Python numpy 模块，cross() 实例源码
我们从Python开源项目中，提取了以下50个代码示例，用于说明如何使用numpy.cross()。
def vector_product(v0, v1, axis=0):
"""Return vector perpendicular to vectors.
>>> v = vector_product([2, 0, 0], [0, 3, 0])
>>> numpy.allclose(v, [0, 0, 6])
True
>>> v0 = [[2, 0, 0, 2], [0, 2, 0, 2], [0, 0, 2, 2]]
>>> v1 = [[3], [0], [0]]
>>> v = vector_product(v0, v1)
>>> numpy.allclose(v, [[0, 0, 0, 0], [0, 0, 6, 6], [0, -6, 0, -6]])
True
>>> v0 = [[2, 0, 0], [2, 0, 0], [0, 2, 0], [2, 0, 0]]
>>> v1 = [[0, 3, 0], [0, 0, 3], [0, 0, 3], [3, 3, 3]]
>>> v = vector_product(v0, v1, axis=1)
>>> numpy.allclose(v, [[0, 0, 6], [0, -6, 0], [6, 0, 0], [0, -6, 6]])
True
"""
return numpy.cross(v0, v1, axis=axis)
def skew(v, return_dv=False):
"""
Returns the skew-symmetric matrix of a vector
Ref: https://github.com/dreamdragon/Solve3Plus1/blob/master/skew3.m
Also known as the cross-product matrix [v]_x such that
the cross product of (v x w) is equivalent to the
matrix multiplication of the cross product matrix of
v ([v]_x) and w
In other words: v x w = [v]_x * w
"""
sk = np.float32([[0, -v[2], v[1]],
[v[2], 0, -v[0]],
[-v[1], v[0], 0]])
if return_dv:
dV = np.float32([[0, 0, 0],
[0, 0, -1],
[0, 1, 0],
[0, 0, 1],
[0, 0, 0],
[-1, 0, 0],
[0, -1, 0],
[1, 0, 0],
[0, 0, 0]])
return sk, dV
else:
return sk
def points_and_normals(self):
"""
Returns the point/normals parametrization for planes,
including clipped zmin and zmax frustums
Note: points need to be in CCW
"""
nv1, fv1 = self._front_back_vertices
nv2 = np.roll(nv1, -1, axis=0)
fv2 = np.roll(fv1, -1, axis=0)
vx = np.vstack([fv1-nv1, nv2[0]-nv1[0], fv1[2]-fv1[1]])
vy = np.vstack([fv2-fv1, nv2[1]-nv2[0], fv1[1]-fv1[0]])
pts = np.vstack([fv1, nv1[0], fv1[1]])
# vx += 1e-12
# vy += 1e-12
vx /= np.linalg.norm(vx, axis=1).reshape(-1,1)
vy /= np.linalg.norm(vy, axis=1).reshape(-1,1)
normals = np.cross(vx, vy)
normals /= np.linalg.norm(normals, axis=1).reshape(-1,1)
return pts, normals
def faceNormals(self, indexed=None):
"""
Return an array (Nf, 3) of normal vectors for each face.
If indexed='faces', then instead return an indexed array
(Nf, 3, 3) (this is just the same array with each vector
copied three times).
"""
if self._faceNormals is None:
v = self.vertexes(indexed='faces')
self._faceNormals = np.cross(v[:,1]-v[:,0], v[:,2]-v[:,0])
if indexed is None:
return self._faceNormals
elif indexed == 'faces':
if self._faceNormalsIndexedByFaces is None:
norms = np.empty((self._faceNormals.shape[0], 3, 3))
norms[:] = self._faceNormals[:,np.newaxis,:]
self._faceNormalsIndexedByFaces = norms
return self._faceNormalsIndexedByFaces
else:
raise Exception("Invalid indexing mode. Accepts: None, 'faces'")
def faceNormals(self, indexed=None):
"""
Return an array (Nf, 3) of normal vectors for each face.
If indexed='faces', then instead return an indexed array
(Nf, 3, 3) (this is just the same array with each vector
copied three times).
"""
if self._faceNormals is None:
v = self.vertexes(indexed='faces')
self._faceNormals = np.cross(v[:,1]-v[:,0], v[:,2]-v[:,0])
if indexed is None:
return self._faceNormals
elif indexed == 'faces':
if self._faceNormalsIndexedByFaces is None:
norms = np.empty((self._faceNormals.shape[0], 3, 3))
norms[:] = self._faceNormals[:,np.newaxis,:]
self._faceNormalsIndexedByFaces = norms
return self._faceNormalsIndexedByFaces
else:
raise Exception("Invalid indexing mode. Accepts: None, 'faces'")
def vector_product(v0, v1, axis=0):
"""Return vector perpendicular to vectors.
