial Theory, Series A,55(1):153 – 160, 1990.[Sze´97] L. A. Sze´kely. Crossing numbers andhard Erdo˝s problems in discrete geometry.Combinatorics, Probability and Computing,6:353–358, 1997.

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Ghost Chimneys

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