String-wrapped Rotating Disks - Semantic Scholar
XIV Spanish Meeting on Computational Geometry, 27–30 June 2011
String-wrapped Rotating Disks Joseph O’Rourke1 1
Department of Computer Science, Smith College, Northampton, MA 01063, USA. [email protected]
Abstract. Let the centers of a finite number of disjoint, closed disks be pinned to the plane, but with each free to rotate about its center. Given an arrangement of such disks with each labeled + or −, we investigate the question of whether they can be all wrapped by a single loop of string so that, when the string is taut and circulates, it rotates by friction all the (+)-disks counterclockwise and all the (−)disks clockwise, without any string-rubbing conflicts. We show that although this is not always possible, natural disk-separation conditions guarantee a solution. This work suggests many open problems.
Introduction Let A be a collection of n disjoint closed disks in the plane, each labeled + or −. We seek to wrap them all in one continuous loop of string so that, were one of the disks rotated by a motor, all the others would spin by friction with the string/belt in a direction consistent with the labeling: counterclockwise (ccw) for + and clockwise (cw) for −. See Figure 1. We call a wrapping proper if it satisfies these conditions:
+ −
− −
+ +
+
+ −
Figure 1. A proper wrapping of disks with a loop of string: each + disk rotates counterclockwise, each − clockwise. (1) The string is taut: it follows arcs of disk boundaries and disk-disk tangents only. (2) Each disk boundary circle has a positive-length arc in contact with the string. (It is fine if the string wraps around a disk more than once.) (3) One of the two possible circulation directions (i.e., orientations) for the string loop rotates each disk in the direction consistent with its labeling. 1 CRM Documents, vol. 8, Centre de Recerca Matem` atica, Bellaterra (Barcelona), 2011
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String-wrapped Rotating Disks
(4) If the string contacts a point of a disk boundary circle, its circulation there must be in the direction consistent with that disk’s label, i.e., there is no rubbing conflict. We permit the string to cross itself at points not on a disk boundary. Indeed such crossings are necessary: for any pair of disks, one + and one −, the string must form a figure-8 shape. Although the conditions for a proper wrapping are suggested by physical analogy, the pursuit here is not driven by any application. Proper string wrappings bear some resemblance to sona sand drawings [DDTT07, LT09], but are more closely related to the conveyor-belt wrappings studied in [DDP10]. Those belts differ from string wrappings in that the rotation directions were not prespecified, and the belt could not self-cross. Although the models are different, the questions raised are analogous. We show that not all arrangements of disks have a proper wrapping, but that various separation conditions guarantee proper wrappings. For example, every collection of unit disks has a proper wrapping when each pair is separated by a distance of 0.31 or more. Characterizing the disk arrangements that admit proper wrappings is posed as an open problem in Section 3.
1 Unwrappable arrangements An example of an unwrappable arrangement is shown in Figure 2.
D2 +
+
+
+
D0
+
D1
+
D0
+
+
+
(a)
+
+
−
D2 +
+ D0
+
+
+
D1 +
+
+
(b)
+
Figure 2. An unwrappable arrangement of seven unit disks.
Figure 3. (a) Proper wrapping with D0 labeled −. (b) Proper wrapping with D1 and D2 displaced slightly.
It consists of one unit disk surrounded by six others, arranged in a hexagonal pennypacking pattern, except the disks are just barely disjoint. We now prove this configuration is unwrappable. The central disk D0 must have a positive-length arc of ccw string touching it. Because a taut string can only leave the boundary of a disk along a tangent, the string follows at least the arc between two adjacent tangents. In order for the string to reach another disk, say D2 , and contribute a ccw arc, it must first touch another disk, D1 in the figure, but now rubbing it in a cw arc. Thus a rubbing conflict is unavoidable. Without moving the disks, this arrangement can be properly wrapped with a different pattern of ± labels. For example, reversing the central disk D0 enables a proper
XIV Spanish Meeting on Computational Geometry, 27–30 June 2011
3
wrapping: Figure 3(a). Indeed all other ± patterns of labels (except all −) in this example are wrappable. Retaining the original + labels but moving two disks slightly also permits the configuration to be wrapped: Figure 3(b).
2 Separation conditions The primary impediment to a proper wrapping is the 4th no-rubbing-conflicts condition. Figure 3(b) indicates that disk-separation conditions may suffice to ensure the existence of a proper wrapping, as separation of the disks separates their tangents and avoids unwanted rubbings. In this section we offer three conditions that ensure a proper string wrapping exits.