>>> v = vector_product([2, 0, 0], [0, 3, 0])
>>> numpy.allclose(v, [0, 0, 6])
True
>>> v0 = [[2, 0, 0, 2], [0, 2, 0, 2], [0, 0, 2, 2]]
>>> v1 = [[3], [0], [0]]
>>> v = vector_product(v0, v1)
>>> numpy.allclose(v, [[0, 0, 0, 0], [0, 0, 6, 6], [0, -6, 0, -6]])
True
>>> v0 = [[2, 0, 0], [2, 0, 0], [0, 2, 0], [2, 0, 0]]
>>> v1 = [[0, 3, 0], [0, 0, 3], [0, 0, 3], [3, 3, 3]]
>>> v = vector_product(v0, v1, axis=1)
>>> numpy.allclose(v, [[0, 0, 6], [0, -6, 0], [6, 0, 0], [0, -6, 6]])
True
"""
return numpy.cross(v0, v1, axis=axis)
def poly_area(poly):
if len(poly) < 3: # not a plane - no area
return 0
total = [0, 0, 0]
N = len(poly)
for i in range(N):
vi1 = poly[i]
vi2 = poly[(i+1) % N]
prod = np.cross(vi1, vi2)
total[0] += prod[0]
total[1] += prod[1]
total[2] += prod[2]
result = np.dot(total, unit_normal(poly[0], poly[1], poly[2]))
return abs(result/2)
#unit normal vector of plane defined by points a, b, and c
def __init__(self, plane_fit, gridsize):
plane = numpy.array(plane_fit)
origin = -plane / numpy.dot(plane, plane)
n = numpy.array([plane[1], plane[2], plane[0]])
u = numpy.cross(plane, n)
v = numpy.cross(plane, u)
u /= numpy.linalg.norm(u)
v /= numpy.linalg.norm(v)
def project_point(point):
return origin + point[0]*u + point[1]*v
vertexes = []
for x in range(-gridsize+1, gridsize):
for y in range(-gridsize+1, gridsize):
vertexes += [project_point((x-1, y-1)),
project_point((x, y-1)),
project_point((x, y)),
project_point((x-1, y))]
super(self, Plane).__init__(vertexes)
def analytic_infinite_wire(obsloc,wireloc,orientation,I=1.):
"""
Compute the response of an infinite wire with orientation 'orientation'
and current I at the obsvervation locations obsloc
Output:
B: magnetic field [Bx,By,Bz]
"""
n,d = obsloc.shape
t,d = wireloc.shape
d = np.sqrt(np.dot(obsloc**2.,np.ones([d,t]))+np.dot(np.ones([n,d]),(wireloc.T)**2.)
- 2.*np.dot(obsloc,wireloc.T))
distr = np.amin(d, axis=1, keepdims = True)
idxmind = d.argmin(axis=1)
r = obsloc - wireloc[idxmind]
orient = np.c_[[orientation for i in range(obsloc.shape[0])]]
B = (mu_0*I)/(2*np.pi*(distr**2.))*np.cross(orientation,r)
return B
def winding_angle(self, path, point):
wa = 0
for i in range(len(path)-1):
p = np.array([path[i].x, path[i].y])
pn = np.array([path[i+1].x, path[i+1].y])
vp = p - point
vpn = pn - point
vp_norm = sqrt(vp[0]**2 + vp[1]**2)
vpn_norm = sqrt(vpn[0]**2 + vpn[1]**2)
assert (vp_norm > 0)
assert (vpn_norm > 0)
z = np.cross(vp, vpn)/(vp_norm * vpn_norm)
z = min(max(z, -1.0), 1.0)
wa += asin(z)
return wa
def rotate_ascii_stl(self, rotation_matrix, content, filename):
"""Rotate the mesh array and save as ASCII STL."""
