def bestfit_circle_numpy(points):
"""Fit a circle through a set of points.
Parameters
----------
points : list
XYZ coordinates of the points.
Returns
-------
tuple
XYZ coordinates of the center of the circle, the normal vector of the
local frame, and the radius of the circle.
Notes
-----
For more information see [1]_.
References
----------
.. [1] Scipy. *Least squares circle*.
Available at: http://scipy-cookbook.readthedocs.io/items/Least_Squares_Circle.html.
Examples
--------
.. code-block:: python
#
"""
o, uvw, _ = pca_numpy(points)
rst = local_coords_numpy(o, uvw, points)
x = rst[:, 0]
y = rst[:, 1]
def dist(xc, yc):
return sqrt((x - xc) ** 2 + (y - yc) ** 2)
def f(c):
Ri = dist(*c)
return Ri - Ri.mean()
xm = mean(x)
ym = mean(y)
c0 = xm, ym
c, ier = leastsq(f, c0)
Ri = dist(*c)
R = Ri.mean()
residu = sum((Ri - R) ** 2)
print(residu)
xyz = global_coords_numpy(o, uvw, [[c[0], c[1], 0.0]])[0]
o = xyz.tolist()
u, v, w = uvw.tolist()
return o, w, R
def oriented_bounding_box_numpy(points):
"""Compute the oriented minimum bounding box of a set of points in 3D space.
Notes
-----
The implementation is based on the convex hull of the points.
Parameters
----------
points : list
XYZ coordinates of the points.
Returns
-------
list
The XYZ coordinates of the corners of the bounding box.
Examples
--------
.. plot::
:include-source:
from numpy.random import randint
from numpy.random import rand
import matplotlib.pyplot as plt
from compas.plotters import Bounds
from compas.plotters import Cloud3D
from compas.plotters import Box
from compas.plotters import create_axes_3d
from compas.geometry import matrix_from_axis_and_angle
from compas.geometry import transform_points
from compas.geometry import oriented_bounding_box_numpy
clouds = []
for i in range(8):
a = randint(1, high=8) * 10 * 3.14159 / 180
d = [1, 1, 1]
cloud = rand(100, 3)
if i in (1, 2, 5, 6):
cloud[:, 0] *= - 10.0
cloud[:, 0] -= 3.0
d[0] = -1
else:
cloud[:, 0] *= 10.0
cloud[:, 0] += 3.0
if i in (2, 3, 6, 7):
cloud[:, 1] *= - 3.0
cloud[:, 1] -= 3.0
d[1] = -1
else:
cloud[:, 1] *= 3.0
cloud[:, 1] += 3.0
if i in (4, 5, 6, 7):
cloud[:, 2] *= - 6.0
cloud[:, 2] -= 3.0
d[2] = -1
else:
cloud[:, 2] *= 6.0
cloud[:, 2] += 3.0
R = matrix_from_axis_and_angle(d, a)
cloud[:] = transform_points(cloud, R)
clouds.append(cloud.tolist())
axes = create_axes_3d()
bounds = Bounds([point for points in clouds for point in points])
bounds.plot(axes)
for cloud in clouds:
bbox = oriented_bounding_box_numpy(cloud)
Cloud3D(cloud).plot(axes)
Box(bbox[1]).plot(axes)
plt.show()
"""
points = asarray(points)
n, dim = points.shape
assert 2 < dim, "The point coordinates should be at least 3D: %i" % dim
points = points[:, :3]
hull = ConvexHull(points)
volume = None
bbox = []
