@@ -1,20 +1,22 @@

-import math

 from collections import Counter

 from collections.abc import Iterable, Sized

 from itertools import chain, combinations

 from math import factorial

-

-import numpy as np

 import scipy.spatial

+from numpy import square, zeros, subtract, array, ones, dot, asarray, concatenate, average, eye, mean, abs, sqrt

+from numpy import sum as nsum

+from numpy.linalg import det as ndet

+from numpy.linalg import slogdet, solve, matrix_rank, norm

+

 def fast_norm(v):

     # notice this method can be even more optimised

     if len(v) == 2:

-        return math.sqrt(v[0] * v[0] + v[1] * v[1])

+        return sqrt(v[0] * v[0] + v[1] * v[1])

     if len(v) == 3:

-        return math.sqrt(v[0] * v[0] + v[1] * v[1] + v[2] * v[2])

-    return math.sqrt(np.dot(v, v))

+        return sqrt(v[0] * v[0] + v[1] * v[1] + v[2] * v[2])

+    return sqrt(dot(v, v))

 def fast_2d_point_in_simplex(point, simplex, eps=1e-8):

@@ -32,12 +34,13 @@ def fast_2d_point_in_simplex(point, simplex, eps=1e-8):

 def point_in_simplex(point, simplex, eps=1e-8):

+    # simplex is list

     if len(point) == 2:

         return fast_2d_point_in_simplex(point, simplex, eps)

-    x0 = np.array(simplex[0], dtype=float)

-    vectors = np.array(simplex[1:], dtype=float) - x0

-    alpha = np.linalg.solve(vectors.T, point - x0)

+    x0 = array(simplex[0], dtype=float)

+    vectors = array(simplex[1:], dtype=float) - x0

+    alpha = solve(vectors.T, point - x0)

     return all(alpha > -eps) and sum(alpha) < 1 + eps

@@ -55,7 +58,7 @@ def fast_2d_circumcircle(points):

     tuple

         (center point : tuple(int), radius: int)

     """

-    points = np.array(points)

+    points = array(points)

     # transform to relative coordinates

     pts = points[1:] - points[0]

@@ -73,7 +76,7 @@ def fast_2d_circumcircle(points):

     # compute center

     x = dx / a

     y = dy / a

-    radius = math.sqrt(x * x + y * y)  # radius = norm([x, y])

+    radius = sqrt(x * x + y * y)  # radius = norm([x, y])

     return (x + points[0][0], y + points[0][1]), radius

@@ -91,7 +94,7 @@ def fast_3d_circumcircle(points):

     tuple

         (center point : tuple(int), radius: int)

     """

-    points = np.array(points)

+    points = array(points)

     pts = points[1:] - points[0]

     (x1, y1, z1), (x2, y2, z2), (x3, y3, z3) = pts

@@ -119,14 +122,14 @@ def fast_3d_circumcircle(points):

 def fast_det(matrix):

-    matrix = np.asarray(matrix, dtype=float)

+    matrix = asarray(matrix, dtype=float)

     if matrix.shape == (2, 2):

         return matrix[0][0] * matrix[1][1] - matrix[1][0] * matrix[0][1]

     elif matrix.shape == (3, 3):

         a, b, c, d, e, f, g, h, i = matrix.ravel()

         return a * (e * i - f * h) - b * (d * i - f * g) + c * (d * h - e * g)

     else:

-        return np.linalg.det(matrix)

+        return ndet(matrix)

 def circumsphere(pts):

@@ -137,20 +140,20 @@ def circumsphere(pts):

         return fast_3d_circumcircle(pts)

     # Modified method from http://mathworld.wolfram.com/Circumsphere.html

-    mat = [[np.sum(np.square(pt)), *pt, 1] for pt in pts]

-

-    center = []

+    mat = array([[nsum(square(pt)), *pt, 1] for pt in pts])

+    center = zeros(dim)

+    a = 1 / (2 * ndet(mat[:, 1:]))

+    factor = a

+    ind = ones((dim + 2,), bool)

     for i in range(1, len(pts)):

-        r = np.delete(mat, i, 1)

-        factor = (-1) ** (i + 1)

-        center.append(factor * fast_det(r))

-

-    a = fast_det(np.delete(mat, 0, 1))

-    center = [x / (2 * a) for x in center]

+        ind[i - 1] = True

+        ind[i] = False

+        center[i - 1] = factor * ndet(mat[:, ind])

+        factor *= -1

     x0 = pts[0]

-    vec = np.subtract(center, x0)

-    radius = fast_norm(vec)

+    vec = subtract(center, x0)

+    radius = sqrt(dot(vec, vec))

     return tuple(center), radius

@@ -174,8 +177,8 @@ def orientation(face, origin):

     If two points lie on the same side of the face, the orientation will

     be equal, if they lie on the other side of the face, it will be negated.

