Illumination of Polygons
with Vertex Lights
Vladimir Estivill-Castro
Joseph O'Rourkey
Jorge Urrutiaz
Dianna Xux
August 28, 1994
Abstract
We show that vertex oodlights of angle suce to illuminate any polygon, and that
no angle less than suces for all polygons.
1 Introduction
There has been recent interest in illumination by oodlights [BGL 93] [CRCU93] [SS94], a new
variation on visibility problems in computational geometry. A oodlight of angle is a light
that projects in a cone of that angle. A oodlight placed in a simple polygon with its apex
at a vertex is called a vertex oodlight; if it has angle , we will call it a vertex -light. The
question of illuminating a polygon with vertex oodlights was rst explored in [ECU94], in
which it was established that every orthogonal polygon can be covered by b3(n ? 1)=8c vertex
=2-lights. This work raised the question of what angle of vertex oodlights would suce to
cover any simple polygon. In [OX94] it was shown that vertex =2-lights do not suce, even
for monotone polygons. In this note we show that vertex -lights suce, and no smaller angle
suces for all polygons.
+
Laboratorio Nacional de Informatica Avanzada, Rebsamen 90, Xalapa, Veracruz 91000, Mexico.
[email protected].
y Dept. of Computer Science, Smith College, Northampton, MA 01063, USA. [email protected]. Supported by NSF grant CCR-9122169.
z Dept. of Computer Science University of Ottawa Ottawa, Ontario CANADA K1N 6N5 [email protected]
x
Dept. of Computer Science, Smith College. [email protected].
1
2
Suces
3
=2
The proof is by induction on the number of vertices of P . If P is a triangle, a oodlight at any
vertex suces to cover it. Suppose P has n > 3 vertices. P must have an ear, by Meister's
Two Ears Theorem: three consecutive vertices a, b, c such that ac is an internal diagonal. The
angle at the ear tip vertex b is strictly convex. Place a oodlight at b and cover the ear triangle
(a; b; c). Removing the ear leaves a polygon of n ? 1 vertices, which can be covered by the
induction hypothesis.
Does Not Suce
Although it is already known that =2 lights do not suce, the polygons we use later employ
variations on the mechanism used in [OX94] to establish this result. We reestablish and simply
this result with a consruction that will form the basis of modications in the next section.
The polygon P is shown in Fig. 1a. It is symmetric about a vertical line through vm . Let
L be the half of P left of the vertical through vm, and let R be the right half. Vertices in L
are labeled to reect their counterparts in R. The internal angles at vertices v , v , and v
all exceed =2; vertices r , r , and vm are reex. The extensions of the edges r v and of r v
(dashed in the gure) intersect the horizontal edge v v0 slightly to the left of v . Vertex vm is
just above the edge v v0 . Floodlights are shown at each vertex by a small shaded =2 wedge.
We now argue that =2 lights cannot cover P . The light at vm may illuminate points in
either L or R but not both. Assume without loss of generality that this light aims leftward as
shown in the gure. Because the angles at vi exceed =2, the lights at these vertices must leave
points on the boundary @P of P near vi locally unilluminated. There are two primary choices
for each vi light: turn clockwise (cw) as far as possible, or turn counterclockwise (ccw) as far
as possible. It is easy to see that intermediate positions are not useful. We now argue that the
light at v must be turned cw or points on v vm will be left uncovered.
Points on @P in a small neighborhood of v are visible from only three vertices: vm , v , and
r (v cannot see v because it is right of the extension of r v ). The light at vm is aiming into
L and of no use, so these points must be covered from v and r . Since the angle at v exceeds
=2, the light at v alone does not suce. So the light at r must be used by turning it cw.
Now consider points on @P in a small neighborhood of v : they are visible from the vertices
vm , r , v , and r . The lights at vm and r are aimed away. The light at v cannot cover points
to both sides of v . Therefore the light at r must be used by turning it cw.
If the light at v is turned ccw ush with v v0 , points on v r in a small neighborhood of
v are left uncovered. Therefore the light at v must be turned cw ush with v r , as shown in
Fig. 1a. We will see in a moment that this still leaves points near v uncovered.
Now we turn attention to the left half of P . Because the light at vm shines into L, seeing
both v0 and v0 , the lights may all be turned cw or ccw: either way covers L entirely. In Fig. 1a
the cw orientation is shown. This arrangement has the light at v0 ush with v0 v , and therefore
shinning down that edge. This allows a thin beam of light to shine into R from v0 .
Finally, we reexamine the vicinity of v , shown magnied in Fig. 1b. Although all the points
0
1
2
1
2
1 1
0 0
2 2
0
0 0
0
2
2
2
0
2
2
2 2
2
2
2
2
2
1
2
1
1
2
1
1
1
0
0 0
0
0 1
0
0 1
0
1
2
0
0 0
0
0
2
r1
r'1
v0
v1
v'1
r2
z
vm
v0
v2
v'2
v'0
(b)
(a)
r'2
3
Figure 1: (a) A polygon for which vertex =2-lights do not suce. (b) A enlargement of the
vicinity of v .
on @P in a neighborhood of v are illuminated, there is a small triangle T of interior points
that remain unilluminated. The light from v0 to a point z 2 T is blocked by vm , from v it is
blocked by r , and from v it is blocked by r . Thus P cannot be covered with =2 lights.
