Problem 392

Enmeshed unit circle

A rectilinear grid is an orthogonal grid where the spacing between the gridlines does not have to be equidistant.
An example of such grid is logarithmic graph paper.

Consider rectilinear grids in the Cartesian coordinate system with the following properties:

The gridlines are parallel to the axes of the Cartesian coordinate system.
There are N+2 vertical and N+2 horizontal gridlines. Hence there are (N+1) x (N+1) rectangular cells.
The equations of the two outer vertical gridlines are x = -1 and x = 1.
The equations of the two outer horizontal gridlines are y = -1 and y = 1.
The grid cells are colored red if they overlap with the unit circle, black otherwise.

For this problem we would like you to find the postions of the remaining N inner horizontal and N inner vertical gridlines so that the area occupied by the red cells is minimized.

E.g. here is a picture of the solution for N = 10:

The area occupied by the red cells for N = 10 rounded to 10 digits behind the decimal point is 3.3469640797.

Find the positions for N = 400.
Give as your answer the area occupied by the red cells rounded to 10 digits behind the decimal point.

单位圆陷阱

所谓直线网格是指网格线间距不一定相等的直角网格。
这类网格的其中一个例子是对数图象纸。

考虑笛卡尔坐标系下满足如下性质的直线网格：

网格线均平行于笛卡尔坐标系的坐标轴。
一共有N+2条竖直网格线和N+2条水平网格线，因此一共有(N+1) x (N+1) 个长方形网格单元。
最外侧两条竖直网格线的方程为x = -1和x = 1。
最外侧两条水平网格线的方程为y = -1和y = 1。
若网格单元和单位圆重叠则染上红色，否则染上黑色。

在这个问题中，我们需要你确定剩下的N条竖直网格线和N条水平网格线，使得红色网格单元占据的总面积最小。

例如，下图是N = 10时最优解的图示：

当N = 10时，红色网格单元占据的总面积四舍五入到小数点后10位小数为3.3469640797。

求N = 400时各条网格线的位置。
你的答案应当是此时红色网格单元占据的总面积四舍五入到小数点后10位小数。