找到旋转矩形的边界矩形 [英] Find the Bounding Rectangle of Rotated Rectangle

问题描述

我有矩阵的坐标（x1，y1）和（x2，y2），我必须使用旋转矩阵将矩形旋转θ的中心值。

  | cosθsinθ| 
 | -sinθcosθ| 
  

我需要找到旋转后的边界矩形的坐标。

  0， 0 
 || 
 | | 
 | x1，y1 | 
 | || | 
 | | | | 
 | | | | 
 | | | | 
 | x2，y2 | 
 | | 
 | | 
W，H 
  

轮播

  0,0 
 |      | 
 | ？，？ | 
 | |/ \| | 
 | | / \ | | 
 | | / \ | | 
 | | / / | | 
 | | / | | 
 | | \ / | | 
 | | \ / | | 
 | | \ / | | 
 | ？，？ | 
 | | 
 | | 
W，H 
  

有没有找到边界矩形坐标的一般公式？ b
$ b

谢谢....

Haris。

解决方案Point（x _{1 ，y 1 ）旋转到（x 1 cos < - >; - y 1 sin（θ1，sinθ）+ y 1cosθ），而点（x 2，y 2，...， sub> 2 ）旋转到（x 2cosθ-y 2 sin +θ2，x 2sinθ+其他两个点可以相应计算。}

边界矩形的坐标是（x _{3，y 3）和（x 4，y 4），其中x 3是所有新的x坐标中最小的，y <3>是所有新y坐标中最小的，x <4>是所有新x坐标中最大的，y 4} >所有新的最伟大的y坐标。

哪个角落产生最小的x（依此类推）取决于您的角度或旋转角度。对于0°以下的角度，到90°，x 3将来自（x 1，y 1），所以x 3 = x ₁ cosθ - y 1 sin&theta ;.对于90°和90°以下的角度，到180°，它将来自（x 2，y 1）等等。
因此，要么根据旋转角度决定使用哪些点，要么只是取所有x和y中的最小者和最大者。

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但我认为您应该在 https://math.stackexchange.com/ 上提出此问题

I have rectangle with co-ordinates(x1,y1) and (x2,y2) and I have to rotate the rectangle an amount of θ about it centre using Rotation Matrix

 |  cosθ  sinθ |
 | -sinθ  cosθ |

I need to find the co-ordinates of bounding rectangle after rotation.

Before rotation

0,0
 |"""""""""""""""""""""""""""""""""""""""""""|
 |                                           |
 |  x1,y1                                    |
 |       |"""""""""""""|                     |
 |       |             |                     |
 |       |             |                     | 
 |       |             |                     |
 |       """""""""""""" x2,y2                |
 |                                           |
 |                                           |
  """"""""""""""""""""""""""""""""""""""""""" W,H

After rotation

 0,0
     |"""""""""""""""""""""""""""""""""""""""""""|
     |           ?,?                             |
     |            |""""/\"""""|                  |      
     |            |   /   \   |                  |
     |            |  /      \ |                  |
     |            | /        /|                  |
     |            |/        / |                  |
     |            |\       /  |                  |
     |            |  \    /   |                  |
     |            |    \ /    |                  |
     |             """""""""""  ?,?              |
     |                                           |
     |                                           |
      """"""""""""""""""""""""""""""""""""""""""" W,H

Is there any general equation for finding the co-ordinates of bounding rectangle?.

Thanks....

Haris.

解决方案

Point (x₁, y₁) rotates to (x₁ cos θ - y₁ sin θ, x₁ sin θ + y₁ cos θ), while point (x₂, y₂) rotates to (x₂ cos θ - y₂ sin θ, x₂ sin θ + y₂ cos θ). The other two points can be calculated accordingly.

The coordinates of the bounding reactangle are (x₃, y₃) and (x₄, y₄), where x₃ is the smallest of all new x coordinates, y₃ the smallest of all new y coordinates, x₄ the greatest of all new x coordinates and y₄ the greatest of all new y coordinates.

Which of the corners produces the smallest x (and so on) depends on your angle or rotation. For angles from 0° to 90°, x₃ will come from (x₁, y₁), so x₃ = x₁ cos θ - y₁ sin θ. For angles from 90° to 180°, it will come from (x₂, y₁), and so on. So either you decide which points to use based on your angle of rotation, or you just take the smallest and greatest of all x's and y's.

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But I think you should probably ask this on https://math.stackexchange.com/

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