Expanding Visibility Polygons by Mirrors upto at least K units

We consider extending visibility polygon (V P ) of a given point q (V P (q)), inside a simple polygon P by converting some edges of P to mirrors. We will show that several variations of the problem of finding mirror-edges to add at least k units of area to V P (q) are NP-complete, or NP-hard. Which k is a given value. We deal with both single and multiple reflecting mirrors, and also specular or diffuse types of reflections. In specular reflection, a single incoming direction is reflected into a single outgoing direction. In this paper diffuse reflection is regarded as reflecting lights at all possible angles from a given surface. The paper deals with finding mirror-edges to add at least k units of area to V P (q). In the case of specular type of reflections we only consider single reflections, and the multiple case is still open. Specular case of the problem is more tricky. We construct a simple polygon for every given instance of a 3-SAT problem. There are some specific spikes which are visible only by some particular mirror-edges. Consequently, to have minimum number of mirror-edges it is required to choose only one of these mirrors to see a particular spike. There is a reduction polygon which contains a clause-gadget corresponding to every clause, and a variable-gadget corresponding to every variable. 3-SAT formula has n variables and m clauses, so the minimum number of mirrors required to add an area of at least k to V P (q) is l = 3m+n+1 if and only if the 3-SAT formula is satisfiable. This reduction works in these two cases: adding at least k vertex of P to V P (q), and expanding V P (q) at least k units of area. 1998 ACM Subject Classification Dummy classification – please refer to http://www.acm.org/ about/class/ccs98-html