使用Numpy我想在坐标系之间转换位置向量。
我有两个三维空间的飞机。
每个平面由其中心定义：C[0] = (X0, Y0, Z0)
C[1] = (X1, Y1, Z1)
(X，Y，Z指全球坐标系)C = np.array([[0,0,0],[-4,2,1]])
以及它的法向量：H[0] = (cos(alpha[0])*sin(A[0]), cos(alpha[0])*cos(A[0]), sin(A[0])
H[1] = (cos(alpha[1])*sin(A[1]), cos(alpha[1])*cos(A[1]), sin(A[1])
alpha=仰角
A=方位角H = np.array([[-0.23, -0.45, 0.86], [-0.12, -0.24, 0.86]])
我有一个点p(xp, yp, 0)位于平面0(xp，yp指的是一个具有中心C[0]的局部坐标系，当alpha = A = 0时，其xyz轴与全局XYZ轴对齐)
我使用以下函数从平面0的局部坐标系转换为全局坐标系：import numpy as np
def rotateAxisX(alpha):
'''
Rotation about x axis
:param alpha: plane altitude angle in degrees
:return: x-axis rotation matrix
'''
rotX = np.array([[1, 0, 0], [0, np.cos(np.deg2rad(alpha)), np.sin(np.deg2rad(alpha))], [0, -np.sin(np.deg2rad(alpha)), np.cos(np.deg2rad(alpha))]])
return rotX
def rotateAxisZ(A):
'''
Rotation about z axis
:param A: plane azimuth angle in degrees
:return: z-axis rotation matrix
'''
rotZ = np.array([[np.cos(np.deg2rad(A)), np.sin(np.deg2rad(A)), 0], [-np.sin(np.deg2rad(A)), np.cos(np.deg2rad(A)), 0], [0, 0, 1]])
return rotZ
def local2Global(positionVector, planeNormalVector, positionVectorLocal):
'''
Convert point from plane's local coordinate system to global coordinate system
:param positionVector: plane center in global coordinates
:param planeNormalVector: the normal vector of the plane
:param positionVectorLocal: a point on plane (xp,yp,0) with respect to the local coordinate system of the plane
:return: the position vector of the point in global coordinates
>>> C = np.array([-10,20,1200])
>>> H = np.array([-0.23, -0.45, 0.86])
>>> p = np.array([-150, -1.5, 0])
>>> P = local2Global(C, H, p)
>>> np.linalg.norm(P-C) == np.linalg.norm(p)
True
'''
alpha = np.rad2deg(np.arcsin(planeNormalVector[2]))
A = np.where(planeNormalVector[1] > 0, np.rad2deg(np.arccos(planeNormalVector[1] / np.cos(np.deg2rad(alpha)))), 360 - np.rad2deg(np.arccos(planeNormalVector[1] / np.cos(np.deg2rad(alpha)))))
positionVectorGlobal = positionVector + np.dot(np.dot(rotateAxisZ(A), rotateAxisX(90 - alpha)), positionVectorLocal)
return positionVectorGlobal
上面的工作似乎和预期的一样。
然后我计算一条线从平面0p(xp,yp,0)上的一个点经过的交点，它的方向向量是S = (0.56, -0.77, 0.3)>>> C = np.array([[0,0,0],[-4,2,1]]) # plane centers
>>> H = np.array([[-0.23, -0.45, 0.86], [-0.12, -0.24, 0.86]]) # plane normal vectors
>>> S = np.array([0.56, -0.77, 0.3]) # a direction vector
>>> p = np.array([-1.5, -1.5, 0]) # a point on a plane
>>> intersectingPlaneIndex = 0 # choose intersecting plane, this plane has the point p on it
>>> intersectedPlaneIndex = 1 # this plane intersects with the line passing from p with direction vector s
>>> P = local2Global(C[intersectingPlaneIndex], H[intersectingPlaneIndex], p) # point p in global coordinates
>>> np.isclose(np.linalg.norm(p), np.linalg.norm(P - C[intersectingPlaneIndex]), 10e-8)
True
所以第一个转变是成功的。
现在让我们在全局坐标系中找到交点E>>> t = np.dot(H[intersectedPlaneIndex], C[intersectedPlaneIndex, :] - P) / np.dot(H[intersectedPlaneIndex], S)
>>> E = P + S * t
>>> np.around(E, 2)
array([ 2.73, -0.67, 1.19])
到目前为止，我找到了位于平面1上的点E(全局坐标)。
问题是：
如何将点E从全局坐标转换为平面1的坐标系并获得e(xe, ye, 0)？
我试过：def global2Local(positionVector, planeNormalVector, positionVectorGlobal):
'''
Convert point from global coordinate system to plane's local coordinate system
:param positionVector: plane center in global coordinates
:param planeNormalVector: the norm快三大小单双稳赚买法ctedPlaneIndex], C[intersectedPlaneIndex, :] - P) / np.dot(H[intersectedPlaneIndex], S)
>>> E = P + S * t
>>> np.around(E, 2)
array([ 2.73, -0.67, 1.19])
到目前为止，我找到了位于平面1上的点E(全局坐标)。
问题是：
如何将点E从全局坐标转换为平面1的坐标系并获得e(xe, ye, 0)？
我试过：def global2Local(positionVector, planeNormalVector, positionVectorGlobal):
'''
Convert point from global coordinate system to plane's local coordinate system
:param positionVector: plane center in global coordinates
:param planeNormalVector: the normal vector of the plane
:param positionVectorGlobal: a point in global coordinates
:note: This function translates the given position vector by the positionVector and rotates the basis axis in order to obtain the positionVectorCoordinates in plane's coordinate system
:warning: it does not function as it should
'''
alpha = np.rad2deg(np.arcsin(planeNormalVector[2]))
A = np.where(planeNormalVector[1] > 0, np.rad2deg(np.arccos(planeNormalVector[1] / np.cos(np.deg2rad(alpha)))), 360 - np.rad2deg(np.arccos(planeNormalVector[1] / np.cos(np.deg2rad(alpha)))))
positionVectorLocal = np.dot(np.dot(np.linalg.inv(rotateAxisZ(A)), np.linalg.inv(rotateAxisX(90 - alpha))), positionVectorGlobal - positionVector) + positionVectorGlobal
return positionVectorLocal
以及：>>> e = global2Local(C[intersectedPlaneIndex], H[intersectedPlaneIndex], E)
>>> e
array([ -2.54839059e+00, -5.48380179e+00, -1.42292121e-03])
首先，只要e[2]接近于零，这看起来是可以的，但是>>> np.linalg.norm(E-C[intersectedPlaneIndex])
7.2440723159783182
>>> np.linalg.norm(e)
6.0470140356703537
所以转换是错误的。有什么想法吗？