wulfric.geometry.absolute_to_relative

wulfric.geometry.absolute_to_relative(vector, basis)

Computes relative coordinates of the vector with respect to the basis.

\[\boldsymbol{v} = v^1 \boldsymbol{e_1} + v^2 \boldsymbol{e_2} + v^3 \boldsymbol{e_3}\]

First scalar products of the vector with the basis vectors are computed

\[\begin{split}\begin{matrix} \boldsymbol{v} \cdot \boldsymbol{e_1} = v^1\, \boldsymbol{e_1} \cdot \boldsymbol{e_1} + v^2\, \boldsymbol{e_2} \cdot \boldsymbol{e_1} + v^3\, \boldsymbol{e_3} \cdot \boldsymbol{e_1} \\ \boldsymbol{v} \cdot \boldsymbol{e_2} = v^1\, \boldsymbol{e_1} \cdot \boldsymbol{e_2} + v^2\, \boldsymbol{e_2} \cdot \boldsymbol{e_2} + v^3\, \boldsymbol{e_3} \cdot \boldsymbol{e_2} \\ \boldsymbol{v} \cdot \boldsymbol{e_3} = v^1\, \boldsymbol{e_1} \cdot \boldsymbol{e_3} + v^2\, \boldsymbol{e_2} \cdot \boldsymbol{e_3} + v^3\, \boldsymbol{e_3} \cdot \boldsymbol{e_3} \end{matrix}\end{split}\]

Then, system of linear equations for \(v^1\), \(v^2\), \(v^3\) is solved.

Parameters:

vector(3,) array-like

Absolute coordinates of a vector.

basis(3, 3) array-like

Basis vectors. Rows are interpreted as vectors. Columns are interpreted as coordinates.

Returns:

relative(3,) numpy.ndarray

Relative coordinates of the vector with respect to the basis. \((v^1, v^2, v^3)\).

Examples

>>> import wulfric as wulf
>>> wulf.geometry.absolute_to_relative([1, 0, 0], [[0, 1, 1], [1, 0, 1], [1, 1, 0]])
array([-0.5,  0.5,  0.5])
>>> wulf.geometry.absolute_to_relative([1, 0, 2.3241], [[0, 1, 1], [1, 0, 1], [1, 1, 0]])
array([ 0.66205,  1.66205, -0.66205])