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tritools.py
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""" Tools for triangular grids. """ import numpy as np from matplotlib import cbook from matplotlib.tri import Triangulation 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])) 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) 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