mesh = np.array(content, dtype=np.float64)
# prefix area vector, if not already done (e.g. in STL format)
if len(mesh[0]) == 3:
row_number = int(len(content)/3)
mesh = mesh.reshape(row_number, 3, 3)
# upgrade numpy with: "pip install numpy --upgrade"
rotated_content = np.matmul(mesh, rotation_matrix)
v0 = rotated_content[:, 0, :]
v1 = rotated_content[:, 1, :]
v2 = rotated_content[:, 2, :]
normals = np.cross(np.subtract(v1, v0), np.subtract(v2, v0)) \
.reshape(int(len(rotated_content)), 1, 3)
rotated_content = np.hstack((normals, rotated_content))
tweaked = list("solid %s" % filename)
tweaked += list(map(self.write_facett, list(rotated_content)))
tweaked.append("\nendsolid %s\n" % filename)
tweaked = "".join(tweaked)
return tweaked
def line_cross_point(line1, line2):
# line1 0= ax+by+c, compute the cross point of line1 and line2
if line1[0] != 0 and line1[0] == line2[0]:
print('Cross point does not exist')
return None
if line1[0] == 0 and line2[0] == 0:
print('Cross point does not exist')
return None
if line1[1] == 0:
x = -line1[2]
y = line2[0] * x + line2[2]
elif line2[1] == 0:
x = -line2[2]
y = line1[0] * x + line1[2]
else:
k1, _, b1 = line1
k2, _, b2 = line2
x = -(b1-b2)/(k1-k2)
y = k1*x + b1
return np.array([x, y], dtype=np.float32)
def test_broadcasting_shapes(self):
u = np.ones((2, 1, 3))
v = np.ones((5, 3))
assert_equal(np.cross(u, v).shape, (2, 5, 3))
u = np.ones((10, 3, 5))
v = np.ones((2, 5))
assert_equal(np.cross(u, v, axisa=1, axisb=0).shape, (10, 5, 3))
assert_raises(ValueError, np.cross, u, v, axisa=1, axisb=2)
assert_raises(ValueError, np.cross, u, v, axisa=3, axisb=0)
u = np.ones((10, 3, 5, 7))
v = np.ones((5, 7, 2))
assert_equal(np.cross(u, v, axisa=1, axisc=2).shape, (10, 5, 3, 7))
assert_raises(ValueError, np.cross, u, v, axisa=-5, axisb=2)
assert_raises(ValueError, np.cross, u, v, axisa=1, axisb=-4)
# gh-5885
u = np.ones((3, 4, 2))
for axisc in range(-2, 2):
assert_equal(np.cross(u, u, axisc=axisc).shape, (3, 4))
def vector_product(v0, v1, axis=0):
"""Return vector perpendicular to vectors.
>>> v = vector_product([2, 0, 0], [0, 3, 0])
>>> numpy.allclose(v, [0, 0, 6])
True
>>> v0 = [[2, 0, 0, 2], [0, 2, 0, 2], [0, 0, 2, 2]]
>>> v1 = [[3], [0], [0]]
>>> v = vector_product(v0, v1)
>>> numpy.allclose(v, [[0, 0, 0, 0], [0, 0, 6, 6], [0, -6, 0, -6]])
True
>>> v0 = [[2, 0, 0], [2, 0, 0], [0, 2, 0], [2, 0, 0]]
>>> v1 = [[0, 3, 0], [0, 0, 3], [0, 0, 3], [3, 3, 3]]
>>> v = vector_product(v0, v1, axis=1)
>>> numpy.allclose(v, [[0, 0, 6], [0, -6, 0], [6, 0, 0], [0, -6, 6]])
True
"""
return numpy.cross(v0, v1, axis=axis)
def intersect(self, segment):
""" point_sur_segment return
p: point d'intersection
u: param t de l'intersection sur le segment courant
v: param t de l'intersection sur le segment segment
"""
v2d = self.vect_2d
c2 = np.cross(segment.vect_2d, (0, 0, 1))
d = np.dot(v2d, c2)
if d == 0:
# segments paralleles
segment._point_sur_segment(self.c0)
segment._point_sur_segment(self.c1)
self._point_sur_segment(segment.c0)
self._point_sur_segment(segment.c1)
return False, 0, 0, 0
c1 = np.cross(v2d, (0, 0, 1))
v3 = self.c0.vect(segment.c0)
v3[2] = 0.0
u = np.dot(c2, v3) / d
v = np.dot(c1, v3) / d
co = self.lerp(u)
return True, co, u, v
def __init__(self, ROI):
bounds = [(ROI[0], ROI[2], 0),
(ROI[1], ROI[2], 0),
(ROI[0], ROI[3], 0),
(ROI[1], ROI[3], 0)]
self.wgs84 = nv.FrameE(name='WGS84')
lon, lat, hei = np.array(bounds).T
geo_points = self.wgs84.GeoPoint(longitude=lon, latitude=lat, z=-hei,
degrees=True)
P = geo_points.to_ecef_vector().pvector.T
dx = normed(P[1] - P[0])
dy = P[2] - P[0]
dy -= dx * dy.dot(dx)
dy = normed(dy)
dz = np.cross(dx, dy)
self.rotation = np.array([dx, dy, dz]).T
self.mu = np.mean(P.dot(self.rotation), axis=0)[np.newaxis, :]
def _signed_volume_of_tri(self, tri):
"""Return the signed volume of the given triangle.
Parameters
----------
tri : :obj:`numpy.ndarray` of int
The triangle for which we wish to compute a signed volume.
Returns
-------
float
The signed volume associated with the triangle.
"""
v1 = self.vertices_[tri[0], :]
v2 = self.vertices_[tri[1], :]
v3 = self.vertices_[tri[2], :]
volume = (1.0 / 6.0) * (v1.dot(np.cross(v2, v3)))
return volume
def _area_of_tri(self, tri):
"""Return the area of the given triangle.
Parameters
----------
tri : :obj:`numpy.ndarray` of int
The triangle for which we wish to compute an area.
Returns
-------
float
The area of the triangle.