2.1 Connected Hull-Visibility Graph Define two disks to be hull-visible to one another (a symmetric relation) iff the (closed) convex hull of the disks does not intersect any other disk; see Figure 4. If two disks can
Figure 4. Two disks are visible to one another if their hull does not intersect any other disk. see one another in this sense, then none of their four bi-tangents are blocked (or even touched) by any other disk. For an arrangement A of disks, define their hull-visibility graph H(A) = H to have a node for each disk, and an arc connecting two disk nodes iff the disks are visible to one another. Call the hull of a pair of disks connected in H to the edge corridor for that edge. Lemma 2.1 (Vis. Gr.) If H(A) is connected, then there is a proper wrapping of A. The conditions of this lemma are by no means necessary for the existence of a proper wrapping: H for the configuration in Figure 3(a) is completely disconnected—seven isolated nodes—and yet it can be properly wrapped.
2.2 Unit disks halo The sufficiency condition of Lemma 2.1 is a global property of the arrangement A of disks, not immediately evident upon inspection. Next we explore local separation conditions that allow us to conclude that H is connected. Define an α-halo, α > 0, for a disk D of radius r to be a concentric disk D0 of radius (1 + α) such that no other disk of A intersects D0 . Lemma 2.2 (Unit Halo) Let Da , Db , and Dc√be three unit disks with centers at a, b, and c respectively, each with α-halos for α = 4/ 3 − 2 ≈ 0.31. Then, if Dc intersects the (Da , Db ) corridor, c is closer to a and to b than is a to b: |ac| < |ab| and |bc| < |ab|.
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String-wrapped Rotating Disks
√ Theorem 2.3 (Unit Disks) An arrangement A of unit disks with α-halos, α = 4/ 3−2, has a connected visibility graph H(A), and so can be properly wrapped.
2.3 Arbitrary radii halo For disks of different radii, we define the distance between them to be the distance between their bounding circles (rather than their centers). The assumption in Theorem 2.3 that all disks are congruent can be removed at the cost of a significant increase in the value of α. Note that, for disks of arbitrary radii, we must allow for a disk radius to be arbitrarily small, effectively a point regardless of α. Thus the strategy to avoid blockage of a (Da , Db ) corridor is to cover it entirely with the α-halos of Da and Db , for no Dc can penetrate these halos by definition. Lemma 2.4 (α=1 Halo) The conclusion of Lemma 2.2 holds for three disks Da , Db , and Dc of arbitrary radius if α = 1. Now the analog of Theorem 2.3 follows immediately by an identical proof, only invoking Lemma 2.4 rather than Lemma 2.2: Theorem 2.5 (α=1 Halo) Any arrangement A disks with α-halos, α = 1, has a connected visibility graph H(A), and so can be properly wrapped.
3 Conclusion There is no question that Theorems 2.3 and 2.5 do not approach a full characterization of the conditions that ensure proper wrapping, as Figure 3 so dramatically indicates. Finding a tighter characterization is one central open question. Surely the α = 1 halo is more generous than needed. An approachable specific version of this question is an arrangement A of unit disks in a hexagonal-packing pattern (as in Figure 2, but with n disks at hexagon lattice points). Which ± labelings are wrappable? I can prove that, if the adjacency graph of like-labeled disks is a forest (a union of disjoint trees), then A is properly wrappable. But this sufficient condition is not necessary. A second central open question is to find a shortest wrapping when proper wrappings exist. For widely spaced disks, this reduces to a version of TSP,1 but the situation is less clear for congested arrangements.
References [DDP10]
E.D. Demaine, M.L. Demaine, and B. Palop. Conveyer-belt alphabet. In H. Aardse and A. van Baalen, eds., Findings in Elasticity, pages 86–89. Pars Foundation, Lars M¨ uller, 2010. [DDTT07] E.D. Demaine, M.L. Demaine, P. Taslakian, and G.T. Toussaint. Sand drawings and Gaussian graphs. J. Math. Arts, 1(2):125–132, 2007. [LT09] Y. Liu and G.T. Toussaint. A simple algorithm for constructing perfect monolinear sona tree drawings, and its application to visual art education. In Proc. 8th Internat. Conf. AI, KE, DB, pages 288–294, 2009.
