将 2D 点反投影为 3D [英] Reverse-projection 2D points into 3D

问题描述

假设我们有一个带有任意方程的平面的 3d 空间:ax+by+cz+d=0现在假设我们在该平面上随机选取 3 个点: (x0,y0,z0) (x1,y1,z1) (x1,y1,z1)

Suppose we have a 3d Space with a plane on it with an arbitary equation : ax+by+cz+d=0 now suppose that we pick 3 random points on that plane: (x0,y0,z0) (x1,y1,z1) (x1,y1,z1)

现在我对这架飞机有了不同的观点(相机).我的意思是我有一个不同的相机可以从不同的角度看这架飞机.从那个相机的角度来看，这些点有不同的位置.例如 (x0,y0,z0) 将是 (x0',y0')并且 (x1,y1,z1) 将是 (x1',y1') 并且 (x2,y2,z2) 从新相机的角度来看将是 (x2',y2').

now i have a different point of view(camera) for this plane. i mean i have a different camera that will look at this plane from a different point of view. From that camera point of view these points have different locations. for example (x0,y0,z0) will be (x0',y0') and (x1,y1,z1) will be (x1',y1') and (x2,y2,z2) will be (x2',y2') from the new camera point of view.

所以这是我的一个难点问题！我想从新相机的角度选择一个点，例如 (X,Y) 并告诉它在那个平面上的位置.我只知道 3 个点及其在 3d 空间上的位置以及它们在新相机视图上的投影位置.

So here is my a little hard question! I want to pick a point for example (X,Y) from the new camera point of view and tell where it will be on that plane. All i know is that 3 points and their locations on 3d space and their projection locations on the new camera view.

你知道平面方程的系数和相机位置(连同投影)，还是只有六个点?- 尼尔斯

Do you know the coefficients of the plane-equation and the camera positions (along with the projection), or do you only have the six points? - Nils

我知道前 3 个点的位置.因此我们可以计算平面的系数.所以我们从 (0,0,0) 的角度确切地知道飞机在哪里.然后我们有了只能看到点的相机！因此，相机唯一能看到的就是 3 个点，并且它知道它们在 3d 空间中的位置(当然也知道它们在 2d 相机视图平面上的位置).毕竟我想查看相机视图，选择一个点(例如 (x1,y1))并告诉该点在该平面上的位置.(当然这个 (X,Y,Z) 点应该适合平面方程).我也对相机位置一无所知.

i know the location of first 3 points. therefore we can calculate the coefficients of the plane. so we know exactly where the plane is from (0,0,0) point of view. and then we have the camera that can only see the points! So the only thing that camera sees is 3 points and also it knows their locations in 3d space (and for sure their locations on 2d camera view plane). and after all i want to look at camera view, pick a point (for example (x1,y1)) and tell where is that point on that plane. (for sure this (X,Y,Z) point should fit on the plane equation). Also i know nothing about the camera location.

推荐答案

你是问直线和平面如何相交?请参阅此处http://paulbourke.net/geometry/pointlineplane/

You are asking how to intersect a line and a plane? See here http://paulbourke.net/geometry/pointlineplane/

ps.你的老师知道这个网站！

ps. Your teacher knows this site!

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