从4个点计算旋转矩阵

Question

I would like to stick a moving car to a curvy terrarian. I can calculate the y coordinate (which is height in my case) for each wheel. These 4 points forms a plane. I don't know how to calculate the rotation matrix from these 4 pints so I can apply it to the car. So this is what I would like to achieve:

calculateRotationMatrix(p1,p2,p3,p4);

BTW I am using c++ and openGL.

Could anybody help me out here?

Answer 1

If you guarantee that all 4 points lie on one plane, then the problem is not that hard to solve: Let's call the points (A,B,C,D) and we define a up vector (UP = [0,1,0])

1) Calculate the plane normal (N)

N = normalize(cross(B-A, C-A));

2) Calculate the rotation axis (R)

R = normalize(cross(N,UP))

3) Calculate rotation angle (alpha)

alpha = dot(N, UP)

The resulting matrix is then the one that rotates around R by an angle of alpha. If your matrix library does not support creating rotation axis around arbitrary axis, you can find the form here .

Note, that there is a singularity when alpha is very small (N will then vanish), so you should only calculate the matrix if alpha is sufficiently large. It might also be that case that some of the vectors point to the opposite direction depending on the winding order in which the points are defined. In this case just switch the two parameters of the cross function.