- Email: [email protected]
Separability of unimodal polygons
Pattern Recognition Letters 7 (1988) 163- 165 North-Holland
March 1988
Separability of unimodalpolygons Arun K. PUJARI School ~)['Mathematics and Computer/Information Sciences, University c f lqvderabad, Hvderabad 500 134, lndia
Received 26 September 1987 Abstract: A conjecture that any three unimodal polygons are sequentially separable is, by means of a counterexamplc, shown to be false. Key words: Movable separability, unimodal polygon, computational geometry.
1. Introduction The problem of movable separability of polygonal objects arises in the context of robotics, CAD, VLSI and several other application areas of computational geometry. Toussaint (1985) presents a classification scheme of polygons and surveys the separability properties of each of the classes. It is shown that the unimodal class of polygon lies between isothetic class and monotone class. The former is sequentially movably separable and the latter is not. Toussaint conjectures that any three unimodal polygons are sequentially movably separable. We present here a counter-example to show that his conjecture is false.
gle translation. More than two objects, PI, P2 . . . . . P, (n > 2) are said to be sequentially movably separable if there exists an ordering P~j, -P~2)..... P~,~ such that for i = 1, 2 ..... n - 1, P,I can be moved an arbitrary distance away from Pli+~) ..... P~,~ without colliding with any of them. A vertex Pi of a polygon P is said to be unimodal if the euclidean distances d(pi, Pi+ 1), d(pi, Pi+2), .... d(pl, Pi- 1) have only one maximum. A polygon P is unimodal if every vertex of P is unimodal. Toussaint, while studying the movable separability of the polygonal objects, conjectures that three unimodal polygons are sequentially movably separable. We show that the conjecture is fal,;e.
2. Definitions
3. Counter-example
We follow the notations and definitions as given by Toussaint (1985). Some of the definitions are reproduced here for reference. Two objects are movably separable if one of them can be moved an arbitrary distance away from the other without colliding with it. If they are not movably separable then they are said to be interlocked. A variety of definitions of movable separability are apparent by specifying the type of motion considered. We are concerned with only motions with sin-
Three identical unimodal polygons are symmetrically arranged in Figure 1. These three polygons are not sequentially movably separable. Any of the polygons can be described as follows: The internal angles at vertices Pl, P2, P3, P4 and P5 are 90 °, 90 °, 120 °, 270 ° and 215 ° respectively. The length of the sides PlP2, P2P3, P3P4, P4Ps, P5P6 and PTPl are 1.0, 5.0, 0.4, 1.0, 0.3 and ll.0 respectively. It is easy to check that the polygon is unimodal.
0167-8655/88/$3.50 © 1988, Elsevier Science Publishers B.V. (North-Holland)
163
Volume 7, Number 3
PATTERN RECOGNITION LETTERS
March 1988
H
/1_ /
/
/
I I /
I
I
P1
134 -- D 7
Figure 1. Three unimodal polygons that are not sequentially separable.
Some of the facts which are not visually clear are as follows: Pl is nearer to P3 than P6 is nearer to P4 than P4 is nearer to P2 than P7 is nearer to P6 than
to to to to
P4; P3; pl; Ps.
It is also easy to see that all the three polygons, arranged in a manner shown in Figure 1, are sequentially interlocked. Without loss of generality, due to symmetry, we consider the movement of P2 in presence of P1 and P3When P3 is absent, P2 can be moved away from P1 in directions defined by the convex cone C D H . In other words, the possible direction in which P2 can be moved is [65 °, 120°]. Similarly when Px is absent, P2 can be moved away from P3 in a set of direction defined by I' M' L' i.e., in [5 °, 60°]. Since both the closed intervals are disjoint, P2 cannot be moved in presence of P1 and P3. Hence none of the 164
polygons can be moved in the presence of other two.
4. Conclusion The unimodal class of polygons lies between the isothetic class and the monotone class. Isothetic polygons are known to be sequentially movably separable. Similarly, three monotone polygons are not always sequentially separable. Contrary to the belief that unimodal polygons are sequentially movably separable like isothetic class or convex class, in this note we showed that any three arbitrary unimodal polygons are not always sequentially movably separable. In the example presented here, it can be noted that even when the type of motion is not restricted only to single translation, but allows rotations too, then also, they are not movably separable. Thus not any arbitrary three unimodal poly-
Volume 7, N u m b e r 3
PATTERN RECOGNITION LETTERS
gons are sequentially movably separable under a single rotation followed by single translation.
Acknowledgement
March 1988
Reference Toussaint, G.T. (1985). Movable separability of sets. In: G.T. Toussaint, Ed., Computational Geometry. North-Holland, Amsterdam.
