计算x / y网格上两个矩形之间的重叠？ [英] Calculate overlap between two rectangles on x/y grid?

问题描述

我需要计算两个矩形在特殊x / y网格上的重叠（数量或是/否）。网格是500x500，但边和角连接（是连续的）。因此，499后的下一个点再次变为0.

在前面的问题中，我要求一种计算网格中两点之间距离的方法。原来是欧几里德距离：

$ $ $ $ $ $ $ $ $ $ sqrt（min（| x1 - x2 |，gridwidth - | x1 - x2 | ）^ 2 + min（| y1 - y2 |，gridheight - | y1-y2 |）^ 2）

计算两个矩形（由点（x，y），宽度和高度定义）在这个网格中是否有重叠的好数学方法是什么？矩形-1（ [x = 0，y = 0]，w = 20，h = 20 ）和矩形-2（ [x = 495，y = 0]，w = 10，h = 10 ）应该有重叠。重叠的矩形（不是真的需要但是）应该是（ [x = 0，y = 0]，w = 5，h = 10 ）

首先，计算每个矩形
的x和y范围（因为你有一个圆环几何做它的mod gridsize）。

所以，给定你的Rectangle-1，计算：
$ b

  x1 = x = 0，x2 = x + w =​​ 20 
 y1 = y = 0，y2 = y + h = 20 
  

与Rectangle-2相同：

  x3 = 495，x4 = 505 mod 500 = 5 
 y3 = 0，y4 = 10 
  

为每个矩形创建x和y区域：

Reactangle-1：x-regions：（0,20）
y-regions：（0,20）

Rectangangle-2：

  x区域：（495,500），（0,5）
 y-regions：（0,10）
  

如果有（两者）x和y区域之间两个矩形有一个非空的交点，那么你的矩形重叠。
这里Rectangle-1的（0，20）x区域和Rectangle-2的（0，5）x区域有一个非空交点，所以（0,20）和（0 ，10）y区域。

I need to calculate the overlap (amount or yes/no) that two rectangles make on a special x/y grid. The grid is 500x500 but the sides and corners connect (are continuous). So the next point after 499 becomes 0 again.

In a previous question I asked for a way to calculate the distance between two points in this grid. This turned out to be the Euclidean distance:

sqrt(min(|x1 - x2|, gridwidth - |x1 - x2|)^2 + min(|y1 - y2|, gridheight - |y1-y2|)^2)

What is the good mathematical way of calculating if two rectangles (defined by a point (x,y), width and height) have overlap in this grid?

Rectangle-1 ([x=0,y=0], w=20, h=20) and Rectangle-2 ([x=495,y=0], w=10, h=10) should have overlap. The overlapping rectangle (not really needed but) should be ([x=0,y=0], w=5, h=10)

解决方案

First, compute the x and y range for each rectangle (because you have a torus geometry do it mod gridsize).

So, given your Rectangle-1, compute:

x1 = x = 0, x2 = x + w = 20
y1 = y = 0, y2 = y + h = 20

Same for Rectangle-2:

x3 = 495, x4 = 505 mod 500 = 5
y3 = 0,   y4 = 10

Create the x and y "regions" for each rectangle:

Reactangle-1: x-regions: (0, 20)
              y-regions: (0, 20)

Rectangle-2:  x-regions: (495, 500), (0, 5)
              y-regions: (0, 10)

If any (both) x and y regions between the two rectangles have a non-null intersection, then your rectangles overlap. Here the (0, 20) x-region of Rectangle-1 and the (0, 5) x-region of Rectangle-2 have a non-null intersection and so do the (0, 20) and (0, 10) y-regions.

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