"""
verts = [self.vertices[i] for i in tri]
ab = verts[1] - verts[0]
ac = verts[2] - verts[0]
return 0.5 * np.linalg.norm(np.cross(ab, ac))
def vector_product(v0, v1, axis=0):
"""Return vector perpendicular to vectors.
>>> v = vector_product([2, 0, 0], [0, 3, 0])
>>> numpy.allclose(v, [0, 0, 6])
True
>>> v0 = [[2, 0, 0, 2], [0, 2, 0, 2], [0, 0, 2, 2]]
>>> v1 = [[3], [0], [0]]
>>> v = vector_product(v0, v1)
>>> numpy.allclose(v, [[0, 0, 0, 0], [0, 0, 6, 6], [0, -6, 0, -6]])
True
>>> v0 = [[2, 0, 0], [2, 0, 0], [0, 2, 0], [2, 0, 0]]
>>> v1 = [[0, 3, 0], [0, 0, 3], [0, 0, 3], [3, 3, 3]]
>>> v = vector_product(v0, v1, axis=1)
>>> numpy.allclose(v, [[0, 0, 6], [0, -6, 0], [6, 0, 0], [0, -6, 6]])
True
"""
return numpy.cross(v0, v1, axis=axis)
def calc_rotation_matrix(q1, q2, ref_q1, ref_q2):
ref_nv = np.cross(ref_q1, ref_q2)
q_nv = np.cross(q1, q2)
if min(norm(ref_nv), norm(q_nv)) == 0.: # avoid 0 degree including angle
return np.identity(3)
axis = np.cross(ref_nv, q_nv)
angle = rad2deg(acos(ref_nv.dot(q_nv) / (norm(ref_nv) * norm(q_nv))))
R1 = axis_angle_to_rotation_matrix(axis, angle)
rot_ref_q1, rot_ref_q2 = R1.dot(ref_q1), R1.dot(ref_q2) # rotate ref_q1,2 plane to q1,2 plane
cos1 = max(min(q1.dot(rot_ref_q1) / (norm(rot_ref_q1) * norm(q1)), 1.), -1.) # avoid math domain error
cos2 = max(min(q2.dot(rot_ref_q2) / (norm(rot_ref_q2) * norm(q2)), 1.), -1.)
angle1 = rad2deg(acos(cos1))
angle2 = rad2deg(acos(cos2))
angle = (angle1 + angle2) / 2.
axis = np.cross(rot_ref_q1, q1)
R2 = axis_angle_to_rotation_matrix(axis, angle)
R = R2.dot(R1)
return R
def __init__(self,origin, pt1, pt2, name=None):
"""
origin: 3x1 vector
pt1: 3x1 vector
pt2: 3x1 vector
"""
self.__origin=origin
vec1 = np.array([pt1[0] - origin[0] , pt1[1] - origin[1] , pt1[2] - origin[2]])
vec2 = np.array([pt2[0] - origin[0] , pt2[1] - origin[1] , pt2[2] - origin[2]])
cos = np.dot(vec1, vec2)/np.linalg.norm(vec1)/np.linalg.norm(vec2)
if cos == 1 or cos == -1:
raise Exception("Three points should not in a line!!")
self.__x = vec1/np.linalg.norm(vec1)
z = np.cross(vec1, vec2)
self.__z = z/np.linalg.norm(z)
self.__y = np.cross(self.z, self.x)
self.__name=uuid.uuid1() if name==None else name
def set_by_3pts(self,origin, pt1, pt2):
"""
origin: tuple 3
pt1: tuple 3
pt2: tuple 3
"""
self.origin=origin
vec1 = np.array([pt1[0] - origin[0] , pt1[1] - origin[1] , pt1[2] - origin[2]])
vec2 = np.array([pt2[0] - origin[0] , pt2[1] - origin[1] , pt2[2] - origin[2]])
cos = np.dot(vec1, vec2)/np.linalg.norm(vec1)/np.linalg.norm(vec2)
if cos == 1 or cos == -1:
raise Exception("Three points should not in a line!!")
self.x = vec1/np.linalg.norm(vec1)
z = np.cross(vec1, vec2)
self.z = z/np.linalg.norm(z)
self.y = np.cross(self.z, self.x)
def normalByCross(vec1,vec2):
r"""Returns normalised normal vectors of plane spanned by two vectors.
Normal vector is computed by:
.. math:: \mathbf{n} = \frac{\mathbf{v_1} \times \mathbf{v_2}}{|\mathbf{v_1} \times \mathbf{v_2}|}
.. note:: Will return zero vector if ``vec1`` and ``vec2`` are colinear.
Args:
vec1 (numpy.ndarray): Vector 1.
vec2 (numpy.ndarray): Vector 2.
Returns:
numpy.ndarray: Normal vector.
"""
if checkColinear(vec1,vec2):
printWarning("Can't compute normal of vectors, they seem to be colinear. Returning zero.")
return np.zeros(np.shape(vec1))
return np.cross(vec1,vec2)/np.linalg.norm(np.cross(vec1,vec2))
def calc_e0(self):
"""
Compute the reference axis for adding dummy atoms.
