3D投影点，2D平面 [英] Projecting 3D points to 2D plane

问题描述

设A是一个点，我有3D坐标x，y，z和我想将它们转换成二维坐标：X，Y。投影应是正交于由给定的正常限定的平面。在平凡的情况下，如果正常的实际上是其中一个轴，很容易解决，只要消除了协调，但如何对其他案件，这是更可能发生？

Let A be a point for which I have the 3D coordinates x, y, z and I want to transform them into 2D coordinates: x, y. The projection shall be orthogonal on a plane defined by a given normal. The trivial case, where the normal is actually one of the axes, it's easy to solve, simply eliminating a coordinate, but how about the other cases, which are more likely to happen?

推荐答案

如果你有你的目标点的 P 的坐标 R_P =（X，Y，Z），并与正常的飞机 N =（NX，NY，NZ）您需要定义一个原点上了飞机，并为两个正交方向 X 和是。例如，如果你原点在 r_O =（牛，OY，盎司）和你的两个坐标轴的平面由 E_1 =定义（ex_1 ，ey_1，ez_1）， E_2 =（ex_2，ey_2，ez_2）然后正交有一个点（N，E_1 ）= 0 ，点（N，E_2）= 0 ，点（E_1，E_2）= 0 （向量积）。

If you have your target point P with coordinates r_P = (x,y,z) and a plane with normal n=(nx,ny,nz) you need to define an origin on the plane, as well as two orthogonal directions for x and y. For example if you origin is at r_O = (ox, oy, oz) and your two coordinate axis in the plane are defined by e_1 = (ex_1,ey_1,ez_1), e_2 = (ex_2,ey_2,ez_2) then orthogonality has that Dot(n,e_1)=0, Dot(n,e_2)=0, Dot(e_1,e_2)=0 (vector dot product).

您目标点的 P 的必须服从方程

Your target point P must obey the equation

r_P = r_O + t_1*e_1 + t_2*e_2 + s*n

其中， T_1 和 T_2 是你的二维坐标沿着 E_1 和 E_2 和取值的面和点之间的正常间隔（距离）。

where t_1 and t_2 are your 2D coordinates along e_1 and e_2 and s the normal separation (distance) between the plane and the point.

有标量是通过推算发现

s = Dot(n, r_P-r_O)
t_1 = Dot(e_1, r_P-r_O)    
t_2 = Dot(e_2, r_P-r_O)

例有平面原点 r_O =（-1,3,1）正常

n = r_O/|r_O| = (-1/√11, 3/√11, 1/√11)

您必须选择垂直方向的二维坐标，例如：

You have to pick orthogonal directions for the 2D coordinates, for example

e_1 = (1/√2, 0 ,1/√2)
e_2 = (-3/√22, -2/√22, 3/√22)

，使得点（N，E_1）= 0 和点（N，E_2）= 0 和点（E_1，E_2）= 0 。

二维的点的坐标的 P 的 R_P =（1,7，-3）是

t_1 = Dot(e_1, r_P-r_O) = ( 1/√2,0,1/√2)·( (1,7,-3)-(-1,3,1) ) =  -√2
t_2 = Dot(e_2, r_P-r_O) = (-3/√22, -2/√22, 3/√22)·( (1,7,-3)-(-1,3,1) ) = -26/√22

和出平面分离的

and the out of plane separation

s = Dot(n, r_P-r_O) = 6/√11

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