`compute_metric()`#

CurvCoords.compute_metric(v1: ndarray[Any, dtype[float64]], v2: ndarray[Any, dtype[float64]]) → ndarray[Any, dtype[float64]]Source #

Compute metric tensor.

The metric tensor \(g_{ij}\) or \(g^{ij}\) is calculated with inner product of covariant or contravariant bases:

\[\begin{split}\begin{aligned} g_{ij} &= \mathbf{b}_i \cdot \mathbf{b}_j \\ g^{ij} &= \mathbf{b}^i \cdot \mathbf{b}^j \end{aligned}\end{split}\]

where \(\mathbf{b}^i\) is the contravariant basis of \(i\)-th coordinate, and \(\mathbf{b}_j\) is the covariant basis of \(j\)-th coordinate.

Parameters:

Returns:

Examples

>>> import numpy as np
>>> from cherab.lhd.emc3 import CenterGrid

>>> grid = CenterGrid("zone0", index_type="coarse")
>>> coords = CurvCoords(grid)

Let compute the metric temsor: \(\mathbf{b}_\rho \cdot \mathbf{b}^\rho\). It should be an array of ones.

>>> g1 = coords.compute_metric(coords.b_rho, coords.b_sup_rho)
>>> g1.shape
(33, 100, 9)
>>> np.allclose(g1, np.ones_like(g1))
True

Let check if the metric tensor: \(\mathbf{b}_\rho\cdot\mathbf{b}_\rho\) is not the same as the before.

>>> g2 = coords.compute_metric(coords.b_rho, coords.b_rho)
>>> np.allclose(g1, g2)
False
>>> g2[0, 0, 0]

compute_metric()#