     """

-    vectors = np.array(face)

-    sign, logdet = np.linalg.slogdet(vectors - origin)

+    vectors = array(face)

+    sign, logdet = slogdet(vectors - origin)

     if logdet < -50:  # assume it to be zero when it's close to zero

         return 0

     return sign

@@ -210,20 +213,20 @@ def simplex_volume_in_embedding(vertices) -> float:

     # Implements http://mathworld.wolfram.com/Cayley-MengerDeterminant.html

     # Modified from https://codereview.stackexchange.com/questions/77593/calculating-the-volume-of-a-tetrahedron

-    vertices = np.asarray(vertices, dtype=float)

+    vertices = asarray(vertices, dtype=float)

     dim = len(vertices[0])

     if dim == 2:

         # Heron's formula

         a, b, c = scipy.spatial.distance.pdist(vertices, metric="euclidean")

         s = 0.5 * (a + b + c)

-        return math.sqrt(s * (s - a) * (s - b) * (s - c))

+        return sqrt(s * (s - a) * (s - b) * (s - c))

     # β_ij = |v_i - v_k|²

     sq_dists = scipy.spatial.distance.pdist(vertices, metric="sqeuclidean")

     # Add border while compressed

     num_verts = scipy.spatial.distance.num_obs_y(sq_dists)

-    bordered = np.concatenate((np.ones(num_verts), sq_dists))

+    bordered = concatenate((ones(num_verts), sq_dists))

     # Make matrix and find volume

     sq_dists_mat = scipy.spatial.distance.squareform(bordered)

@@ -236,7 +239,7 @@ def simplex_volume_in_embedding(vertices) -> float:

             return 0

         raise ValueError("Provided vertices do not form a simplex")

-    return np.sqrt(vol_square)

+    return sqrt(vol_square)

 class Triangulation:

@@ -287,8 +290,8 @@ class Triangulation:

             raise ValueError("Please provide at least one simplex")

         coords = list(map(tuple, coords))

-        vectors = np.subtract(coords[1:], coords[0])

-        if np.linalg.matrix_rank(vectors) < dim:

+        vectors = subtract(coords[1:], coords[0])

+        if matrix_rank(vectors) < dim:

             raise ValueError(

                 "Initial simplex has zero volumes "

                 "(the points are linearly dependent)"

@@ -338,9 +341,9 @@ class Triangulation:

         if len(simplex) != self.dim + 1:

             # We are checking whether point belongs to a face.

             simplex = self.containing(simplex).pop()

-        x0 = np.array(self.vertices[simplex[0]])

-        vectors = np.array(self.get_vertices(simplex[1:])) - x0

-        alpha = np.linalg.solve(vectors.T, point - x0)

+        x0 = array(self.vertices[simplex[0]])

+        vectors = array(self.get_vertices(simplex[1:])) - x0

+        alpha = solve(vectors.T, point - x0)

         if any(alpha < -eps) or sum(alpha) > 1 + eps:

             return []

@@ -403,7 +406,7 @@ class Triangulation:

         # we do not really need the center, we only need a point that is

         # guaranteed to lie strictly within the hull

         hull_points = self.get_vertices(self.hull)

-        pt_center = np.average(hull_points, axis=0)

+        pt_center = average(hull_points, axis=0)

         pt_index = len(self.vertices)

         self.vertices.append(new_vertex)

@@ -447,7 +450,7 @@ class Triangulation:

         tuple (center point, radius)

             The center and radius of the circumscribed circle

         """

-        pts = np.dot(self.get_vertices(simplex), transform)

+        pts = dot(self.get_vertices(simplex), transform)

         return circumsphere(pts)

     def point_in_cicumcircle(self, pt_index, simplex, transform):

@@ -455,13 +458,13 @@ class Triangulation:

         eps = 1e-8

         center, radius = self.circumscribed_circle(simplex, transform)

-        pt = np.dot(self.get_vertices([pt_index]), transform)[0]

+        pt = dot(self.get_vertices([pt_index]), transform)[0]

-        return np.linalg.norm(center - pt) < (radius * (1 + eps))

+        return norm(center - pt) < (radius * (1 + eps))

     @property

     def default_transform(self):

-        return np.eye(self.dim)

+        return eye(self.dim)

     def bowyer_watson(self, pt_index, containing_simplex=None, transform=None):

         """Modified Bowyer-Watson point adding algorithm.

@@ -532,9 +535,9 @@ class Triangulation:

         volume is only dependent on the shape of the simplex and not on the

         absolute size. Due to the weird scaling, the only use of this method

         is to check that a simplex is almost flat."""

-        vertices = np.array(self.get_vertices(simplex))

+        vertices = array(self.get_vertices(simplex))

         vectors = vertices[1:] - vertices[0]

-        average_edge_length = np.mean(np.abs(vectors))

+        average_edge_length = mean(abs(vectors))

         return self.volume(simplex) / (average_edge_length ** self.dim)

     def add_point(self, point, simplex=None, transform=None):

@@ -587,8 +590,8 @@ class Triangulation:

             return self.bowyer_watson(pt_index, actual_simplex, transform)

     def volume(self, simplex):

-        prefactor = np.math.factorial(self.dim)

-        vertices = np.array(self.get_vertices(simplex))

+        prefactor = factorial(self.dim)

+        vertices = array(self.get_vertices(simplex))

         vectors = vertices[1:] - vertices[0]

         return float(abs(fast_det(vectors)) / prefactor)