"""
Tools for triangular grids.
"""
import numpy as np
from matplotlib import cbook
from matplotlib.tri import Triangulation
[docs]class TriAnalyzer:
"""
Define basic tools for triangular mesh analysis and improvement.
A TriAnalyzer encapsulates a `.Triangulation` object and provides basic
tools for mesh analysis and mesh improvement.
Attributes
----------
scale_factors
Parameters
----------
triangulation : `~matplotlib.tri.Triangulation`
The encapsulated triangulation to analyze.
"""
def __init__(self, triangulation):
cbook._check_isinstance(Triangulation, triangulation=triangulation)
self._triangulation = triangulation
@property
def scale_factors(self):
"""
Factors to rescale the triangulation into a unit square.
Returns
-------
(float, float)
Scaling factors (kx, ky) so that the triangulation
``[triangulation.x * kx, triangulation.y * ky]``
fits exactly inside a unit square.
"""
compressed_triangles = self._triangulation.get_masked_triangles()
node_used = (np.bincount(np.ravel(compressed_triangles),
minlength=self._triangulation.x.size) != 0)
return (1 / np.ptp(self._triangulation.x[node_used]),
1 / np.ptp(self._triangulation.y[node_used]))
[docs] def circle_ratios(self, rescale=True):
"""
Return a measure of the triangulation triangles flatness.
The ratio of the incircle radius over the circumcircle radius is a
widely used indicator of a triangle flatness.
It is always ``<= 0.5`` and ``== 0.5`` only for equilateral
triangles. Circle ratios below 0.01 denote very flat triangles.
To avoid unduly low values due to a difference of scale between the 2
axis, the triangular mesh can first be rescaled to fit inside a unit
square with `scale_factors` (Only if *rescale* is True, which is
its default value).
Parameters
----------
rescale : bool, default: True
If True, internally rescale (based on `scale_factors`), so that the
(unmasked) triangles fit exactly inside a unit square mesh.
Returns
-------
masked array
Ratio of the incircle radius over the circumcircle radius, for
each 'rescaled' triangle of the encapsulated triangulation.
Values corresponding to masked triangles are masked out.
"""
# Coords rescaling
if rescale:
(kx, ky) = self.scale_factors
else:
(kx, ky) = (1.0, 1.0)
pts = np.vstack([self._triangulation.x*kx,
self._triangulation.y*ky]).T
tri_pts = pts[self._triangulation.triangles]
# Computes the 3 side lengths
a = tri_pts[:, 1, :] - tri_pts[:, 0, :]
b = tri_pts[:, 2, :] - tri_pts[:, 1, :]
c = tri_pts[:, 0, :] - tri_pts[:, 2, :]
a = np.hypot(a[:, 0], a[:, 1])
b = np.hypot(b[:, 0], b[:, 1])
c = np.hypot(c[:, 0], c[:, 1])
# circumcircle and incircle radii
s = (a+b+c)*0.5
prod = s*(a+b-s)*(a+c-s)*(b+c-s)
# We have to deal with flat triangles with infinite circum_radius
bool_flat = (prod == 0.)
if np.any(bool_flat):
# Pathologic flow
ntri = tri_pts.shape[0]
circum_radius = np.empty(ntri, dtype=np.float64)
circum_radius[bool_flat] = np.inf
abc = a*b*c
circum_radius[~bool_flat] = abc[~bool_flat] / (
4.0*np.sqrt(prod[~bool_flat]))
else:
# Normal optimized flow
circum_radius = (a*b*c) / (4.0*np.sqrt(prod))
in_radius = (a*b*c) / (4.0*circum_radius*s)
circle_ratio = in_radius/circum_radius
mask = self._triangulation.mask
if mask is None:
return circle_ratio
else:
return np.ma.array(circle_ratio, mask=mask)
[docs] def get_flat_tri_mask(self, min_circle_ratio=0.01, rescale=True):
"""
Eliminate excessively flat border triangles from the triangulation.
Returns a mask *new_mask* which allows to clean the encapsulated
triangulation from its border-located flat triangles
(according to their :meth:`circle_ratios`).
This mask is meant to be subsequently applied to the triangulation
using `.Triangulation.set_mask`.
*new_mask* is an extension of the initial triangulation mask
in the sense that an initially masked triangle will remain masked.
The *new_mask* array is computed recursively; at each step flat
triangles are removed only if they share a side with the current mesh
border. Thus no new holes in the triangulated domain will be created.