# this can be vectorised!
for simplex in hull.simplices:
abc = points[simplex]
uvw = local_axes(abc[0], abc[1], abc[2])
xyz = points[hull.vertices]
rst = local_coords_numpy(abc[0], uvw, xyz)
dr, ds, dt = ptp(rst, axis=0)
v = dr * ds * dt
if volume is None or v < volume:
rmin, smin, tmin = argmin(rst, axis=0)
rmax, smax, tmax = argmax(rst, axis=0)
bbox = [
[rst[rmin, 0], rst[smin, 1], rst[tmin, 2]],
[rst[rmax, 0], rst[smin, 1], rst[tmin, 2]],
[rst[rmax, 0], rst[smax, 1], rst[tmin, 2]],
[rst[rmin, 0], rst[smax, 1], rst[tmin, 2]],
[rst[rmin, 0], rst[smin, 1], rst[tmax, 2]],
[rst[rmax, 0], rst[smin, 1], rst[tmax, 2]],
[rst[rmax, 0], rst[smax, 1], rst[tmax, 2]],
[rst[rmin, 0], rst[smax, 1], rst[tmax, 2]],
]
bbox = global_coords_numpy(abc[0], uvw, bbox)
volume = v
return hull, bbox, volume
def oriented_bounding_box_numpy(points):
"""Compute the oriented minimum bounding box of a set of points in 3D space.
Parameters
----------
points : array-like
XYZ coordinates of the points.
Returns
-------
array
XYZ coordinates of 8 points defining a box.
Raises
------
QhullError
If the data is essentially 2D.
Notes
-----
The *oriented (minimum) bounding box* (OBB) of a given set of points
is computed using the following procedure:
1. Compute the convex hull of the points.
2. For each of the faces on the hull:
1. Compute face frame.
2. Compute coordinates of other points in face frame.
3. Find "peak-to-peak" (PTP) values of point coordinates along local axes.
4. Compute volume of box formed with PTP values.
3. Select the box with the smallest volume.
Examples
--------
Generate a random set of points with
:math:`x \in [0, 10]`, :math:`y \in [0, 1]` and :math:`z \in [0, 3]`.
Add the corners of the box such that we now the volume is supposed to be :math:`30.0`.
>>> points = numpy.random.rand(10000, 3)
>>> bottom = numpy.array([[0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [1.0, 1.0, 0.0]])
>>> top = numpy.array([[0.0, 0.0, 1.0], [1.0, 0.0, 1.0], [0.0, 1.0, 1.0], [1.0, 1.0, 1.0]])
>>> points = numpy.concatenate((points, bottom, top))
>>> points[:, 0] *= 10
>>> points[:, 2] *= 3
Rotate the points around an arbitrary axis by an arbitrary angle.
>>> from compas.geometry import Rotation
>>> from compas.geometry import transform_points_numpy
>>> R = Rotation.from_axis_and_angle([1.0, 1.0, 0.0], 0.3 * 3.14159)
>>> points = transform_points_numpy(points, R)
Compute the volume of the oriented bounding box.
>>> bbox = oriented_bounding_box_numpy(points)
>>> a = length_vector(subtract_vectors(bbox[1], bbox[0]))
>>> b = length_vector(subtract_vectors(bbox[3], bbox[0]))
>>> c = length_vector(subtract_vectors(bbox[4], bbox[0]))
>>> a * b * c
30.0
"""
points = asarray(points)
n, dim = points.shape
assert 2 < dim, "The point coordinates should be at least 3D: %i" % dim
points = points[:, :3]
# try:
# hull = ConvexHull(points)
# except Exception as e:
# if 'QH6154' in str(e):
# hull = ConvexHull(points, qhull_options='Qb2:0B2:0')
# else:
# raise e
hull = ConvexHull(points)
volume = None
bbox = []
# this can be vectorised!
for simplex in hull.simplices:
a, b, c = points[simplex]
uvw = local_axes(a, b, c)
xyz = points[hull.vertices]
rst = local_coords_numpy(a, uvw, xyz)
dr, ds, dt = ptp(rst, axis=0)
v = dr * ds * dt
if volume is None or v < volume:
rmin, smin, tmin = amin(rst, axis=0)
rmax, smax, tmax = amax(rst, axis=0)
bbox = [
[rmin, smin, tmin],
[rmax, smin, tmin],
[rmax, smax, tmin],
[rmin, smax, tmin],
[rmin, smin, tmax],
[rmax, smin, tmax],
[rmax, smax, tmax],
[rmin, smax, tmax],
]
bbox = global_coords_numpy(a, uvw, bbox)
volume = v
return bbox