# Source code for pde.grids.cartesian

"""
Cartesian grids of arbitrary dimension.

.. codeauthor:: David Zwicker <david.zwicker@ds.mpg.de>

"""

from __future__ import annotations

import itertools
from typing import (  # @UnusedImport
TYPE_CHECKING,
Any,
Dict,
Generator,
List,
Optional,
Sequence,
Tuple,
Union,
)

import numpy as np

from ..tools.cuboid import Cuboid
from ..tools.plotting import plot_on_axes
from .base import DimensionError, GridBase, _check_shape

if TYPE_CHECKING:
from .boundaries.axes import Boundaries, BoundariesData  # @UnusedImport

[docs]class CartesianGrid(GridBase):  # lgtm [py/missing-equals]
r""" d-dimensional Cartesian grid with uniform discretization for each axis

The grids can be thought of as a collection of n-dimensional boxes, called cells, of
equal length in each dimension. The bounds then defined the total volume covered by
these cells, while the cell coordinates give the location of the box centers. We
index the boxes starting from 0 along each dimension. Consequently, the cell
:math:i-\frac12 corresponds to the left edge of the  covered interval and the
index :math:i+\frac12 corresponds to the right edge, when the dimension is covered
by d boxes.

In particular, the discretization along dimension :math:k is defined as

.. math::
x^{(k)}_i &= x^{(k)}_\mathrm{min} + \left(i + \frac12\right)
\Delta x^{(k)}
\\
\Delta x^{(k)} &= \frac{x^{(k)}_\mathrm{max} -
x^{(k)}_\mathrm{min}}{N^{(k)}}

where :math:N^{(k)} is the number of cells along this dimension. Consequently,
cells have dimension :math:\Delta x^{(k)} and cover the interval
:math:[x^{(k)}_\mathrm{min}, x^{(k)}_\mathrm{max}].
"""

cuboid: Cuboid

boundary_names = {  # name all the boundaries
"left": (0, False),
"right": (0, True),
"bottom": (1, False),
"top": (1, True),
"back": (2, False),
"front": (2, True),
}

def __init__(
self,
bounds: Sequence[Tuple[float, float]],
shape: Union[int, Sequence[int]],
periodic: Union[Sequence[bool], bool] = False,
):
"""
Args:
bounds (list of tuple):
Give the coordinate range for each axis. This should be a tuple of two
number (lower and upper bound) for each axis. The length of bounds
thus determines the grid dimension.
shape (list):
The number of support points for each axis. The length of shape needs
to match the grid dimension.
periodic (bool or list):
Specifies which axes possess periodic boundary conditions. This is
either a list of booleans defining periodicity for each individual axis
or a single boolean value specifying the same periodicity for all axes.
"""
bounds_arr = np.array(bounds, ndmin=1, dtype=np.double)
if bounds_arr.shape == (2,):
raise ValueError(
"bounds with shape (2,) are ambiguous. Either use shape (1, 2) to set "
"up a 1d system with two bounds or shape (2, 1) for a 2d system with "
"only the upper bounds specified"
)

if bounds_arr.ndim == 1 or bounds_arr.shape == 1:
# only set the upper bounds
bounds_arr = np.atleast_1d(np.squeeze(bounds_arr))
self.cuboid = Cuboid(np.zeros_like(bounds_arr), bounds_arr, mutable=False)

elif bounds_arr.ndim == 2 and bounds_arr.shape == 2:
# upper and lower bounds of the grid are given
self.cuboid = Cuboid.from_bounds(bounds_arr, mutable=False)

else:
raise ValueError(
f"Do not know how to interpret shape {bounds_arr.shape} for bounds"
)

# handle the shape array
shape = _check_shape(shape)
if len(shape) == 1 and self.cuboid.dim > 1:
shape = (int(shape),) * self.cuboid.dim
if self.cuboid.dim != len(shape):
raise DimensionError("Dimension of bounds and shape are not compatible")

