# Source code for wulffpack.core.facet

```
from typing import List, Tuple, Union
import numpy as np
from .geometry import get_angle, get_tetrahedral_volume
[docs]class Facet(object):
"""
A `Facet` carries information about a particular facet, i.e.,
only one facet out of potentially several that are equivalent
by symmetry.
Parameters
----------
normal
The normal to the surface in Cartesian coordintes
energy
Surface energy (in units of energy per surface area)
rotation
Symmetry operation (acting on scaled coordinates) that
carried an arbtriraily defined "parent facet" to this facet
original_grain
If True, this facet is part of an "original grain", i.e.,
in the case of decahedron/icosahedron, it is part of
the first created tetrahedron
tol
Numerical tolerance
"""
def __init__(self, normal: Union[List[float], Tuple[float]],
energy: float,
symmetry: np.ndarray,
original_grain: bool=True,
tol: float=1e-5):
self.normal = np.array(normal) / np.linalg.norm(normal)
self.original_normal = self.normal.copy()
self.energy = energy
self.symmetries = [symmetry]
self.vertices = []
self.original_grain = original_grain
self.original_vertices = []
self.tol = tol
[docs] def add_vertex(self, vertex: Union[List, tuple]):
"""
Add a vertex to the list of vertices. This function checks
that there is not already such a vertex, to ensure that
no vertex is duplicated. It is also checked that the vertex
lies in the expected plane (as defined by the first vertex and
the surface normal).
Parameters
----------
vertex
The vertex in Cartesian coordinates
"""
# Need to check that the vertex is not already added
# (relevant if more than three planes cross at the
# same point, such as in a non-truncated octahedron)
for comp_vertex in self.vertices:
if np.linalg.norm(vertex - comp_vertex) < self.tol:
return None
# Check that it is in the right plane
if self.vertices:
if abs(np.dot(vertex, self.normal) - self.distance_from_origin) > self.tol:
raise ValueError('Vertex {} does not lie in the same plane '
'as previously added vertices.'.format(vertex))
self.vertices.append(vertex)
@property
def distance_from_origin(self) -> float:
"""Returns the distance from the origin to the facet."""
return np.dot(self.vertices[0], self.normal)
[docs] def remove_redundant_vertices(self):
"""
Remove any vertex which is midway between two other vertices.
This would not work if the polygon were concave, but that is
not allowed anyway.
"""
to_pop = []
for i, vertex_i in enumerate(self.vertices):
for j, vertex_j in enumerate(self.vertices):
if j <= i:
continue
v1 = vertex_j - vertex_i
for k, vertex_k in enumerate(self.vertices):
if k <= j:
continue
v2 = vertex_k - vertex_i
angleish = abs(get_angle(v2, v1)) / np.pi
if abs(angleish - round(angleish)) > self.tol / 100:
continue
# Then we have found three vertices on a line
# Now we need to remove the middle one
v3 = vertex_k - vertex_j
d1 = np.linalg.norm(v1)
d2 = np.linalg.norm(v2)
d3 = np.linalg.norm(v3)
_, indices = zip(
*sorted(zip([d1, d2, d3], list(range(3)))))
to_pop.append([k, j, i][indices[-1]])
for i in reversed(sorted(set(to_pop))):
self.vertices.pop(i)
@property
def ordered_vertices(self) -> List[np.ndarray]:
"""
Returns the vertices ordered such that they enclose a polygon.
The first/last vertex occurs twice.
It is assumed (but not checked) that the coordinates lie in
a plane and that they form a convex polygon.
"""
# Find closest pairs by comparing angles between each triplet of
# vertices
best_pairs = []
for i, vertex_i in enumerate(self.vertices):
# Its closest neighbors are those with which the largest angle is
# formed
best_angle = 0
for j, vertex_j in enumerate(self.vertices):
if i == j:
continue
for k, vertex_k in enumerate(self.vertices):
if k <= j or i == k:
continue
angle = get_angle(vertex_j - vertex_i, vertex_k - vertex_i)
if angle > best_angle:
best_angle = angle
neighbors = (j, k)
for neighbor in neighbors:
pair = tuple(sorted([i, neighbor]))
if pair not in best_pairs:
best_pairs.append(pair)
# Make a new list of vertices, now in order (as implicitly
# defined by the pairs of nearest neighbors)
ordered_vertices = [self.vertices[0]]
current_vertex_index = 0
while len(ordered_vertices) < len(self.vertices):
for pair_index, pair in enumerate(best_pairs):
if current_vertex_index in pair:
for new_vertex_index in pair:
if new_vertex_index != current_vertex_index:
ordered_vertices.append(
self.vertices[new_vertex_index])
break
best_pairs.pop(pair_index)
current_vertex_index = new_vertex_index
break
ordered_vertices.append(self.vertices[0])
return ordered_vertices
@property
def face_as_triangles(self) -> List[List[np.ndarray]]:
"""
Given N 3D coordinates, ABCD...N, split them into triangles
that do not overlap. Two of the vertices of each triangle
will be vertices of the face, the third vertex will be
the origin.
"""
vertices = self.ordered_vertices[:-1]
nvertices = len(vertices)
center = np.mean(vertices, axis=0) # Mid point
# Standardize order because some applications
# (such as threejs) expect that
normal = np.cross(vertices[1] - center,
vertices[0] - center)
reverse = (np.dot(normal, self.normal) > 0)
# Construct all triangles
triangles = []
for i in range(0, len(vertices)):
if reverse:
triangle = [center,
vertices[(i + 1) % nvertices],
vertices[i]]
else:
triangle = [center,
vertices[i],
vertices[(i + 1) % nvertices]]
triangles.append(triangle)
assert len(triangles) == len(vertices)
return triangles
@property
def area(self) -> float:
"""Returns the total area of the facet."""
area = 0
for triangle in self.face_as_triangles:
v1 = triangle[1] - triangle[0]
v2 = triangle[2] - triangle[0]
area += np.linalg.norm(np.cross(v1, v2))
return area / 2
@property
def surface_energy(self) -> float:
"""Returns the total surface energy of the facet."""
return self.area * self.energy
[docs] def get_volume(self, origin: Union[List, tuple]=None) -> float:
"""
Calculate the volume formed by the pyramid having the face as
its base and the specified origin as its tip.
Parameters
----------
origin : list of 3 ints
The origin
Returns
-------
The total volume
"""
if origin is None:
origin = np.array([0, 0, 0])
else:
origin = np.array(origin)
volume = 0
for triangle in self.face_as_triangles:
volume += get_tetrahedral_volume(triangle, origin)
return volume
@property
def perimeter_length(self) -> float:
"""Returns the length of the perimeter"""
length = 0.0
ordered_vertices = self.ordered_vertices
for i, vertex in enumerate(ordered_vertices[1:]):
length += np.linalg.norm(vertex - ordered_vertices[i])
return length
```