Only used in the case of linear molecules.
We first find the Cartesian axis that is "most perpendicular" to the molecular axis.
Next we take the cross product with the molecular axis to create a perpendicular vector.
Finally, this perpendicular vector is normalized to make a unit vector.
"""
ysel = self.x0[self.a, :]
vy = ysel[-1]-ysel[0]
ev = vy / np.linalg.norm(vy)
# Cartesian axes.
ex = np.array([1.0,0.0,0.0])
ey = np.array([0.0,1.0,0.0])
ez = np.array([0.0,0.0,1.0])
self.e0 = np.cross(vy, [ex, ey, ez][np.argmin([np.dot(i, ev)**2 for i in [ex, ey, ez]])])
self.e0 /= np.linalg.norm(self.e0)
def normal_vector(self, xyz):
xyz = xyz.reshape(-1,3)
a = np.array(self.a)
b = self.b
c = np.array(self.c)
xyza = np.mean(xyz[a], axis=0)
xyzc = np.mean(xyz[c], axis=0)
# vector from first atom to central atom
vector1 = xyza - xyz[b]
# vector from last atom to central atom
vector2 = xyzc - xyz[b]
# norm of the two vectors
norm1 = np.sqrt(np.sum(vector1**2))
norm2 = np.sqrt(np.sum(vector2**2))
crs = np.cross(vector1, vector2)
crs /= np.linalg.norm(crs)
return crs
def value(self, xyz):
xyz = xyz.reshape(-1,3)
a = np.array(self.a)
b = self.b
c = self.c
d = np.array(self.d)
xyza = np.mean(xyz[a], axis=0)
xyzd = np.mean(xyz[d], axis=0)
vec1 = xyz[b] - xyza
vec2 = xyz[c] - xyz[b]
vec3 = xyzd - xyz[c]
cross1 = np.cross(vec2, vec3)
cross2 = np.cross(vec1, vec2)
arg1 = np.sum(np.multiply(vec1, cross1)) * \
np.sqrt(np.sum(vec2**2))
arg2 = np.sum(np.multiply(cross1, cross2))
answer = np.arctan2(arg1, arg2)
return answer
def value(self, xyz):
xyz = xyz.reshape(-1,3)
a = self.a
b = self.b
c = self.c
d = self.d
vec1 = xyz[b] - xyz[a]
vec2 = xyz[c] - xyz[b]
vec3 = xyz[d] - xyz[c]
cross1 = np.cross(vec2, vec3)
cross2 = np.cross(vec1, vec2)
arg1 = np.sum(np.multiply(vec1, cross1)) * \
np.sqrt(np.sum(vec2**2))
arg2 = np.sum(np.multiply(cross1, cross2))
answer = np.arctan2(arg1, arg2)
return answer
def getProjectedAngleInXYPlane(self, z=0, ref_axis=[0,1], centre=[0,0], inDeg=True):
'''
Project the OA vector to z=z, calculate the XY position, construct a
2D vector from [centre] to this XY and measure the angle subtended by
this vector from [ref_axis] (clockwise).
'''
ref_axis = np.array(ref_axis)
centre = np.array(centre)
point_vector_from_fit_centre = np.array(self.getXY(z=z)) - centre
dotP = np.dot(ref_axis, point_vector_from_fit_centre)
crossP = np.cross(ref_axis, point_vector_from_fit_centre)
angle = np.arccos(dotP/(np.linalg.norm(ref_axis)*np.linalg.norm(point_vector_from_fit_centre)))
if np.sign(crossP) > 0:
angle = (np.pi-angle) + np.pi
if inDeg:
dir_v = self._eval_direction_vector()
return np.degrees(angle)
else:
return angle
def compute_normals(self):
"""Compute vertex and face normals of the triangular mesh."""
# Compute face normals, easy as cake.
for fi, face in enumerate(self.faces):
self.face_normals[fi] = np.cross(self.vertices[face[2]] - self.vertices[face[0]],
self.vertices[face[1]] - self.vertices[face[0]])
# Next, compute the vertex normals.
for fi, face in enumerate(self.faces):
self.vertex_normals[face[0]] += self.face_normals[fi]
self.vertex_normals[face[1]] += self.face_normals[fi]
self.vertex_normals[face[2]] += self.face_normals[fi]
# Normalize all vectors
for i, f_norm in enumerate(self.face_normals):
self.face_normals[i] = normalize(f_norm)
for i, v_norm in enumerate(self.vertex_normals):
self.vertex_normals[i] = normalize(v_norm)
def rotate_coord_sys(old_u, old_v, new_norm):
"""Rotate a coordinate system to be perpendicular to the given normal."""
new_u = old_u
new_v = old_v
old_norm = np.cross(old_u, old_v)
# Project old normal onto new normal
ndot = np.dot(old_norm, new_norm)
# If projection is leq to -1, simply reverse
if ndot <= -1:
new_u = -new_u
new_v = -new_v
return new_u, new_v
# Otherwise, compute new normal
perp_old = new_norm - ndot * old_norm
dperp = (old_norm + new_norm) / (1 + ndot)
new_u -= dperp * np.dot(new_u, perp_old)
new_v -= dperp * np.dot(new_v, perp_old)
return new_u, new_v
def tensor_spherical_to_cartesian(theta, phi, psi):
"""Calculate the eigenvectors for a Tensor given the three angles.
