Area Bounds of Rectilinear Polygons Realized by Angle Sequences

Minimum Rectilinear Polygons for Given Angle Sequences

“…If we did prolong the bottom extreme edge e of P 1 (which can happen only in Case (ii) for B TL ), then the polygon P 2 is of minimum area by our discussion of Case (2). Given that r has the smallest size among all steps in S \ {s} and given that s is greater than s, we conclude that r is a smallest step in S .…”

“…To this end, we therefore assume that (a) we did prolong an extreme edge of P 1 (it has to be the bottom one), or that (b) the step r has size b/(a + 1) + 1. We cut P * and P in the same way as in Case (2) and we define, for 1 ≤ i ≤ 3, the variables P i and P * i , as well as BL , S, S , s and s in the same way as in Case (2). Note that r is in S and in S .…”

“…Culberson and Rawlins' algorithm, when constrained to ±90 • angles, produces polygons with no colinear edges, implying that any n-vertex polygon can be drawn with area approximately (n/2 − 1) 2 . However, as Bae et al [2] showed, the bound is not tight.…”

“…Related Work. Bae et al [2] considered, for a given angle sequence S, the polygon P (S) that realizes S and minimizes its area. They studied the following question: Given a number n, find an angle sequence S of length n such that the area of P (S) is minimized, or maximized.…”