vector

Routines for working with vectors These routines can be used with vectors, as well as with matrices containing a vector in each row.

Functions

Details

Routines for working with vectors These routines can be used with vectors, as well as with matrices containing a vector in each row.

vector.GramSchmidt(p1, p2, p3)[source]

Gram-Schmidt orthogonalization

Parameters:
  • p1 (array (3,) or (M,3)) – coordinates of Point 1
  • p2 (array (3,) or (M,3)) – coordinates of Point 2
  • p3 (array (3,) or (M,3)) – coordinates of Point 3
Returns:

Rmat – flattened rotation matrix

Return type:

array (9,) or (M,9)

Example

>>> P1 = np.array([[0, 0, 0], [1,2,3]])
>>> P2 = np.array([[1, 0, 0], [4,1,0]])
>>> P3 = np.array([[1, 1, 0], [9,-1,1]])
>>> GramSchmidt(P1,P2,P3)
array([[ 1.        ,  0.        ,  0.        ,  0.        ,  1.        ,
     0.        ,  0.        ,  0.        ,  1.        ],
   [ 0.6882472 , -0.22941573, -0.6882472 ,  0.62872867, -0.28470732,
     0.72363112, -0.36196138, -0.93075784, -0.05170877]])

Notes

The flattened rotation matrix corresponds to

\[\mathbf{R} = [ \vec{e}_1 \, \vec{e}_2 \, \vec{e}_3 ]\]
vector.angle(v1, v2)[source]

Angle between two vectors

Parameters:
  • v1 (array (N,) or (M,N)) – vector 1
  • v2 (array (N,) or (M,N)) – vector 2
Returns:

angle – angle between v1 and v2

Return type:

double or array(M,)

Example

>>> v1 = np.array([[1,2,3],
>>>       [4,5,6]])
>>> v2 = np.array([[1,0,0],
>>>       [0,1,0]])
>>> thLib.vector.angle(v1,v2)
array([ 1.30024656,  0.96453036])

Notes

\[\alpha =arccos(\frac{\vec{v_1} \cdot \vec{v_2}}{| \vec{v_1} | \cdot | \vec{v_2}|})\]
vector.normalize(v)[source]

Normalization of a given vector

Parameters:v (array (N,) or (M,N)) – input vector
Returns:v_normalized – normalized input vector
Return type:array (N,) or (M,N)

Example

>>> thLib.vector.normalize([3, 0, 0])
array([[ 1.,  0.,  0.]])
>>> v = [[pi, 2, 3], [2, 0, 0]]
>>> thLib.vector.normalize(v)
array([[ 0.6569322 ,  0.41821602,  0.62732404],
   [ 1.        ,  0.        ,  0.        ]])

Notes

\[\vec{n} = \frac{\vec{v}}{|\vec{v}|}\]
vector.plane_orientation(p1, p2, p3)[source]

The vector perpendicular to the plane defined by three points.

Parameters:
  • p1 (array (3,) or (M,3)) – coordinates of Point 1
  • p2 (array (3,) or (M,3)) – coordinates of Point 2
  • p3 (array (3,) or (M,3)) – coordinates of Point 3
Returns:

n – vector perpendicular to the plane

Return type:

array (3,) or (M,3)

Example

>>> P1 = np.array([[0, 0, 0], [1,2,3]])
>>> P2 = np.array([[1, 0, 0], [4,1,0]])
>>> P3 = np.array([[1, 1, 0], [9,-1,1]])
>>> plane_orientation(P1,P2,P3)
array([[ 0.        ,  0.        ,  1.        ],
       [-0.36196138, -0.93075784, -0.05170877]])

Notes

\[\vec{n} = \frac{ \vec{a} \times \vec{b}} {| \vec{a} \times \vec{b}|}\]
vector.project(v1, v2)[source]

Project one vector onto another

Parameters:
  • v1 (array (N,) or (M,N)) – projected vector
  • v2 (array (N,) or (M,N)) – target vector
Returns:

v_projected – projection of v1 onto v2

Return type:

array (N,) or (M,N)

Example

>>> v1 = np.array([[1,2,3],
>>>       [4,5,6]])
>>> v2 = np.array([[1,0,0],
>>>       [0,1,0]])
>>> thLib.vector.project(v1,v2)
array([[ 1.,  0.,  0.],
   [ 0.,  5.,  0.]])

Notes

\[ \begin{align}\begin{aligned}\vec{n} = \frac{ \vec{a} }{| \vec{a} |}\\\vec{v}_{proj} = \vec{n} (\vec{v} \cdot \vec{n})\end{aligned}\end{align} \]
vector.q_shortest_rotation(v1, v2)[source]

Quaternion indicating the shortest rotation from one vector into another. You can read “qrotate” as either “quaternion rotate” or as “quick rotate”.

Parameters:
  • v1 (ndarray (3,)) – first vector
  • v2 (ndarray (3,)) – second vector
Returns:

q – quaternion rotating v1 into v2

Return type:

ndarray (3,)

Example

>>> v1 = np.r_[1,0,0]
>>> v2 = np.r_[1,1,0]
>>> q = qrotate(v1, v2)
>>> print(q)
[ 0.          0.          0.38268343]
vector.rotate_vector(vector, q)[source]

Rotates a vector, according to the given quaternions. Note that a single vector can be rotated into many orientations; or a row of vectors can all be rotated by a single quaternion.

Parameters:
  • vector (array, shape (3,) or (N,3)) – vector(s) to be rotated.
  • q (array_like, shape ([3,4],) or (N,[3,4])) – quaternions or quaternion vectors.
Returns:

rotated – rotated vector(s)

Return type:

array, shape (3,) or (N,3)

Notes

\[q \circ \left( {\vec x \cdot \vec I} \right) \circ {q^{ - 1}} = \left( {{\bf{R}} \cdot \vec x} \right) \cdot \vec I\]

More info under http://en.wikipedia.org/wiki/Quaternion

Examples

>>> mymat = eye(3)
>>> myVector = r_[1,0,0]
>>> quats = array([[0,0, sin(0.1)],[0, sin(0.2), 0]])
>>> quat.rotate_vector(myVector, quats)
array([[ 0.98006658,  0.19866933,  0.        ],
       [ 0.92106099,  0.        , -0.38941834]])
>>> quat.rotate_vector(mymat, [0, 0, sin(0.1)])
array([[ 0.98006658,  0.19866933,  0.        ],
       [-0.19866933,  0.98006658,  0.        ],
       [ 0.        ,  0.        ,  1.        ]])