The author is grateful to the referee for useful suggestions and corrections.
165
March 1988
Separability of unimodalpolygons Arun K. PUJARI School ~)['Mathematics and Computer/Information Sciences, University c f lqvderabad, Hvderabad 500 134, lndia
Received 26 September 1987 Abstract: A conjecture that any three unimodal polygons are sequentially separable is, by means of a counterexamplc, shown to be false. Key words: Movable separability, unimodal polygon, computational geometry.
1. Introduction The problem of movable separability of polygonal objects arises in the context of robotics, CAD, VLSI and several other application areas of computational geometry. Toussaint (1985) presents a classification scheme of polygons and surveys the separability properties of each of the classes. It is shown that the unimodal class of polygon lies between isothetic class and monotone class. The former is sequentially movably separable and the latter is not. Toussaint conjectures that any three unimodal polygons are sequentially movably separable. We present here a counter-example to show that his conjecture is false.
gle translation. More than two objects, PI, P2 . . . . . P, (n > 2) are said to be sequentially movably separable if there exists an ordering P~j, -P~2)..... P~,~ such that for i = 1, 2 ..... n - 1, P,I can be moved an arbitrary distance away from Pli+~) ..... P~,~ without colliding with any of them. A vertex Pi of a polygon P is said to be unimodal if the euclidean distances d(pi, Pi+ 1), d(pi, Pi+2), .... d(pl, Pi- 1) have only one maximum. A polygon P is unimodal if every vertex of P is unimodal. Toussaint, while studying the movable separability of the polygonal objects, conjectures that three unimodal polygons are sequentially movably separable. We show that the conjecture is fal,;e.
2. Definitions
3. Counter-example
We follow the notations and definitions as given by Toussaint (1985). Some of the definitions are reproduced here for reference. Two objects are movably separable if one of them can be moved an arbitrary distance away from the other without colliding with it. If they are not movably separable then they are said to be interlocked. A variety of definitions of movable separability are apparent by specifying the type of motion considered. We are concerned with only motions with sin-
Three identical unimodal polygons are symmetrically arranged in Figure 1. These three polygons are not sequentially movably separable. Any of the polygons can be described as follows: The internal angles at vertices Pl, P2, P3, P4 and P5 are 90 °, 90 °, 120 °, 270 ° and 215 ° respectively. The length of the sides PlP2, P2P3, P3P4, P4Ps, P5P6 and PTPl are 1.0, 5.0, 0.4, 1.0, 0.3 and ll.0 respectively. It is easy to check that the polygon is unimodal.
0167-8655/88/$3.50 © 1988, Elsevier Science Publishers B.V. (North-Holland)
163
Volume 7, Number 3
PATTERN RECOGNITION LETTERS
March 1988
H
/1_ /
/
/
I I /
I
I
P1
134 -- D 7
Figure 1. Three unimodal polygons that are not sequentially separable.
Some of the facts which are not visually clear are as follows: Pl is nearer to P3 than P6 is nearer to P4 than P4 is nearer to P2 than P7 is nearer to P6 than
to to to to
P4; P3; pl; Ps.
It is also easy to see that all the three polygons, arranged in a manner shown in Figure 1, are sequentially interlocked. Without loss of generality, due to symmetry, we consider the movement of P2 in presence of P1 and P3When P3 is absent, P2 can be moved away from P1 in directions defined by the convex cone C D H . In other words, the possible direction in which P2 can be moved is [65 °, 120°]. Similarly when Px is absent, P2 can be moved away from P3 in a set of direction defined by I' M' L' i.e., in [5 °, 60°]. Since both the closed intervals are disjoint, P2 cannot be moved in presence of P1 and P3. Hence none of the 164
polygons can be moved in the presence of other two.
4. Conclusion The unimodal class of polygons lies between the isothetic class and the monotone class. Isothetic polygons are known to be sequentially movably separable. Similarly, three monotone polygons are not always sequentially separable. Contrary to the belief that unimodal polygons are sequentially movably separable like isothetic class or convex class, in this note we showed that any three arbitrary unimodal polygons are not always sequentially movably separable. In the example presented here, it can be noted that even when the type of motion is not restricted only to single translation, but allows rotations too, then also, they are not movably separable. Thus not any arbitrary three unimodal poly-
Volume 7, N u m b e r 3
PATTERN RECOGNITION LETTERS
gons are sequentially movably separable under a single rotation followed by single translation.
Acknowledgement
March 1988
Reference Toussaint, G.T. (1985). Movable separability of sets. In: G.T. Toussaint, Ed., Computational Geometry. North-Holland, Amsterdam.
The author is grateful to the referee for useful suggestions and corrections.
165