# initialize the base class
super().__init__()
self._shape = _check_shape(shape)
self.dim = len(self.shape)
self.num_axes = self.dim

if isinstance(periodic, (bool, np.bool_)):
self.periodic = [bool(periodic)] * self.dim
elif len(periodic) != self.dim:
raise DimensionError(
"Number of axes with specified periodicity does not match grid "
f"dimension ({len(periodic)} != {self.dim})"
)
else:
self.periodic = list(periodic)

if self.dim <= 3:
self.axes = list("xyz"[: self.dim])
else:
self.axes = [chr(97 + i) for i in range(self.dim)]

# determine the coordinates
p1, p2 = self.cuboid.corners
axes_coords, discretization = [], []
for d in range(self.dim):
num = self.shape[d]
c, dc = np.linspace(p1[d], p2[d], num, endpoint=False, retstep=True)
if self.shape[d] == 1:
# correct for singular dimension
dc = p2[d] - p1[d]
c += dc / 2
axes_coords.append(c)
discretization.append(dc)
self._discretization = np.array(discretization)
self._axes_coords = tuple(axes_coords)
self._axes_bounds = tuple(self.cuboid.bounds)

@property
def state(self) -> Dict[str, Any]:
"""dict: the state of the grid"""
return {
"bounds": self.axes_bounds,
"shape": self.shape,
"periodic": self.periodic,
}

[docs]    @classmethod
def from_state(cls, state: Dict[str, Any]) -> CartesianGrid:  # type: ignore
"""create a field from a stored state.

Args:
state (dict):
The state from which the grid is reconstructed.
"""
state_copy = state.copy()
obj = cls(
bounds=state_copy.pop("bounds"),
shape=state_copy.pop("shape"),
periodic=state_copy.pop("periodic"),
)
if state_copy:
raise ValueError(f"State items {state_copy.keys()} were not used")
return obj

[docs]    @classmethod
def from_bounds(
cls,
bounds: Sequence[Tuple[float, float]],
shape: Sequence[int],
periodic: Sequence[bool],
) -> CartesianGrid:
"""
Args:
bounds (tuple):
Give the coordinate range for each axis. This should be a tuple of two
number (lower and upper bound) for each axis. The length of bounds
thus determines the grid dimension.
shape (tuple):
The number of support points for each axis. The length of shape needs
to match the grid dimension.
periodic (bool or list):
Specifies which axes possess periodic boundary conditions. This is
either a list of booleans defining periodicity for each individual axis
or a single boolean value specifying the same periodicity for all axes.

Returns:
:class:CartesianGrid representing the region chosen by bounds
"""
return CartesianGrid(bounds, shape, periodic)

@property
def volume(self) -> float:
"""float: total volume of the grid"""
return float(self.cuboid.volume)

@property
def cell_volume_data(self):
"""size associated with each cell"""
return tuple(self.discretization)

[docs]    def iter_mirror_points(
self, point: np.ndarray, with_self: bool = False, only_periodic: bool = True
) -> Generator:
"""generates all mirror points corresponding to point

Args:
point (:class:~numpy.ndarray): the point within the grid
with_self (bool): whether to include the point itself
only_periodic (bool): whether to only mirror along periodic axes

Returns:
A generator yielding the coordinates that correspond to mirrors
"""
point = np.asanyarray(point, dtype=np.double)

# find all offsets of the individual axes
offsets = []
for i in range(self.dim):
if only_periodic and not self.periodic[i]:
offsets.append()
else:
s = self.cuboid.size[i]
offsets.append([-s, 0, s])

# produce the respective mirrored points
for offset in itertools.product(*offsets):
if with_self or np.linalg.norm(offset) != 0:
yield point + offset

[docs]    def get_random_point(
self,
*,
boundary_distance: float = 0,
coords: str = "cartesian",
rng: Optional[np.random.Generator] = None,
) -> np.ndarray:
"""return a random point within the grid

