A rectilinear polygon is a simple polygon whose edges are axis-aligned. Walking counterclockwise on the boundary of such a polygon yields a sequence of left turns and right turns. The number of left turns always equals the number of right turns plus four. It is known that any such sequence can be realized by a rectilinear polygon.

In this paper, we consider the problem of finding realizations that minimize the perimeter or the area of the polygon or the area of the bounding box of the polygon. We show that all three problems are $\mathsf{NP}$ -hard in general. This answers an open question of Patrignani (2001) [13], who showed that it is $\mathsf{NP}$ -hard to minimize the area of the bounding box of an orthogonal drawing of a given planar graph. We also show that realizing a polyline within a bounding box of minimum area (or within a fixed given rectangle) is $\mathsf{NP}$ -hard. Then we consider the special cases of x-monotone and xy-monotone rectilinear polygons. For these, we can optimize the three objectives efficiently.

甲直线多边形是一个简单的多边形，其边缘是轴对齐。在这种多边形的边界上逆时针行走会产生一系列左转和右转。左转的次数总是等于右转的次数加四。众所周知，任何这样的序列都可以通过直线多边形来实现。

在本文中，我们考虑找到最小化多边形周长或面积或多边形边界框面积的实现问题。我们证明所有三个问题都是$\mathsf{NP}$- 一般很难。这回答了 Patrignani (2001) [13] 的一个悬而未决的问题，他表明$\mathsf{NP}$- 难以最小化给定平面图的正交图的边界框面积。我们还表明，在最小面积的边界框内（或在固定的给定矩形内）实现折线是$\mathsf{NP}$-硬。然后我们考虑x单调和xy单调直线多边形的特殊情况。对于这些，我们可以有效地优化三个目标。