1I thank Erik Demaine for this observation.
String-wrapped Rotating Disks Joseph O’Rourke1 1
Department of Computer Science, Smith College, Northampton, MA 01063, USA. [email protected]
Abstract. Let the centers of a finite number of disjoint, closed disks be pinned to the plane, but with each free to rotate about its center. Given an arrangement of such disks with each labeled + or −, we investigate the question of whether they can be all wrapped by a single loop of string so that, when the string is taut and circulates, it rotates by friction all the (+)-disks counterclockwise and all the (−)disks clockwise, without any string-rubbing conflicts. We show that although this is not always possible, natural disk-separation conditions guarantee a solution. This work suggests many open problems.
Introduction Let A be a collection of n disjoint closed disks in the plane, each labeled + or −. We seek to wrap them all in one continuous loop of string so that, were one of the disks rotated by a motor, all the others would spin by friction with the string/belt in a direction consistent with the labeling: counterclockwise (ccw) for + and clockwise (cw) for −. See Figure 1. We call a wrapping proper if it satisfies these conditions:
+ −
− −
+ +
+
+ −
Figure 1. A proper wrapping of disks with a loop of string: each + disk rotates counterclockwise, each − clockwise. (1) The string is taut: it follows arcs of disk boundaries and disk-disk tangents only. (2) Each disk boundary circle has a positive-length arc in contact with the string. (It is fine if the string wraps around a disk more than once.) (3) One of the two possible circulation directions (i.e., orientations) for the string loop rotates each disk in the direction consistent with its labeling. 1 CRM Documents, vol. 8, Centre de Recerca Matem` atica, Bellaterra (Barcelona), 2011
2
String-wrapped Rotating Disks
(4) If the string contacts a point of a disk boundary circle, its circulation there must be in the direction consistent with that disk’s label, i.e., there is no rubbing conflict. We permit the string to cross itself at points not on a disk boundary. Indeed such crossings are necessary: for any pair of disks, one + and one −, the string must form a figure-8 shape. Although the conditions for a proper wrapping are suggested by physical analogy, the pursuit here is not driven by any application. Proper string wrappings bear some resemblance to sona sand drawings [DDTT07, LT09], but are more closely related to the conveyor-belt wrappings studied in [DDP10]. Those belts differ from string wrappings in that the rotation directions were not prespecified, and the belt could not self-cross. Although the models are different, the questions raised are analogous. We show that not all arrangements of disks have a proper wrapping, but that various separation conditions guarantee proper wrappings. For example, every collection of unit disks has a proper wrapping when each pair is separated by a distance of 0.31 or more. Characterizing the disk arrangements that admit proper wrappings is posed as an open problem in Section 3.
1 Unwrappable arrangements An example of an unwrappable arrangement is shown in Figure 2.
D2 +
+
+
+
D0
+
D1
+
D0
+
+
+
(a)
+
+
−
D2 +
+ D0
+
+
+
D1 +
+
+
(b)
+
Figure 2. An unwrappable arrangement of seven unit disks.
Figure 3. (a) Proper wrapping with D0 labeled −. (b) Proper wrapping with D1 and D2 displaced slightly.
It consists of one unit disk surrounded by six others, arranged in a hexagonal pennypacking pattern, except the disks are just barely disjoint. We now prove this configuration is unwrappable. The central disk D0 must have a positive-length arc of ccw string touching it. Because a taut string can only leave the boundary of a disk along a tangent, the string follows at least the arc between two adjacent tangents. In order for the string to reach another disk, say D2 , and contribute a ccw arc, it must first touch another disk, D1 in the figure, but now rubbing it in a cw arc. Thus a rubbing conflict is unavoidable. Without moving the disks, this arrangement can be properly wrapped with a different pattern of ± labels. For example, reversing the central disk D0 enables a proper
XIV Spanish Meeting on Computational Geometry, 27–30 June 2011
3
wrapping: Figure 3(a). Indeed all other ± patterns of labels (except all −) in this example are wrappable. Retaining the original + labels but moving two disks slightly also permits the configuration to be wrapped: Figure 3(b).
2 Separation conditions The primary impediment to a proper wrapping is the 4th no-rubbing-conflicts condition. Figure 3(b) indicates that disk-separation conditions may suffice to ensure the existence of a proper wrapping, as separation of the disks separates their tangents and avoids unwanted rubbings. In this section we offer three conditions that ensure a proper string wrapping exits.