Args:
boundary_distance (float): The minimal distance this point needs to
have from all boundaries.
coords (str):
Determines the coordinate system in which the point is specified. Valid
values are cartesian, cell, and grid;
see :meth:~pde.grids.base.GridBase.transform.
rng (:class:~numpy.random.Generator):
Random number generator (default: :func:~numpy.random.default_rng())

Returns:
:class:~numpy.ndarray: The coordinates of the point
"""
if rng is None:
rng = np.random.default_rng()

# handle the boundary distance
cuboid = self.cuboid
if boundary_distance != 0:
if any(cuboid.size <= 2 * boundary_distance):
raise RuntimeError("Random points would be too close to boundary")
cuboid = cuboid.buffer(-boundary_distance)

# create random point
point = cuboid.pos + rng.random(self.dim) * cuboid.size

if coords == "cartesian" or coords == "grid":
return point  # type: ignore
elif coords == "cell":
return self.transform(point, "grid", "cell")
else:
raise ValueError(f"Unknown coordinate system {coords}")

[docs]    def get_line_data(self, data: np.ndarray, extract: str = "auto") -> Dict[str, Any]:
"""return a line cut through the given data

Args:
data (:class:~numpy.ndarray):
The values at the grid points
extract (str):
Determines which cut is done through the grid. Possible choices
are (default is cut_0):

* cut_#: return values along the axis specified by # and use
the mid point along all other axes.
* project_#: average values for all axes, except axis #.

Here, # can either be a zero-based index (from 0 to dim-1) or
a letter denoting the axis.

Returns:
A dictionary with information about the line cut, which is
convenient for plotting.
"""
if data.shape[-self.dim :] != self.shape:
raise ValueError(
f"Shape {data.shape} of the data array is not compatible with grid "
f"shape {self.shape}"
)

def _get_axis(axis):
"""determine the axis from a given specifier"""
try:
axis = int(axis)
except ValueError:
try:
axis = self.axes.index(axis)
except ValueError:
raise ValueError(f"Axis {axis} not defined")
return axis

if extract == "auto":
extract = "cut_0"  # use a cut along first axis

if extract.startswith("cut_"):
# consider a cut along a given axis
axis = _get_axis(extract[4:])
data_y = data
rank = data.ndim - self.dim  # rank of data
for ax in reversed(range(self.dim)):
if ax != axis:
mid_point = self.shape[ax] // 2
data_y = np.take(data_y, mid_point, axis=ax + rank)
label_y = f"Cut along {self.axes[axis]}"

elif extract.startswith("project_"):
# consider a projection along a given axis
axis = _get_axis(extract[8:])
avg_axes = [ax - self.dim for ax in range(self.dim) if ax != axis]
data_y = data.mean(axis=tuple(avg_axes))
label_y = f"Projection onto {self.axes[axis]}"

else:
raise ValueError(f"Unknown extraction method {extract}")

if self.dim == 1:
label_y = ""

# return the data with the respective labels
return {
"data_x": self.axes_coords[axis],
"data_y": data_y,
"extent_x": self.axes_bounds[axis],
"label_x": self.axes[axis],
"label_y": label_y,
}

[docs]    def get_image_data(self, data: np.ndarray) -> Dict[str, Any]:
"""return a 2d-image of the data

Args:
data (:class:~numpy.ndarray): The values at the grid points

Returns:
A dictionary with information about the image, which is  convenient
for plotting.
"""
if data.shape[-self.dim :] != self.shape:
raise ValueError(
f"Shape {data.shape} of the data array is not compatible with grid "
f"shape {self.shape}"
)

if self.dim == 2:
image_data = data
elif self.dim == 3:
image_data = data[:, :, self.shape[-1] // 2]
else:
raise NotImplementedError(
"Creating images is only implemented for 2d and 3d grids"
)

extent: List[float] = []
for c in self.axes_bounds[:2]:
extent.extend(c)

return {
"data": image_data,
"x": self.axes_coords,
"y": self.axes_coords,
"extent": extent,
"label_x": self.axes,
"label_y": self.axes,
}