This will return the eigenvectors unsorted, since this function knows nothing about the eigenvalues. The caller
of this function will have to sort them by eigenvalue if necessary.
Args:
theta (ndarray): matrix of list of theta's
phi (ndarray): matrix of list of phi's
psi (ndarray): matrix of list of psi's
Returns:
tuple: The three eigenvector for every voxel given. The return matrix for every eigenvector is of the given
shape + [3].
"""
v0 = spherical_to_cartesian(theta, phi)
v1 = rotate_orthogonal_vector(v0, spherical_to_cartesian(theta + np.pi / 2.0, phi), psi)
v2 = np.cross(v0, v1)
return v0, v1, v2
def rotate_vector(basis, to_rotate, psi):
"""Uses Rodrigues' rotation formula to rotate the given vector v by psi around k.
If a matrix is given the operation will by applied on the last dimension.
Args:
basis: the unit vector defining the rotation axis (k)
to_rotate: the vector to rotate by the angle psi (v)
psi: the rotation angle (psi)
Returns:
vector: the rotated vector
"""
cross_product = np.cross(basis, to_rotate)
dot_product = np.sum(np.multiply(basis, to_rotate), axis=-1)[..., None]
cos_psi = np.cos(psi)[..., None]
sin_psi = np.sin(psi)[..., None]
return to_rotate * cos_psi + cross_product * sin_psi + basis * dot_product * (1 - cos_psi)
def rotate_orthogonal_vector(basis, to_rotate, psi):
"""Uses Rodrigues' rotation formula to rotate the given vector v by psi around k.
If a matrix is given the operation will by applied on the last dimension.
This function assumes that the given two vectors (or matrix of vectors) are orthogonal for every voxel.
This assumption allows for some speedup in the rotation calculation.
Args:
basis: the unit vector defining the rotation axis (k)
to_rotate: the vector to rotate by the angle psi (v)
psi: the rotation angle (psi)
Returns:
vector: the rotated vector
"""
cross_product = np.cross(basis, to_rotate)
cos_psi = np.cos(psi)[..., None]
sin_psi = np.sin(psi)[..., None]
return to_rotate * cos_psi + cross_product * sin_psi
def signed_angle(v1, v2, look):
'''
Compute the signed angle between two vectors.
Returns a number between -180 and 180. A positive number indicates a
clockwise sweep from v1 to v2. A negative number is counterclockwise.
'''
# The sign of (A x B) dot look gives the sign of the angle.
# > 0 means clockwise, < 0 is counterclockwise.
sign = np.sign(np.cross(v1, v2).dot(look))
# 0 means collinear: 0 or 180. Let's call that clockwise.
if sign == 0:
sign = 1
return sign * angle(v1, v2, look)
def rotation_from_up_and_look(up, look):
'''
Rotation matrix to rotate a mesh into a canonical reference frame. The
result is a rotation matrix that will make up along +y and look along +z
(i.e. facing towards a default opengl camera).
Note that if you're reorienting a mesh, you can use its `reorient` method
to accomplish this.
up: The foot-to-head direction.
look: The direction the eyes are facing, or the heel-to-toe direction.
'''
up, look = np.array(up, dtype=np.float64), np.array(look, dtype=np.float64)
if np.linalg.norm(up) == 0:
raise ValueError("Singular up")
if np.linalg.norm(look) == 0:
raise ValueError("Singular look")
y = up / np.linalg.norm(up)
z = look - np.dot(look, y)*y
if np.linalg.norm(z) == 0:
raise ValueError("up and look are colinear")
z = z / np.linalg.norm(z)
x = np.cross(y, z)
return np.array([x, y, z])
def from_points(cls, p1, p2, p3):
'''
If the points are oriented in a counterclockwise direction, the plane's
normal extends towards you.
'''
from blmath.numerics import as_numeric_array
p1 = as_numeric_array(p1, shape=(3,))
p2 = as_numeric_array(p2, shape=(3,))
p3 = as_numeric_array(p3, shape=(3,))
v1 = p2 - p1
v2 = p3 - p1
normal = np.cross(v1, v2)
return cls(point_on_plane=p1, unit_normal=normal)
def from_points_and_vector(cls, p1, p2, vector):
'''
Compute a plane which contains two given points and the given
vector. Its reference point will be p1.
