def exp_domain(self, tangent_vec, base_point):
"""Compute the domain of the Euclidean exponential map.
Compute the real interval of time where the Euclidean geodesic starting
at point `base_point` in direction `tangent_vec` is defined.
Parameters
----------
tangent_vec : array-like, shape=[n_samples, n, n]
base_point : array-like, shape=[n_samples, n, n]
Returns
-------
exp_domain : array-like, shape=[n_samples, 2]
"""
base_point = gs.to_ndarray(base_point, to_ndim=3)
tangent_vec = gs.to_ndarray(tangent_vec, to_ndim=3)
invsqrt_base_point = gs.linalg.powerm(base_point, -.5)
reduced_vec = gs.matmul(invsqrt_base_point, tangent_vec)
reduced_vec = gs.matmul(reduced_vec, invsqrt_base_point)
eigvals = gs.linalg.eigvalsh(reduced_vec)
min_eig = gs.amin(eigvals, axis=1)
max_eig = gs.amax(eigvals, axis=1)
inf_value = gs.where(max_eig <= 0, -math.inf, -1 / max_eig)
inf_value = gs.to_ndarray(inf_value, to_ndim=2)
sup_value = gs.where(min_eig >= 0, math.inf, -1 / min_eig)
sup_value = gs.to_ndarray(sup_value, to_ndim=2)
domain = gs.concatenate((inf_value, sup_value), axis=1)
return domain
def exp_domain(tangent_vec, base_point):
"""Compute the domain of the Euclidean exponential map.
Compute the real interval of time where the Euclidean geodesic starting
at point `base_point` in direction `tangent_vec` is defined.
Parameters
----------
tangent_vec : array-like, shape=[..., n, n]
Tangent vector at base point.
base_point : array-like, shape=[..., n, n]
Base point.
Returns
-------
exp_domain : array-like, shape=[..., 2]
Interval of time where the geodesic is defined.
"""
invsqrt_base_point = SymmetricMatrices.powerm(base_point, -.5)
reduced_vec = gs.matmul(invsqrt_base_point, tangent_vec)
reduced_vec = gs.matmul(reduced_vec, invsqrt_base_point)
eigvals = gs.linalg.eigvalsh(reduced_vec)
min_eig = gs.amin(eigvals, axis=1)
max_eig = gs.amax(eigvals, axis=1)
inf_value = gs.where(
max_eig <= 0., gs.array(-math.inf), - 1. / max_eig)
inf_value = gs.to_ndarray(inf_value, to_ndim=2)
sup_value = gs.where(
min_eig >= 0., gs.array(-math.inf), - 1. / min_eig)
sup_value = gs.to_ndarray(sup_value, to_ndim=2)
domain = gs.concatenate((inf_value, sup_value), axis=1)
return domain
def diameter(self, points):
"""Give the distance between two farthest points.
Distance between the two points that are farthest away from each other
in points.
Parameters
----------
points : array-like, shape=[..., dim]
Points.
Returns
-------
diameter : float
Distance between two farthest points.
"""
diameter = 0.0
n_points = points.shape[0]
for i in range(n_points - 1):
dist_to_neighbors = self.dist(points[i, :], points[i + 1:, :])
dist_to_farthest_neighbor = gs.amax(dist_to_neighbors)
diameter = gs.maximum(diameter, dist_to_farthest_neighbor)
return diameter
def plot(self, points, ax=None, space=None, **point_draw_kwargs):
if space == "SE3_GROUP":
ax_s = AX_SCALE * gs.amax(gs.abs(points[:, 3:6]))
elif space == "SO3_GROUP":
ax_s = AX_SCALE * gs.amax(gs.abs(points[:, :3]))
ax_s = float(ax_s)
bounds = (-ax_s, ax_s)
plt.setp(
ax,
xlim=bounds,
ylim=bounds,
zlim=bounds,
xlabel="X",
ylabel="Y",
zlabel="Z",
)
trihedrons = convert_to_trihedron(points, space=space)
for t in trihedrons:
t.draw(ax, **point_draw_kwargs)
def diameter(self, points):
"""
Distance between the two points that are farthest away from each other
in points.
"""
diameter = 0.0
n_points = points.shape[0]
for i in range(n_points - 1):
dist_to_neighbors = self.dist(points[i, :], points[i + 1:, :])
dist_to_farthest_neighbor = gs.amax(dist_to_neighbors)
diameter = gs.maximum(diameter, dist_to_farthest_neighbor)
return diameter
def exp_domain(self, tangent_vec, base_point):
base_point = gs.to_ndarray(base_point, to_ndim=3)
tangent_vec = gs.to_ndarray(tangent_vec, to_ndim=3)
invsqrt_base_point = gs.linalg.powerm(base_point, -.5)
reduced_vec = gs.matmul(invsqrt_base_point, tangent_vec)
reduced_vec = gs.matmul(reduced_vec, invsqrt_base_point)
eigvals = gs.linalg.eigvalsh(reduced_vec)
min_eig = gs.amin(eigvals, axis=1)
max_eig = gs.amax(eigvals, axis=1)
inf_value = gs.where(max_eig <= 0, -math.inf, -1 / max_eig)
inf_value = gs.to_ndarray(inf_value, to_ndim=2)
sup_value = gs.where(min_eig >= 0, math.inf, -1 / min_eig)
sup_value = gs.to_ndarray(sup_value, to_ndim=2)
domain = gs.concatenate((inf_value, sup_value), axis=1)
return domain
def plot(points, ax=None, space=None, **point_draw_kwargs):
"""Plot trihedrons."""
ax_s = AX_SCALE * gs.amax(gs.abs(points[:, :3]))
ax_s = float(ax_s)
bounds = (-ax_s, ax_s)
plt.setp(
ax,
xlim=bounds,
ylim=bounds,
zlim=bounds,
xlabel="X",
ylabel="Y",
zlabel="Z",
)
trihedrons = convert_to_trihedron(points, space=space)
for t in trihedrons:
t.draw(ax, **point_draw_kwargs)
def diameter(self, points):
"""Compute distance between the two points farthest from each other.