[docs]    def point_to_cartesian(
self, points: np.ndarray, *, full: bool = False
) -> np.ndarray:
"""convert coordinates of a point to Cartesian coordinates

Args:
points (:class:~numpy.ndarray): Points given in grid coordinates
full (bool): Compatibility option not used in this method

Returns:
:class:~numpy.ndarray: The Cartesian coordinates of the point
"""
assert points.shape[-1] == self.dim, f"Point must have {self.dim} coordinates"
return points

[docs]    def point_from_cartesian(self, coords: np.ndarray) -> np.ndarray:
"""convert points given in Cartesian coordinates to this grid

Args:
coords (:class:~numpy.ndarray): Points in Cartesian coordinates.

Returns:
:class:~numpy.ndarray: Points given in the coordinates of the grid
"""
assert coords.shape[-1] == self.dim, f"Point must have {self.dim} coordinates"
return coords

[docs]    def polar_coordinates_real(
self, origin: np.ndarray, *, ret_angle: bool = False
) -> Union[np.ndarray, Tuple[np.ndarray, np.ndarray, np.ndarray]]:
"""return polar coordinates associated with the grid

Args:
origin (:class:~numpy.ndarray):
Coordinates of the origin at which the polar coordinate system is
anchored.
ret_angle (bool):
Determines whether angles are returned alongside the distance. If False
only the distance to the origin is returned for each support point of the
grid. If True, the distance and angles are returned. For a 1d system
system, the angle is defined as the sign of the difference between the
point and the origin, so that angles can either be 1 or -1. For 2d
systems and 3d systems, polar coordinates and spherical coordinates are
used, respectively.
"""
origin = np.array(origin, dtype=np.double, ndmin=1)
if len(origin) != self.dim:
raise DimensionError("Dimensions are not compatible")

# calculate the difference vector between all cells and the origin
diff = self.difference_vector_real(origin, self.cell_coords)
dist: np.ndarray = np.linalg.norm(diff, axis=-1)  # get distance

# determine distance and optionally angles for these vectors
if ret_angle:
if self.dim == 1:
return dist, np.sign(diff)[..., 0]  # type: ignore

elif self.dim == 2:
return dist, np.arctan2(diff[:, :, 0], diff[:, :, 1])  # type: ignore

elif self.dim == 3:
theta = np.arccos(diff[..., 2] / dist)
phi = np.arctan2(diff[..., 0], diff[..., 1])
return dist, theta, phi

else:
raise NotImplementedError(
f"Cannot calculate angles for dimension {self.dim}"
)
else:
return dist

[docs]    def from_polar_coordinates(
self,
distance: np.ndarray,
angle: np.ndarray,
origin: Optional[np.ndarray] = None,
) -> np.ndarray:
"""convert polar coordinates to Cartesian coordinates

This function is currently only implemented for 1d and 2d systems.

Args:
distance (:class:~numpy.ndarray):
angle (:class:~numpy.ndarray):
The angle with respect to the origin
origin (:class:~numpy.ndarray, optional):
Sets the origin of the coordinate system. If omitted, the zero point is
assumed as the origin.

Returns:
:class:~numpy.ndarray: The Cartesian coordinates corresponding to the given
polar coordinates.
"""
distance = np.asarray(distance)
angle = np.asarray(angle)
if origin is None:
origin = np.zeros(self.dim)
else:
origin = np.atleast_1d(origin)

if self.dim == 1:
diff = distance * angle
coords = origin + diff[..., None]

elif self.dim == 2:
unit_vector = np.moveaxis(np.array([np.sin(angle), np.cos(angle)]), 0, -1)
diff = distance[..., None] * unit_vector
coords = origin + diff

else:
raise NotImplementedError(
f"Cannot calculate coordinates for dimension {self.dim}"
)

return self.normalize_point(coords, reflect=False)

