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icosphere.py
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'''
Creating geodesic icosahedron with given (integer) subdivision frequency (and
not by recursively applying Loop-like subdivision).
Advantage of subdivision frequency compared to the recursive subdivision is in
controlling the mesh resolution. Mesh resolution grows quadratically with
subdivision frequencies while it grows exponentially with iterations of the
recursive subdivision. To be precise, using the recursive
subdivision (each iteration being a subdivision with frequency nu=2), the
possible number of vertices grows with iterations i as
[12+10*(2**i+1)*(2**i-1) for i in range(10)]
which gives
[12, 42, 162, 642, 2562, 10242, 40962, 163842, 655362, 2621442].
Notice for example there is no mesh having between 2562 and 10242 vertices.
Using subdivision frequency, possible number of vertices grows with nu as
[12+10*(nu+1)*(nu-1) for nu in range(1,33)]
which gives
[12, 42, 92, 162, 252, 362, 492, 642, 812, 1002, 1212, 1442, 1692, 1962,
2252, 2562, 2892, 3242, 3612, 4002, 4412, 4842, 5292, 5762, 6252, 6762,
7292, 7842, 8412, 9002, 9612, 10242]
where nu = 32 gives 10242 vertices, and there are 15 meshes having between
2562 and 10242 vertices. The advantage is even more pronounced when using
higher resolutions.
Author: [email protected], 2014, 2017, 2021.
Originally developed in connectiton with
https://ieeexplore.ieee.org/document/7182720
'''
import numpy as np
def icosphere(nu = 1, nr_verts = None):
'''
Returns a geodesic icosahedron with subdivision frequency nu. Frequency
nu = 1 returns regular unit icosahedron, and nu>1 preformes subdivision.
If nr_verts is given, nu will be adjusted such that icosphere contains
at least nr_verts vertices. Returned faces are zero-indexed!
Parameters
----------
nu : subdivision frequency, integer (larger than 1 to make a change).
nr_verts: desired number of mesh vertices, if given, nu may be increased.
Returns
-------
subvertices : vertex list, numpy array of shape (20+10*(nu+1)*(nu-1)/2, 3)
subfaces : face list, numpy array of shape (10*n**2, 3)
'''
# Unit icosahedron
(vertices,faces) = icosahedron()
# If nr_verts given, computing appropriate subdivision frequency nu.
# We know nr_verts = 12+10*(nu+1)(nu-1)
if not nr_verts is None:
nu_min = np.ceil(np.sqrt(max(1+(nr_verts-12)/10, 1)))
nu = max(nu, nu_min)
# Subdividing
if nu>1:
(vertices,faces) = subdivide_mesh(vertices, faces, nu)
vertices = vertices/np.sqrt(np.sum(vertices**2, axis=1, keepdims=True))
return(vertices, faces)
def icosahedron():
'''' Regular unit icosahedron. '''
# 12 principal directions in 3D space: points on an unit icosahedron
phi = (1+np.sqrt(5))/2
vertices = np.array([
[0, 1, phi], [0,-1, phi], [1, phi, 0],
[-1, phi, 0], [phi, 0, 1], [-phi, 0, 1]])/np.sqrt(1+phi**2)
vertices = np.r_[vertices,-vertices]
# 20 faces
faces = np.array([
[0,5,1], [0,3,5], [0,2,3], [0,4,2], [0,1,4],
[1,5,8], [5,3,10], [3,2,7], [2,4,11], [4,1,9],
[7,11,6], [11,9,6], [9,8,6], [8,10,6], [10,7,6],
[2,11,7], [4,9,11], [1,8,9], [5,10,8], [3,7,10]], dtype=int)
return (vertices, faces)
def subdivide_mesh(vertices, faces, nu):
'''
Subdivides mesh by adding vertices on mesh edges and faces. Each edge
will be divided in nu segments. (For example, for nu=2 one vertex is added
on each mesh edge, for nu=3 two vertices are added on each mesh edge and
one vertex is added on each face.) If V and F are number of mesh vertices
and number of mesh faces for the input mesh, the subdivided mesh contains
V + F*(nu+1)*(nu-1)/2 vertices and F*nu^2 faces.
Parameters
----------
vertices : vertex list, numpy array of shape (V,3)
faces : face list, numby array of shape (F,3). Zero indexed.
nu : subdivision frequency, integer (larger than 1 to make a change).