0
1
0
1
4
2
2
? Does Not Suce
We now generalize the design in the preceeding section to show that any angle < does
not suce. We rst describe the polygon for a particular , and second dicuss the general
construction.
4.1 A Particular Polygon
The polygon P is a direct generalization of Fig. 1, again symmetrical with respect to a vertical
line through vm . The right half is composed of vertices v ; r ; v ; . . . ; vk ; rk . Vertices v ; . . . ; vk all
have internal angles equal to . Fig. 2 shows a portion of this half for k = 5, = 3=4 = 135.
The edge vk vm slopes downward to the left, and (as in Fig. 1) vm is just above the v v0 edge.
The vertices ri , i = 0; . . . ; k are placed very near to vi , (so near as to be indiscernable in
Fig. 2), and are slightly reex. A magnied view of the vicinity of one vertex vi is shown in
the circle in Fig. 2. The edges v r ; v r ; v r ; . . . increase in length in a manner that will be
detailed later. The extensions of the edges ri vi (dashed in the gure) intersect v v0 slightly left
of v . For concreteness we arrange that they all meet v v0 at the same point x (so close to v
as to be indistinguishable in Fig. 2).
We now argue that that P cannot be covered by an angle smaller than . The proof
is exactly analogous to that in the previous section. Assume without loss of generality that
the light at vm aims into L. Then points on @P in a small neighborhood of vk can only be
illuminated from vk and rk , since all of the vertices with indices k ? 1; . . . ; 0 lie right of the
extension of rk vk . Because the light at vk cannot cover all of such a neighborhood, the light
at rk must be turned cw. Now the argument is repeated at vk? ; . . . ; v , concluding that the
lights at ri must all be turned cw. Finally this requires that the light at v be turned cw to
cover points on v r . Just as in the previous section, this leaves a small interior region of points
(analogous to the region in Fig. 1b) unilluminated, regardless of how the left half L is covered.
0
1
1
0
0 0
0 1
1 2
2 3
0 0
0
0
0 0
1
1
0
0 1
4.2 General Construction
For arbitary < , the main challenge is to arrange that the extensions of the edges ri vi all
intersect v v0 at point x left of v , while maintaining the angles at vi to be . One can base a
general construction on the properties of the logarithmic spiral, r = aeb . This spiral has the
property that the angle between the radial vector to a point p on the curve and the tangent at
p is a constant. If b = cot , this constant angle is .
Now we show how to use the family of spirals r = ae , a = 1; 2; . . . to construct the
polygon P . Call these spirals Sa . We illustrate the technique for = 3=4 = 135 in Fig. 3.
The origin of the spirals becomes x, the point just left of v in P . Draw an edge from v (just
0 0
0
(cot
0
4
)
0
v3
135o
v5
r5
v4
r3
r4
vm
v3
r3
v2
v1
v0 r1
v'0
x
Figure 2: Vertex (135 ? )-lights do not suce. Only a portion of the right half R of P is
shown. The vicinity of v is magnied in the circle.
3
5
r2
S5
v5
v4
S4
5
S3
v3
S2
S1
-5
Figure 3: Log spiral construction for = 3=4 = 135.
6
v2
v1
right of x) to S forming an angle with the horizontal; the intersection point becomes v .
From v follow the tangent line of S to S ; this intersection is v . This process is continued
until the tangent line slopes downward. If Fig. 3, this is achieved at v
By the properties of the logarithmic spiral, we are guaranteed that the angle between xvi
and vi vi is . Because < , we are guaranteed that advances on the spirals from vi to
vi . This advancement in turn guarantees that for some k, the tangent will slope downwards.
With this basic structure established, it is easy to insert the reex vertices ri close to vi by a
slight expansion along the xvi line (as illustrated for v in Fig. 2). The result is a polygon P
for which oodlights of angle smaller than do not suce.
This construction works for any < . (Note that the logarithmic spiral is not dened for
= , when cot = ?1.) Thus, given any angle < , we can construct a polygon that
cannot be illuminated by vertex lights of angle .
1
1
1
1
2
2
5
+1
1
+1
3
5 Discussion
Now that we know that no angle less than suces, it becomes an interesting question to ask
how many vertex -lights are ever needed as a function of n, the number of vertices of P .
References
[BGL 93] P. Bose, L. Guibas, A. Lubiw, M. Overmars, D. Souvaine, and J. Urrutia. The
oodlight problem. In Proc. 5th Canad. Conf. Comput. Geom., pages 399{404,
Waterloo, Canada, 1993.
[CRCU93] J. Czyzowicz, E. Rivera-Campo, and J. Urrutia. Optimal oodlight illumination
in polynomial time. In Proc. 5th Canad. Conf. Comput. Geom., pages 393{398,
Waterloo, Canada, 1993.
[ECU94] V. Estivill-Castro and J. Urrutia. Optimal oodlight illumination of orthogonal art
galleries. In Proc. 6th Canad. Conf. Comput. Geom., pages 81{86, 1994.
[OX94] J. O'Rourke and D. Xu. Illumination of polygons with 90 vertex lights. In Snapshots
in Comput. Geom. Univ. Saskatchewan, August 1994. Technical Report 034, Dept.
Comput. Sci., Smith College, July 1 994.
[SS94]
W. Steiger and I. Streinu. Positive and negative results on the oodlight problem.
In Proc. 6th Canad. Conf. Comput. Geom., pages 87{92, 1994.
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1
An argument that we will not detail shows that k = =( ? ) suces.
7