2.1 Connected Hull-Visibility Graph Define two disks to be hull-visible to one another (a symmetric relation) iff the (closed) convex hull of the disks does not intersect any other disk; see Figure 4. If two disks can
Figure 4. Two disks are visible to one another if their hull does not intersect any other disk. see one another in this sense, then none of their four bi-tangents are blocked (or even touched) by any other disk. For an arrangement A of disks, define their hull-visibility graph H(A) = H to have a node for each disk, and an arc connecting two disk nodes iff the disks are visible to one another. Call the hull of a pair of disks connected in H to the edge corridor for that edge. Lemma 2.1 (Vis. Gr.) If H(A) is connected, then there is a proper wrapping of A. The conditions of this lemma are by no means necessary for the existence of a proper wrapping: H for the configuration in Figure 3(a) is completely disconnected—seven isolated nodes—and yet it can be properly wrapped.
2.2 Unit disks halo The sufficiency condition of Lemma 2.1 is a global property of the arrangement A of disks, not immediately evident upon inspection. Next we explore local separation conditions that allow us to conclude that H is connected. Define an α-halo, α > 0, for a disk D of radius r to be a concentric disk D0 of radius (1 + α) such that no other disk of A intersects D0 . Lemma 2.2 (Unit Halo) Let Da , Db , and Dc√be three unit disks with centers at a, b, and c respectively, each with α-halos for α = 4/ 3 − 2 ≈ 0.31. Then, if Dc intersects the (Da , Db ) corridor, c is closer to a and to b than is a to b: |ac| < |ab| and |bc| < |ab|.
4
String-wrapped Rotating Disks
√ Theorem 2.3 (Unit Disks) An arrangement A of unit disks with α-halos, α = 4/ 3−2, has a connected visibility graph H(A), and so can be properly wrapped.
2.3 Arbitrary radii halo For disks of different radii, we define the distance between them to be the distance between their bounding circles (rather than their centers). The assumption in Theorem 2.3 that all disks are congruent can be removed at the cost of a significant increase in the value of α. Note that, for disks of arbitrary radii, we must allow for a disk radius to be arbitrarily small, effectively a point regardless of α. Thus the strategy to avoid blockage of a (Da , Db ) corridor is to cover it entirely with the α-halos of Da and Db , for no Dc can penetrate these halos by definition. Lemma 2.4 (α=1 Halo) The conclusion of Lemma 2.2 holds for three disks Da , Db , and Dc of arbitrary radius if α = 1. Now the analog of Theorem 2.3 follows immediately by an identical proof, only invoking Lemma 2.4 rather than Lemma 2.2: Theorem 2.5 (α=1 Halo) Any arrangement A disks with α-halos, α = 1, has a connected visibility graph H(A), and so can be properly wrapped.
3 Conclusion There is no question that Theorems 2.3 and 2.5 do not approach a full characterization of the conditions that ensure proper wrapping, as Figure 3 so dramatically indicates. Finding a tighter characterization is one central open question. Surely the α = 1 halo is more generous than needed. An approachable specific version of this question is an arrangement A of unit disks in a hexagonal-packing pattern (as in Figure 2, but with n disks at hexagon lattice points). Which ± labelings are wrappable? I can prove that, if the adjacency graph of like-labeled disks is a forest (a union of disjoint trees), then A is properly wrappable. But this sufficient condition is not necessary. A second central open question is to find a shortest wrapping when proper wrappings exist. For widely spaced disks, this reduces to a version of TSP,1 but the situation is less clear for congested arrangements.
References [DDP10]
E.D. Demaine, M.L. Demaine, and B. Palop. Conveyer-belt alphabet. In H. Aardse and A. van Baalen, eds., Findings in Elasticity, pages 86–89. Pars Foundation, Lars M¨ uller, 2010. [DDTT07] E.D. Demaine, M.L. Demaine, P. Taslakian, and G.T. Toussaint. Sand drawings and Gaussian graphs. J. Math. Arts, 1(2):125–132, 2007. [LT09] Y. Liu and G.T. Toussaint. A simple algorithm for constructing perfect monolinear sona tree drawings, and its application to visual art education. In Proc. 8th Internat. Conf. AI, KE, DB, pages 288–294, 2009.
1I thank Erik Demaine for this observation.