Returns
-------
subvertices : vertex list, numpy array of shape (V + F*(nu+1)*(nu-1)/2, 3)
subfaces : face list, numpy array of shape (F*n**2, 3)
Author: vand at dtu.dk, 8.12.2017. Translated to python 6.4.2021
'''
edges = np.r_[faces[:,:-1], faces[:,1:],faces[:,[0,2]]]
edges = np.unique(np.sort(edges, axis=1),axis=0)
F = faces.shape[0]
V = vertices.shape[0]
E = edges.shape[0]
subfaces = np.empty((F*nu**2, 3), dtype = int)
subvertices = np.empty((V+E*(nu-1)+F*(nu-1)*(nu-2)//2, 3))
subvertices[:V] = vertices
# Dictionary for accessing edge index from indices of edge vertices.
edge_indices = dict()
for i in range(V):
edge_indices[i] = dict()
for i in range(E):
edge_indices[edges[i,0]][edges[i,1]] = i
edge_indices[edges[i,1]][edges[i,0]] = -i
template = faces_template(nu)
ordering = vertex_ordering(nu)
reordered_template = ordering[template]
# At this point, we have V vertices, and now we add (nu-1) vertex per edge
# (on-edge vertices).
w = np.arange(1,nu)/nu # interpolation weights
for e in range(E):
edge = edges[e]
for k in range(nu-1):
subvertices[V+e*(nu-1)+k] = (w[-1-k] * vertices[edge[0]]
+ w[k] * vertices[edge[1]])
# At this point we have E(nu-1)+V vertices, and we add (nu-1)*(nu-2)/2
# vertices per face (on-face vertices).
r = np.arange(nu-1)
for f in range(F):
# First, fixing connectivity. We get hold of the indices of all
# vertices invoved in this subface: original, on-edges and on-faces.
T = np.arange(f*(nu-1)*(nu-2)//2+E*(nu-1)+V,
(f+1)*(nu-1)*(nu-2)//2+E*(nu-1)+V) # will be added
eAB = edge_indices[faces[f,0]][faces[f,1]]
eAC = edge_indices[faces[f,0]][faces[f,2]]
eBC = edge_indices[faces[f,1]][faces[f,2]]
AB = reverse(abs(eAB)*(nu-1)+V+r, eAB<0) # already added
AC = reverse(abs(eAC)*(nu-1)+V+r, eAC<0) # already added
BC = reverse(abs(eBC)*(nu-1)+V+r, eBC<0) # already added
VEF = np.r_[faces[f], AB, AC, BC, T]
subfaces[f*nu**2:(f+1)*nu**2, :] = VEF[reordered_template]
# Now geometry, computing positions of face vertices.
subvertices[T,:] = inside_points(subvertices[AB,:],subvertices[AC,:])
return (subvertices, subfaces)
def reverse(vector, flag):
'''' For reversing the direction of an edge. '''
if flag:
vector = vector[::-1]
return(vector)
def faces_template(nu):
'''
Template for linking subfaces 0
in a subdivision of a face. / \
Returns faces with vertex 1---2
indexing given by reading order / \ / \
(as illustratated). 3---4---5
/ \ / \ / \
6---7---8---9
/ \ / \ / \ / \
10--11--12--13--14
'''
faces = []
# looping in layers of triangles
for i in range(nu):
vertex0 = i*(i+1)//2
skip = i+1
for j in range(i): # adding pairs of triangles, will not run for i==0
faces.append([j+vertex0, j+vertex0+skip, j+vertex0+skip+1])
faces.append([j+vertex0, j+vertex0+skip+1, j+vertex0+1])
# adding the last (unpaired, rightmost) triangle
faces.append([i+vertex0, i+vertex0+skip, i+vertex0+skip+1])
return (np.array(faces))
def vertex_ordering(nu):
'''
Permutation for ordering of 0
face vertices which transformes / \
reading-order indexing into indexing 3---6
first corners vertices, then on-edges / \ / \
vertices, and then on-face vertices 4---12--7
(as illustrated). / \ / \ / \
5---13--14--8
/ \ / \ / \ / \
1---9--10--11---2
'''
left = [j for j in range(3, nu+2)]
right = [j for j in range(nu+2, 2*nu+1)]
bottom = [j for j in range(2*nu+1, 3*nu)]
inside = [j for j in range(3*nu,(nu+1)*(nu+2)//2)]
o = [0] # topmost corner
for i in range(nu-1):
o.append(left[i])
o = o + inside[i*(i-1)//2:i*(i+1)//2]
o.append(right[i])
o = o + [1] + bottom + [2]
return(np.array(o))
def inside_points(vAB,vAC):
'''
Returns coordinates of the inside .
(on-face) vertices (marked by star) / \
for subdivision of the face ABC when vAB0---vAC0
given coordinates of the on-edge / \ / \
vertices AB[i] and AC[i]. vAB1---*---vAC1
/ \ / \ / \
vAB2---*---*---vAC2
/ \ / \ / \ / \
.---.---.---.---.
'''
v = []
for i in range(1,vAB.shape[0]):
w = np.arange(1,i+1)/(i+1)
for k in range(i):
v.append(w[-1-k]*vAB[i,:] + w[k]*vAC[i,:])
return(np.array(v).reshape(-1,3)) # reshape needed for empty return