For example, to find the vertical plane that passes through
two landmarks:
from_points_and_normal(p1, p2, vector)
Another way to think about this: identify the plane to which
your result plane should be perpendicular, and specify vector
as its normal vector.
'''
from blmath.numerics import as_numeric_array
p1 = as_numeric_array(p1, shape=(3,))
p2 = as_numeric_array(p2, shape=(3,))
v1 = p2 - p1
v2 = as_numeric_array(vector, shape=(3,))
normal = np.cross(v1, v2)
return cls(point_on_plane=p1, unit_normal=normal)
def rpy(self):
acc = self.acceleration()
yaw = self.yaw()
norm = np.linalg.norm(acc)
# print(acc)
if norm < 1e-6:
return (0.0, 0.0, yaw)
else:
thrust = acc + np.array([0, 0, 9.81])
z_body = thrust / np.linalg.norm(thrust)
x_world = np.array([math.cos(yaw), math.sin(yaw), 0])
y_body = np.cross(z_body, x_world)
x_body = np.cross(y_body, z_body)
pitch = math.asin(-x_body[2])
roll = math.atan2(y_body[2], z_body[2])
return (roll, pitch, yaw)
# "private" methods
def triangleArea(triangleSet):
"""
Calculate areas of subdivided triangles
Input: the set of subdivided triangles
Output: a list of the areas with corresponding idices with the the triangleSet
"""
triangleAreaSet = []
for i in range(len(triangleSet)):
v1 = triangleSet[i][1] - triangleSet[i][0]
v2 = triangleSet[i][2] - triangleSet[i][0]
area = np.linalg.norm(np.cross(v1, v2))/2
triangleAreaSet.append(area)
return triangleAreaSet
def crossArea(forceVecs,triangleAreaSet,triNormVecs):
"""
Preparation for Young's Modulus
Calculate the cross sectional areas perpendicular to the force vectors
Input: forceVecs = a list of force vectors
triangleAreaSet = area of triangles
triNormVecs = a list of normal vectors for each triangle (should be given by the stl file)
Output: A list of cross sectional area, approximated by the area of the triangle perpendicular to the force vector
"""
crossAreaSet = np.zeros(len(triangleAreaSet))
for i in range(len(forceVecs)):
costheta = np.dot(forceVecs[i],triNormVecs[i])/(np.linalg.norm(forceVecs[i])*np.linalg.norm(triNormVecs[i]))
crossAreaSet[i] = abs(costheta*triangleAreaSet[i])
return crossAreaSet
def computeNormals(vtx, idx):
nrml = numpy.zeros(vtx.shape, numpy.float32)
# compute normal per triangle
triN = numpy.cross(vtx[idx[:,1]] - vtx[idx[:,0]], vtx[idx[:,2]] - vtx[idx[:,0]])
# sum normals at vtx
nrml[idx[:,0]] += triN[:]
nrml[idx[:,1]] += triN[:]
nrml[idx[:,2]] += triN[:]
# compute norms
nrmlNorm = numpy.sqrt(nrml[:,0]*nrml[:,0]+nrml[:,1]*nrml[:,1]+nrml[:,2]*nrml[:,2])
return nrml/nrmlNorm.reshape(-1,1)
def convex_hull(points, vind, nind, tind, obj):
"super ineffective"
cnt = len(points)
for a in range(cnt):
for b in range(a+1,cnt):
for c in range(b+1,cnt):
vec1 = points[a] - points[b]
vec2 = points[a] - points[c]
n = np.cross(vec1, vec2)
n /= np.linalg.norm(n)
C = np.dot(n, points[a])
inner = np.inner(n, points)
pos = (inner <= C+0.0001).all()
neg = (inner >= C-0.0001).all()
if not pos and not neg: continue
obj.out.write("f %i//%i %i//%i %i//%i\n" % (
(vind[a], nind[a], vind[b], nind[b], vind[c], nind[c])
if (inner - C).sum() < 0 else
(vind[a], nind[a], vind[c], nind[c], vind[b], nind[b]) ) )
#obj.out.write("f %i/%i/%i %i/%i/%i %i/%i/%i\n" % (
# (vind[a], tind[a], nind[a], vind[b], tind[b], nind[b], vind[c], tind[c], nind[c])
# if (inner - C).sum() < 0 else
# (vind[a], tind[a], nind[a], vind[c], tind[c], nind[c], vind[b], tind[b], nind[b]) ) )
def test_broadcasting_shapes(self):
u = np.ones((2, 1, 3))
v = np.ones((5, 3))
assert_equal(np.cross(u, v).shape, (2, 5, 3))
u = np.ones((10, 3, 5))
v = np.ones((2, 5))
assert_equal(np.cross(u, v, axisa=1, axisb=0).shape, (10, 5, 3))
assert_raises(ValueError, np.cross, u, v, axisa=1, axisb=2)
assert_raises(ValueError, np.cross, u, v, axisa=3, axisb=0)
u = np.ones((10, 3, 5, 7))
v = np.ones((5, 7, 2))
assert_equal(np.cross(u, v, axisa=1, axisc=2).shape, (10, 5, 3, 7))
assert_raises(ValueError, np.cross, u, v, axisa=-5, axisb=2)
assert_raises(ValueError, np.cross, u, v, axisa=1, axisb=-4)
# gh-5885
u = np.ones((3, 4, 2))
for axisc in range(-2, 2):
assert_equal(np.cross(u, u, axisc=axisc).shape, (3, 4))
def _setup_normalized_vectors(self, normal_vector, north_vector):
normal_vector, north_vector = _validate_unit_vectors(normal_vector,
north_vector)
mylog.debug('Setting normalized vectors' + str(normal_vector)
+ str(north_vector))
# Now we set up our various vectors
normal_vector /= np.sqrt(np.dot(normal_vector, normal_vector))
if north_vector is None:
vecs = np.identity(3)
t = np.cross(normal_vector, vecs).sum(axis=1)
ax = t.argmax()
east_vector = np.cross(vecs[ax, :], normal_vector).ravel()
