我的平面不是垂直的，如何将点云的坐标更新为垂直平面

Question

I have a bunch of points lying on a vertical plane. In reality this plane should be exactly vertical. But, when I visualize the point cloud, there is a slight inclination (nearly 2 degrees) from the verticality. At the moment, I can calculate this inclination only. Concerning other errors, I assume there are no shifts or something like that.

So, I want to update coordinates of my point data so that they lie on the vertical plane. I think, I should do some kind of transformation. It may be only via rotation along X-axis. Not sure what it would be.

I guess, you understood my question. Honestly, I am poor at mathematics. So, please let me know how to update my point coordinates to lie on the exact vertical plane.

Note: AS I am implementing this in c++ and there are many programmers who have sound knowledge on these things, I am posting this question under c++.

UPDATES

If I say exactly what I have done so far; I have point cloud data representing a vertical object + its surroundings things. (The data is collected by a moving scanner and may have axes deviations from the correct world axes). The problem is, I cannot say exactly that there is an error on my data or not. Therefore, I checked this with a vertical planar object (which is the dominated object in my data as well). In reality that plane is truly vertical. But, when I fit a plane by removing outliers, then that plane is not truly vertical and has nearly 2 degree inclination. Therefore, I am suspecting that my data has some error. So I want to update all my point clouds (including points on the plane and points which represent other objects) in a way to lay that particular planar points exactly on the vertical plane. Then, I guess, all the points will be updated into their correct positions as in the reality. That is all (x,y,z) coordinates should be updated.

As an example please refer the below figure.

left-represents original point cloud (as you can see, points themselves are not vertical) and back line tells the vertical plane which I fitted and red is the zenith line. as you can see, there is an inclination of the vertical plane. So, I want to update whole my data in the right figure. then, after updating if i fit a plane again (removing outliers), then it is exactly parallel to the zenith line. please help me.

Answer 1

I may be able to help you out, considering I worked with planes recently. First of all, how come the points aren't coplanar from the get go? I'd make the points coplanar in the first place instead of them being at an inclination (from what origin?), and then having to fix them. Also, having the points be coplanar on your first go would increase efficiency.

Sorry if this is the answer you're not looking for, but I need more information before I can help you out. Also, 3D math is hard. If you work with it enough, it starts to get pounded into your head, where you will NEVER forget it, especially if you went through the headaches I had to go through.

Answer 2

I did a bit of thinking on it, and since you want to rotate along the x-axis, your rotation will be done on the xz-plane, which means we can make this a 2D problem. After doing a bit of research on Wikipedia, this may be your solution.

new z = ((x - intended x) * sin(angle)) + (z * cos(angle)) + intended x

What I'm doing here is subtracting our intended x value from our current x value, so that we make (intended x, 0) our point of origin to rotate around. After the point is rotated, I add (intended x, 0) back to our coordinate so that we get the correct result.

Answer 3

Depending on where you got your points from (some kind of measurement, I guess) and what you want to do with them, there are several different things you could do with your data.

The search keyword "regression plane" might help - there are several ways of finding planes approximating point clouds, and several ways to "snap" points to planes.

Edit : You want to apply a rotation around the axis defined by the cross product of the normal vector on your regression plane and the normal of your desired plane, and a point your choice. From your illustration I take it that you probably want the bottom of your vertical planar object to be the point of reference for the rotation.

So you've got your point of reference, you now the axis around which you want to rotate, and the angle. All you need to do is:

1. Translation (to get to your point of reference)
2. Rotation

Answer 4

I read your question again, and hopefully this answer will help you out. If there's anything else I need to know, please tell me.

Now, In order to rotate anything, there must be a center point to rotate around. Now you've already been able to detect the angle of inclination, so now we need a formula for rotating a point a certain angle around an origin. In addition, since this problem only occurs on a 2D plane, we can use this basic formula to readjust the points. For any two axis x and y:

Theta is the angle that you will rotate around in a counter-clockwise direction. x' and y' are your new points. x.origin and y.origin are the coordinates for the point you will be going around. Now I don't know if my math is 100% correct on this but if it's not, hopefully you can change a thing or two and it will work.