将3d平面上的点转换为2d坐标 [英] Convert points on a 3d Plane to 2d Coordinates

问题描述

我有一个点云，所有点都位于3D空间中的平面上.我需要将每个点转换为2D坐标，反之亦然.

I have a point cloud, all the points lie on a plane in 3D Space. I need to convert each point to 2D Coordinates and vice versa.

(x，y，z)在坐标系 A =>变换矩阵( T1 )=>(x，y)在坐标系 B

(x,y,z) in Coordinate System A => Transformation Matrix (T1) => (x,y) in Coordinate System B

(x，y)=>坐标系 A

(x,y) in Coordinate System B => Transformation Matrix (T2) => (x,y,z) in Coordinate System A

我需要 T1 和 T2 . 坐标系 B 可以是任意参考系.

I need T1 and T2. The coordinate system B can be any arbitrary reference frame.

推荐答案

据我了解，所有点都位于同一平面上，因此您希望减小尺寸并稍后恢复坐标.

As far as I understand, all points lie in the same plane, and you want to reduce dimension and later restore coordinates.

获得三个非共线的点A，B，C.制作矢量AB和AC.
垂直于该平面的是

Get three non-collinear points A, B, C. Make vectors AB and AC.
Normal to that plane is

 N = AB x AC //cross product

现在将向量AB和N归一化，得到单位U = uAB和uN.构建第二个基本矢量(它是单位，位于平面中)

Now normalize vectors AB and N getting unit U = uAB and uN. Build the second base vector (it is unit and lies in the plane)

 V = U x uN

现在您有四个基点A, u=A+U, v=A+V, n=A+uN

变形应将这些点相应地映射到四元组(0,0,0), (1,0,0), (0,1,0), (0,0,1)中.

Tranformation should map these points into quadruplet (0,0,0), (1,0,0), (0,1,0), (0,0,1) correspondingly.

现在使用仿射变换矩阵进行此映射:

Now about affine transformation matrix to make this mapping:

      [Ax ux vx nx]   [0 1 0 0]
 M *  [Ay uy vy ny] = [0 0 1 0]
      [Az uz vz nz]   [0 0 0 1]
      [1  1  1  1 ]   [1 1 1 1]

或

  M * S = D
  M * S * Sinv = D * Sinv
  M = D * Sinv

因此计算S=[Ax ux...]的逆矩阵并获得所需的矩阵M.

So calculate inverse matrix for S=[Ax ux...] and get needed matrix M.

将M应用于平面中的任何点将给出z分量为零的新坐标.

Application of M to any point in the plane gives new coordinates with zero z-component.

将M的逆应用于(x，y，0)会在给定平面中获得3D坐标.

Application of inverse of M to (x,y,0) results 3D coordinates in given plane.

槭树，其点为A=1,1,1 B=2,1,1 C=1,1,2 (in plane Y=1)

新坐标AA，BB，CC的z分量为零.

new coordinates AA, BB, CC have zero z-component.

对于映射后同一平面z分量中的任意点，也为零.

For arbitrary point in the same plane z-component after mapping is zero too.

 P:=vector([-2,1,7,1]);
 > PP := multiply(M, P);
 PP := [-3, 6, 0, 1]

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