**Geometric Structures, Mathematics 244**

Guards and Visibility

prepared by:

Joseph Malkevitch

Department of Mathematics and Computing

York College (CUNY)

Jamaica, New York 11451

1. a. How many vertices and edges does the polygon P shown have?

b. Triangulate polygon P in two different ways. How many triangles does each triangulation have? If the lines used to carry out the triangulation are called *diagonals* (see q. below) do all triangulations require equal numbers of diagonals?

c. List all of the ears of the polygon P.

d. Determine a 3-coloring of the vertices of polygon P.

e. Find the smallest number of vertex guards that are necessary to guard polygon P? Call the set of vertices at which this minimal set of guards is located M.

f. Two vertices are said to *guard each other* if the guards are mutually visible. A set of guards is called a *guarded* if for each guard g there is another (different) guard h so that g and h guard each other. Does the set M you found in e. constitute a guarded set of guards? If not find the smallest set of guarded guards for this polygon.

g. For each vertex w of P find number of vertices of P which is visible from w.

h. Draw the visibility polygons for vertices 6 and 2 of polygon p.

i. A vertex u of a polygon Q is called a *reflex vertex* if the interior angle at u is greater than 180 degrees. Determine all of the reflex vertices of polygon P.

j. A *diagonal* of a plane polygon Q is a segment s joining two vertices u and v, which are not end points of a side of the polygon, such that all the points of s other than u and v are interior points of Q.

How many diagonals does P have? How many diagonals does P have which do not intersect at any interior point of the polygon?

k. How many vertices does the convex hull of P have?

m. If Q is a plane convex polygon, show that Q has a guarded set of guards with 2 elements.

n. Draw a 9 sided polygon that needs 3 guards.

o. Draw a 12 sided polygon that needs 4 guards.

p. Can you find an equilateral polygon with 3k sides that needs k guards, for each value of k?

r. Can a plane polygon which is not self-intersecting and not convex be equiangular?

s. A plane polygon Q is called *orthogonal* or *rectilinear* if all of its interior angles are either 90 or 270 degrees. For which values of n can you draw a rectilinear polygon?

t. Show there is a polygon with 6 sides which requires two guards but no rectilinear polygon with 6 sides requires 2 guards. Find the smallest number of sides for a rectilinear polygon which requires 2 guards.