# self.north_vector must remain None otherwise rotations about a fixed axis will break.
# The north_vector calculated here will still be included in self.unit_vectors.
north_vector = np.cross(normal_vector, east_vector).ravel()
else:
if self.steady_north or (np.dot(north_vector, normal_vector) != 0.0):
north_vector = north_vector - np.dot(north_vector,normal_vector)*normal_vector
east_vector = np.cross(north_vector, normal_vector).ravel()
north_vector /= np.sqrt(np.dot(north_vector, north_vector))
east_vector /= np.sqrt(np.dot(east_vector, east_vector))
self.normal_vector = normal_vector
self.north_vector = north_vector
self.unit_vectors = YTArray([east_vector, north_vector, normal_vector], "")
self.inv_mat = np.linalg.pinv(self.unit_vectors)
def base_vectors(self):
""" Returns 3 orthognal base vectors, the first one colinear to
the axis of the loop.
"""
# normalize n
n = self.direction / (self.direction**2).sum(axis=-1)
# choose two vectors perpendicular to n
# choice is arbitrary since the coil is symetric about n
if np.abs(n[0])==1 :
l = np.r_[n[2], 0, -n[0]]
else:
l = np.r_[0, n[2], -n[1]]
l /= (l**2).sum(axis=-1)
m = np.cross(n, l)
return n, l, m
def base_vectors(n):
""" Returns 3 orthognal base vectors, the first one colinear to n.
"""
# normalize n
n = n / np.sqrt(np.square(n).sum(axis=-1))
# choose two vectors perpendicular to n
# choice is arbitrary since the coil is symetric about n
if abs(n[0]) == 1 :
l = np.r_[n[2], 0, -n[0]]
else:
l = np.r_[0, n[2], -n[1]]
l = l / np.sqrt(np.square(l).sum(axis=-1))
m = np.cross(n, l)
return n, l, m
def normal(self, t, above=True):
""" Evaluate the normal of the curve at the given parametric value(s).
This function returns an *n* × 3 array, where *n* is the number of
evaluation points.
The normal is computed as the cross product between the binormal and
the tangent of the curve.
:param t: Parametric coordinates in which to evaluate
:type t: float or [float]
:param bool above: Evaluation in the limit from above
:return: Derivative array
:rtype: numpy.array
"""
# error test input
if self.dimension != 3:
raise RuntimeError('Normals require dimension = 3')
# compute derivative
T = self.tangent(t, above=above)
B = self.binormal(t, above=above)
return np.cross(B,T)
def test_curvature(self):
# linear curves have zero curvature
crv = Curve()
self.assertAlmostEqual(crv.curvature(.3), 0.0)
# test multiple evaluation points
t = np.linspace(0,1, 10)
k = crv.curvature(t)
self.assertTrue(np.allclose(k, 0.0))
# test circle
crv = CurveFactory.circle(r=3) + [1,1]
t = np.linspace(0,2*pi, 10)
k = crv.curvature(t)
self.assertTrue(np.allclose(k, 1.0/3.0)) # circles: k = 1/r
# test 3D (np.cross has different behaviour in 2D/3D)
crv.set_dimension(3)
k = crv.curvature(t)
self.assertTrue(np.allclose(k, 1.0/3.0)) # circles: k = 1/r
def thru_plane_position(dcm):
"""Gets spatial coordinate of image origin whose axis
is perpendicular to image plane.
"""
orientation = tuple((float(o) for o in dcm.ImageOrientationPatient))
position = tuple((float(p) for p in dcm.ImagePositionPatient))
rowvec, colvec = orientation[:3], orientation[3:]
normal_vector = np.cross(rowvec, colvec)
slice_pos = np.dot(position, normal_vector)
return slice_pos