Parameters
----------
min_circle_ratio : float, default: 0.01
Border triangles with incircle/circumcircle radii ratio r/R will
be removed if r/R < *min_circle_ratio*.
rescale : bool, default: True
If True, first, internally rescale (based on `scale_factors`) so
that the (unmasked) triangles fit exactly inside a unit square
mesh. This rescaling accounts for the difference of scale which
might exist between the 2 axis.
Returns
-------
bool array-like
Mask to apply to encapsulated triangulation.
All the initially masked triangles remain masked in the
*new_mask*.
Notes
-----
The rationale behind this function is that a Delaunay
triangulation - of an unstructured set of points - sometimes contains
almost flat triangles at its border, leading to artifacts in plots
(especially for high-resolution contouring).
Masked with computed *new_mask*, the encapsulated
triangulation would contain no more unmasked border triangles
with a circle ratio below *min_circle_ratio*, thus improving the
mesh quality for subsequent plots or interpolation.
"""
# Recursively computes the mask_current_borders, true if a triangle is
# at the border of the mesh OR touching the border through a chain of
# invalid aspect ratio masked_triangles.
ntri = self._triangulation.triangles.shape[0]
mask_bad_ratio = self.circle_ratios(rescale) < min_circle_ratio
current_mask = self._triangulation.mask
if current_mask is None:
current_mask = np.zeros(ntri, dtype=bool)
valid_neighbors = np.copy(self._triangulation.neighbors)
renum_neighbors = np.arange(ntri, dtype=np.int32)
nadd = -1
while nadd != 0:
# The active wavefront is the triangles from the border (unmasked
# but with a least 1 neighbor equal to -1
wavefront = (np.min(valid_neighbors, axis=1) == -1) & ~current_mask
# The element from the active wavefront will be masked if their
# circle ratio is bad.
added_mask = wavefront & mask_bad_ratio
current_mask = added_mask | current_mask
nadd = np.sum(added_mask)
# now we have to update the tables valid_neighbors
valid_neighbors[added_mask, :] = -1
renum_neighbors[added_mask] = -1
valid_neighbors = np.where(valid_neighbors == -1, -1,
renum_neighbors[valid_neighbors])
return np.ma.filled(current_mask, True)
def _get_compressed_triangulation(self):
"""
Compress (if masked) the encapsulated triangulation.
Returns minimal-length triangles array (*compressed_triangles*) and
coordinates arrays (*compressed_x*, *compressed_y*) that can still
describe the unmasked triangles of the encapsulated triangulation.
Returns
-------
compressed_triangles : array-like
the returned compressed triangulation triangles
compressed_x : array-like
the returned compressed triangulation 1st coordinate
compressed_y : array-like
the returned compressed triangulation 2nd coordinate
tri_renum : int array
renumbering table to translate the triangle numbers from the
encapsulated triangulation into the new (compressed) renumbering.
-1 for masked triangles (deleted from *compressed_triangles*).
node_renum : int array
renumbering table to translate the point numbers from the
encapsulated triangulation into the new (compressed) renumbering.
-1 for unused points (i.e. those deleted from *compressed_x* and
*compressed_y*).
"""
# Valid triangles and renumbering
tri_mask = self._triangulation.mask
compressed_triangles = self._triangulation.get_masked_triangles()
ntri = self._triangulation.triangles.shape[0]
if tri_mask is not None:
tri_renum = self._total_to_compress_renum(~tri_mask)
else:
tri_renum = np.arange(ntri, dtype=np.int32)
# Valid nodes and renumbering
valid_node = (np.bincount(np.ravel(compressed_triangles),
minlength=self._triangulation.x.size) != 0)
compressed_x = self._triangulation.x[valid_node]
compressed_y = self._triangulation.y[valid_node]
node_renum = self._total_to_compress_renum(valid_node)
# Now renumbering the valid triangles nodes
compressed_triangles = node_renum[compressed_triangles]
return (compressed_triangles, compressed_x, compressed_y, tri_renum,
node_renum)
@staticmethod
def _total_to_compress_renum(valid):
"""
Parameters
----------
valid : 1d bool array
Validity mask.
Returns
-------
int array
Array so that (`valid_array` being a compressed array
based on a `masked_array` with mask ~*valid*):
- For all i with valid[i] = True:
valid_array[renum[i]] = masked_array[i]
- For all i with valid[i] = False:
renum[i] = -1 (invalid value)
"""
renum = np.full(np.size(valid), -1, dtype=np.int32)
n_valid = np.sum(valid)
renum[valid] = np.arange(n_valid, dtype=np.int32)
return renum
