将3d平面旋转到xy平面

Question

我想将同一3D平面ax + by + cz = d中的多个点旋转到2D平面xy。我能够旋转飞机，但无法完全旋转。 如您在这张图片中所看到的（红色点是xy平面上的圆，黄色是我想要旋转的圆，红色是我能够旋转的圆。

我试图创建旋转矩阵，计算平面（a，b，c）的法线向量，然后计算this rotation matrix！ ，通过垂直于（a，b，c）和（0,0,1）的向量旋转平面。

def grid2d_perplane(plane): # creates a 2d grid in a plane ax+by+cz=d by calculating a,b,c,d

    p1 = copy.copy(plane[0])
    p2 = copy.copy(plane[13])
    p3 = copy.copy(plane[30])



    # These two vectors are in the plane
    v1 = p3 - p1
    v2 = p2 - p1

    # the cross product is a vector normal to the plane
    cp = np.cross(v1, v2)
    cp /= np.sqrt(cp[0]*cp[0] + cp[1]*cp[1] +cp[2]*cp[2])
    a, b, c = cp


    # This evaluates a * x3 + b * y3 + c * z3 which equals d
    d = np.dot(cp, p3)

    maxx = np.max(plane[:,0:1])
    maxy = np.max(plane[:,1:2])
    minx = np.min(plane[:,0:1])
    miny = np.min(plane[:,1:2])


    x = np.arange(minx*2,maxx*2,0.25)
    y = np.arange(miny*2,maxy*2,0.25)

    xx,yy= np.meshgrid(x,y)

    zz = (d - a * xx - b * yy) / c

    return xx, yy, zz, a , b , c, d

def transfor2d(p,a,b,c):
    "The function will transform a set of points from a 3d plane to  xy plane we have to keep the indexes intact for us to use later"
    #TODO make it only rotation, and try to translate after wards

    norm = np.sqrt(a**2+ b**2 +c **2)
    cos = c /norm
    theta = np.arccos(cos)

    # if cos < 0:
    #     cos = -cos # this is a hack I have to test it again.
    sen = np.sqrt((a**2+ b**2)/(norm**2))
    theta1 = np.arcsin(sen)
    print(theta,theta1)
    u_one = b /norm
    u_two = -a /norm

    u_one_s = u_one**2
    u_two_s = u_two**2


    T11 = cos + (u_one_s*(1-cos))
    T12 = ((u_one)*(u_two))*(1-cos)
    T13 = u_two * sen
    T21 = ((u_one)*(u_two))*(1-cos)
    T22 = cos + (u_two_s*(1-cos))
    T23 = -u_one * sen
    T31 = -u_two * sen
    T32 = u_one * sen
    T33 = cos


    matrix = np.zeros((3,3),np.float32)
    matrix[0][0] = T11
    matrix[0][1] = T12
    matrix[0][2] = T13
    matrix[1][0] = T21
    matrix[1][1] = T22
    matrix[1][2] = T23
    matrix[2][0] = T31
    matrix[2][1] = T32
    matrix[2][2] = T33


    m = matrix.dot(p)
    return np.array([m[0],m[1],m[2]])

预期结果应使黄色点变为红色点。但是，旋转矩阵没有这样做。您能看到我犯的任何错误吗？我认为从理论上讲应该可以。