tensors

FabricTensor

Fabric tensor of directional density function.

density classmethod

density(
    *coords: ndarray,
    deg: int,
    kind: Literal["second", "third"] = "third",
    theta_den: ndarray | None = None,
    phi_den: ndarray | None = None,
    csys: Literal["cartesian", "polar"] = "polar"
) -> Tuple[ndarray, ndarray, ndarray]

Directional density function, defined as for the second kind:

\[ f(\mathbf{n}) = \frac{1}{4\pi}F_{i_1i_2...i_n} \mathbf{n}_{i_1}\mathbf{n}_{i_2}...\mathbf{n}_{i_n} \]

and for the third kind:

\[ f(\mathbf{n}) = \frac{1}{4\pi}\sum_{k=0}^{\infty}D_{i_1i_2...i_k} \mathbf{n}_{i_1}\mathbf{n}_{i_2}...\mathbf{n}_{i_k} \]

PARAMETER	DESCRIPTION
coords	Tuple of (theta, phi, radius) in polar coordinates, or tuple of (x, y, z) in cartesian coordinates. TYPE: Tuple[ndarray, ndarray, ndarray] DEFAULT: ()
deg	Degree of the fabric tensor, must be even and greater or equal to 0, by default 2. TYPE: int
kind	Kind of the fabric tensor to use, by default "third". TYPE: Literal['second', 'third'] DEFAULT: 'third'
theta_den	Meshgrid of theta for the directional density function, by default theta is derived from coords. TYPE: ndarray DEFAULT: None
phi_den	Meshgrid of phi for the directional density function, by default phi is derived from coords. TYPE: ndarray DEFAULT: None
csys	Coordinate system of the directional data, by default "polar". TYPE: Literal['cartesian', 'polar'] DEFAULT: 'polar'

RETURNS	DESCRIPTION
Tuple[ndarray, ndarray, ndarray]	Tuple of meshgrid of theta, meshgrid of phi, and directional density function.

first_kind classmethod

first_kind(
    *coords: ndarray, deg: int, csys: Literal["cartesian", "polar"] = "polar"
) -> ndarray

Fabric tensor of the first kind, defined as the average of the product of directional coordinates:

\[ N_{i_1i_2...i_n} = \frac{1}{n} \sum_{j=1}^n x_{i_1}^jx_{i_2}^j...x_{i_n}^j \]

PARAMETER	DESCRIPTION
coords	Tuple of (theta, phi, radius) in polar coordinates, or tuple of (x, y, z) in cartesian coordinates. TYPE: Tuple[ndarray, ndarray, ndarray] DEFAULT: ()
deg	Degree of the fabric tensor, must be greater or equal to 0. TYPE: int
csys	Coordinate system of the directional data, by default "polar". TYPE: Literal['cartesian', 'polar'] DEFAULT: 'polar'

RETURNS	DESCRIPTION
ndarray	Fabric tensor of the first kind of shape 3^order.

second_kind classmethod

second_kind(
    *coords: ndarray, deg: int, csys: Literal["cartesian", "polar"] = "polar"
) -> ndarray

Fabric tensor of the second kind.

PARAMETER	DESCRIPTION
coords	Tuple of (theta, phi, radius) in polar coordinates, or tuple of (x, y, z) in cartesian coordinates. TYPE: Tuple[ndarray, ndarray, ndarray] DEFAULT: ()
deg	Degree of the fabric tensor, must be even and greater or equal to 0. TYPE: int
csys	Coordinate system of the directional data, by default "polar". TYPE: Literal['cartesian', 'polar'] DEFAULT: 'polar'

RETURNS	DESCRIPTION
ndarray	Fabric tensor of the first kind of shape 3^order.

tensor classmethod

tensor(
    *coords: ndarray,
    deg: int,
    kind: Literal["first", "second", "third"],
    csys: Literal["cartesian", "polar"] = "polar"
) -> ndarray

Calculate the fabric tensor of the directional density function.

PARAMETER	DESCRIPTION
coords	Tuple of (theta, phi, radius) in polar coordinates, or tuple of (x, y, z) in cartesian coordinates. TYPE: Tuple[ndarray, ndarray, ndarray] DEFAULT: ()
deg	Degree of the fabric tensor, must be even and greater or equal to 0. TYPE: int
kind	Kind of the fabric tensor to use. TYPE: Literal['first', 'second', 'third']
csys	Coordinate system of the directional data, by default "polar". TYPE: Literal['cartesian', 'polar'] DEFAULT: 'polar'

RETURNS	DESCRIPTION
ndarray	Fabric tensor of the directional density function of shape 3^order.

third_kind classmethod

third_kind(
    *coords: ndarray, deg: int, csys: Literal["cartesian", "polar"] = "polar"
) -> ndarray

Fabric tensor of the third kind.

PARAMETER	DESCRIPTION
coords	Tuple of (theta, phi, radius) in polar coordinates, or tuple of (x, y, z) in cartesian coordinates. TYPE: Tuple[ndarray, ndarray, ndarray] DEFAULT: ()
deg	Degree of the fabric tensor, must be even and greater or equal to 0. TYPE: int
csys	Coordinate system of the directional data, by default "polar". TYPE: Literal['cartesian', 'polar'] DEFAULT: 'polar'

RETURNS	DESCRIPTION
ndarray	Fabric tensor of the first kind of shape 3^order.