[docs]    @plot_on_axes()
def plot(self, ax, **kwargs):
r"""visualize the grid

Args:
{PLOT_ARGS}
\**kwargs: Extra arguments are passed on the to the matplotlib
plotting routines, e.g., to set the color of the lines
"""
if self.dim not in {1, 2}:
raise NotImplementedError(
f"Plotting is not implemented for grids of dimension {self.dim}"
)

kwargs.setdefault("color", "k")
xb = self.axes_bounds
for x in np.linspace(*xb, self.shape + 1):
ax.axvline(x, **kwargs)
ax.set_xlim(*xb)
ax.set_xlabel(self.axes)

if self.dim == 2:
yb = self.axes_bounds
for y in np.linspace(*yb, self.shape + 1):
ax.axhline(y, **kwargs)
ax.set_ylim(*yb)
ax.set_ylabel(self.axes)

ax.set_aspect(1)

[docs]    def slice(self, indices: Sequence[int]) -> CartesianGrid:
"""return a subgrid of only the specified axes

Args:
indices (list):
Indices indicating the axes that are retained in the subgrid

Returns:
:class:CartesianGrid: The subgrid
"""
subgrid = self.__class__(
bounds=[self.axes_bounds[i] for i in indices],
shape=tuple(self.shape[i] for i in indices),
periodic=[self.periodic[i] for i in indices],
)
subgrid.axes = [self.axes[i] for i in indices]
return subgrid

[docs]class UnitGrid(CartesianGrid):
r"""d-dimensional Cartesian grid with unit discretization in all directions

The grids can be thought of as a collection of d-dimensional cells of unit length.
The shape parameter determines how many boxes there are in each direction. The
cells are enumerated starting with 0, so the last cell has index :math:n-1 if
there are :math:n cells along a dimension. A given cell :math:i extends from
coordinates :math:i to :math:i + 1, so the midpoint is at :math:i + \frac12,
which is the cell coordinate. Taken together, the cells covers the interval
:math:[0, n] along this dimension.
"""

def __init__(
self, shape: Sequence[int], periodic: Union[Sequence[bool], bool] = False
):
"""
Args:
shape (list):
The number of support points for each axis. The dimension of the grid is
given by len(shape).
periodic (bool or list):
Specifies which axes possess periodic boundary conditions. This is
either a list of booleans defining periodicity for each individual axis
or a single boolean value specifying the same periodicity for all axes.
"""
if isinstance(shape, int):
shape = [shape]
super().__init__([(0, s) for s in shape], shape, periodic)
self.cuboid = Cuboid(np.zeros(self.dim), self.shape)
self._discretization = np.ones(self.dim)

# determine the cell center coordinates
self._axes_coords = tuple(np.arange(n) + 0.5 for n in self.shape)
self._axes_bounds = tuple(self.cuboid.bounds)

@property
def state(self) -> Dict[str, Any]:
"""dict: the state of the grid"""
return {"shape": self.shape, "periodic": self.periodic}

[docs]    @classmethod
def from_state(cls, state: Dict[str, Any]) -> UnitGrid:  # type: ignore
"""create a field from a stored state.

Args:
state (dict):
The state from which the grid is reconstructed.
"""
state_copy = state.copy()
obj = cls(shape=state_copy.pop("shape"), periodic=state_copy.pop("periodic"))
if state_copy:
raise ValueError(f"State items {state_copy.keys()} were not used")
return obj

[docs]    def to_cartesian(self) -> CartesianGrid:
"""convert unit grid to :class:CartesianGrid"""
return CartesianGrid(
self.cuboid.bounds, shape=self.shape, periodic=self.periodic
)

[docs]    def slice(self, indices: Sequence[int]) -> UnitGrid:
"""return a subgrid of only the specified axes

Args:
indices (list):
Indices indicating the axes that are retained in the subgrid

Returns:
:class:UnitGrid: The subgrid
"""
subgrid = self.__class__(
shape=[self.shape[i] for i in indices],
periodic=[self.periodic[i] for i in indices],
)
subgrid.axes = [self.axes[i] for i in indices]
return subgrid
`