1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
|
"""
Various transforms used for by the 3D code
"""
import numpy as np
from matplotlib import _api
def world_transformation(xmin, xmax,
ymin, ymax,
zmin, zmax, pb_aspect=None):
"""
Produce a matrix that scales homogeneous coords in the specified ranges
to [0, 1], or [0, pb_aspect[i]] if the plotbox aspect ratio is specified.
"""
dx = xmax - xmin
dy = ymax - ymin
dz = zmax - zmin
if pb_aspect is not None:
ax, ay, az = pb_aspect
dx /= ax
dy /= ay
dz /= az
return np.array([[1/dx, 0, 0, -xmin/dx],
[ 0, 1/dy, 0, -ymin/dy],
[ 0, 0, 1/dz, -zmin/dz],
[ 0, 0, 0, 1]])
def _rotation_about_vector(v, angle):
"""
Produce a rotation matrix for an angle in radians about a vector.
"""
vx, vy, vz = v / np.linalg.norm(v)
s = np.sin(angle)
c = np.cos(angle)
t = 2*np.sin(angle/2)**2 # more numerically stable than t = 1-c
R = np.array([
[t*vx*vx + c, t*vx*vy - vz*s, t*vx*vz + vy*s],
[t*vy*vx + vz*s, t*vy*vy + c, t*vy*vz - vx*s],
[t*vz*vx - vy*s, t*vz*vy + vx*s, t*vz*vz + c]])
return R
def _view_axes(E, R, V, roll):
"""
Get the unit viewing axes in data coordinates.
Parameters
----------
E : 3-element numpy array
The coordinates of the eye/camera.
R : 3-element numpy array
The coordinates of the center of the view box.
V : 3-element numpy array
Unit vector in the direction of the vertical axis.
roll : float
The roll angle in radians.
Returns
-------
u : 3-element numpy array
Unit vector pointing towards the right of the screen.
v : 3-element numpy array
Unit vector pointing towards the top of the screen.
w : 3-element numpy array
Unit vector pointing out of the screen.
"""
w = (E - R)
w = w/np.linalg.norm(w)
u = np.cross(V, w)
u = u/np.linalg.norm(u)
v = np.cross(w, u) # Will be a unit vector
# Save some computation for the default roll=0
if roll != 0:
# A positive rotation of the camera is a negative rotation of the world
Rroll = _rotation_about_vector(w, -roll)
u = np.dot(Rroll, u)
v = np.dot(Rroll, v)
return u, v, w
def _view_transformation_uvw(u, v, w, E):
"""
Return the view transformation matrix.
Parameters
----------
u : 3-element numpy array
Unit vector pointing towards the right of the screen.
v : 3-element numpy array
Unit vector pointing towards the top of the screen.
w : 3-element numpy array
Unit vector pointing out of the screen.
E : 3-element numpy array
The coordinates of the eye/camera.
"""
Mr = np.eye(4)
Mt = np.eye(4)
Mr[:3, :3] = [u, v, w]
Mt[:3, -1] = -E
M = np.dot(Mr, Mt)
return M
def _persp_transformation(zfront, zback, focal_length):
e = focal_length
a = 1 # aspect ratio
b = (zfront+zback)/(zfront-zback)
c = -2*(zfront*zback)/(zfront-zback)
proj_matrix = np.array([[e, 0, 0, 0],
[0, e/a, 0, 0],
[0, 0, b, c],
[0, 0, -1, 0]])
return proj_matrix
def _ortho_transformation(zfront, zback):
# note: w component in the resulting vector will be (zback-zfront), not 1
a = -(zfront + zback)
b = -(zfront - zback)
proj_matrix = np.array([[2, 0, 0, 0],
[0, 2, 0, 0],
[0, 0, -2, 0],
[0, 0, a, b]])
return proj_matrix
def _proj_transform_vec(vec, M):
vecw = np.dot(M, vec.data)
w = vecw[3]
txs, tys, tzs = vecw[0]/w, vecw[1]/w, vecw[2]/w
if np.ma.isMA(vec[0]): # we check each to protect for scalars
txs = np.ma.array(txs, mask=vec[0].mask)
if np.ma.isMA(vec[1]):
tys = np.ma.array(tys, mask=vec[1].mask)
if np.ma.isMA(vec[2]):
tzs = np.ma.array(tzs, mask=vec[2].mask)
return txs, tys, tzs
def _proj_transform_vec_clip(vec, M, focal_length):
vecw = np.dot(M, vec.data)
w = vecw[3]
txs, tys, tzs = vecw[0] / w, vecw[1] / w, vecw[2] / w
if np.isinf(focal_length): # don't clip orthographic projection
tis = np.ones(txs.shape, dtype=bool)
else:
tis = (-1 <= txs) & (txs <= 1) & (-1 <= tys) & (tys <= 1) & (tzs <= 0)
if np.ma.isMA(vec[0]):
tis = tis & ~vec[0].mask
if np.ma.isMA(vec[1]):
tis = tis & ~vec[1].mask
if np.ma.isMA(vec[2]):
tis = tis & ~vec[2].mask
txs = np.ma.masked_array(txs, ~tis)
tys = np.ma.masked_array(tys, ~tis)
tzs = np.ma.masked_array(tzs, ~tis)
return txs, tys, tzs, tis
def inv_transform(xs, ys, zs, invM):
"""
Transform the points by the inverse of the projection matrix, *invM*.
"""
vec = _vec_pad_ones(xs, ys, zs)
vecr = np.dot(invM, vec)
if vecr.shape == (4,):
vecr = vecr.reshape((4, 1))
for i in range(vecr.shape[1]):
if vecr[3][i] != 0:
vecr[:, i] = vecr[:, i] / vecr[3][i]
return vecr[0], vecr[1], vecr[2]
def _vec_pad_ones(xs, ys, zs):
if np.ma.isMA(xs) or np.ma.isMA(ys) or np.ma.isMA(zs):
return np.ma.array([xs, ys, zs, np.ones_like(xs)])
else:
return np.array([xs, ys, zs, np.ones_like(xs)])
def proj_transform(xs, ys, zs, M):
"""
Transform the points by the projection matrix *M*.
"""
vec = _vec_pad_ones(xs, ys, zs)
return _proj_transform_vec(vec, M)
@_api.deprecated("3.10")
def proj_transform_clip(xs, ys, zs, M):
return _proj_transform_clip(xs, ys, zs, M, focal_length=np.inf)
def _proj_transform_clip(xs, ys, zs, M, focal_length):
"""
Transform the points by the projection matrix
and return the clipping result
returns txs, tys, tzs, tis
"""
vec = _vec_pad_ones(xs, ys, zs)
return _proj_transform_vec_clip(vec, M, focal_length)
def _proj_points(points, M):
return np.column_stack(_proj_trans_points(points, M))
def _proj_trans_points(points, M):
points = np.asanyarray(points)
xs, ys, zs = points[:, 0], points[:, 1], points[:, 2]
return proj_transform(xs, ys, zs, M)
|