Parameters
----------
points
Returns
-------
diameter
"""
diameter = 0.0
n_points = points.shape[0]
for i in range(n_points - 1):
dist_to_neighbors = self.dist(points[i, :], points[i + 1:, :])
dist_to_farthest_neighbor = gs.amax(dist_to_neighbors)
diameter = gs.maximum(diameter, dist_to_farthest_neighbor)
return diameter
def test_geodesic(self):
"""Test geodesic.
Check that the norm of the velocity is constant.
"""
initial_point = self.categorical.random_point()
end_point = self.categorical.random_point()
n_steps = 100
geod = self.metric.geodesic(initial_point=initial_point,
end_point=end_point)
t = gs.linspace(0.0, 1.0, n_steps)
geod_at_t = geod(t)
velocity = n_steps * (geod_at_t[1:, :] - geod_at_t[:-1, :])
velocity_norm = self.metric.norm(velocity, geod_at_t[:-1, :])
result = (1 / gs.amin(velocity_norm) *
(gs.amax(velocity_norm) - gs.amin(velocity_norm)))
expected = 0.0
self.assertAllClose(expected, result, rtol=1.0)
def plot(points,
ax=None,
space=None,
point_type='extrinsic',
**point_draw_kwargs):
"""Plot points in the 3D Special Euclidean Group.
Plot points in the 3D Special Euclidean Group,
by showing them as trihedrons.
"""
if space not in IMPLEMENTED:
raise NotImplementedError(
'The plot function is not implemented'
' for space {}. The spaces available for visualization'
' are: {}.'.format(space, IMPLEMENTED))
if points is None:
raise ValueError("No points given for plotting.")
points = gs.to_ndarray(points, to_ndim=2)
if space in ('SO3_GROUP', 'SE3_GROUP'):
if ax is None:
ax = plt.subplot(111, projection='3d')
if space == 'SE3_GROUP':
ax_s = AX_SCALE * gs.amax(gs.abs(points[:, 3:6]))
elif space == 'SO3_GROUP':
ax_s = AX_SCALE * gs.amax(gs.abs(points[:, :3]))
plt.setp(ax,
xlim=(-ax_s, ax_s),
ylim=(-ax_s, ax_s),
zlim=(-ax_s, ax_s),
xlabel='X',
ylabel='Y',
zlabel='Z')
trihedrons = convert_to_trihedron(points, space=space)
for t in trihedrons:
t.draw(ax, **point_draw_kwargs)
elif space == 'S1':
circle = Circle()
ax = circle.set_ax(ax=ax)
circle.add_points(points)
circle.draw(ax, **point_draw_kwargs)
elif space == 'S2':
sphere = Sphere()
ax = sphere.set_ax(ax=ax)
sphere.add_points(points)
sphere.draw(ax, **point_draw_kwargs)
elif space == 'H2_poincare_disk':
poincare_disk = PoincareDisk(point_type=point_type)
ax = poincare_disk.set_ax(ax=ax)
poincare_disk.add_points(points)
poincare_disk.draw(ax, **point_draw_kwargs)
elif space == 'poincare_polydisk':
n_disks = points.shape[1]
poincare_poly_disk = PoincarePolyDisk(point_type=point_type,
n_disks=n_disks)
n_columns = gs.ceil(n_disks**0.5)
n_rows = gs.ceil(n_disks / n_columns)
axis_list = []
for i_disk in range(n_disks):
axis_list.append(ax.add_subplot(n_rows, n_columns, i_disk + 1))
for i_disk, ax in enumerate(axis_list):
ax = poincare_poly_disk.set_ax(ax=ax)
poincare_poly_disk.clear_points()
poincare_poly_disk.add_points(points[:, i_disk, ...])
poincare_poly_disk.draw(ax, **point_draw_kwargs)
elif space == 'H2_poincare_half_plane':
poincare_half_plane = PoincareHalfPlane()
ax = poincare_half_plane.set_ax(ax=ax)
poincare_half_plane.add_points(points)
poincare_half_plane.draw(ax, **point_draw_kwargs)
elif space == 'H2_klein_disk':
klein_disk = KleinDisk()
ax = klein_disk.set_ax(ax=ax)
klein_disk.add_points(points)
klein_disk.draw(ax, **point_draw_kwargs)
return ax
def plot(points, ax=None, space=None, point_type=None, **point_draw_kwargs):
"""Plot points in one of the implemented manifolds.
The implemented manifolds are:
- the special orthogonal group SO(3)
- the special Euclidean group SE(3)
- the circle S1 and the sphere S2
- the hyperbolic plane (the Poincare disk, the Poincare
half plane and the Klein disk)
- the Poincare polydisk
- the Kendall shape space of 2D triangles
- the Kendall shape space of 3D triangles
Parameters
----------
points : array-like, shape=[..., dim]
Points to be plotted.
space: str, optional, {'SO3_GROUP', 'SE3_GROUP', 'S1', 'S2',
'H2_poincare_disk', 'H2_poincare_half_plane', 'H2_klein_disk',
'poincare_polydisk', 'S32', 'M32', 'S33', 'M33'}
point_type: str, optional, {'extrinsic', 'ball', 'half-space', 'pre-shape'}
"""
if space not in IMPLEMENTED:
raise NotImplementedError(
'The plot function is not implemented'
' for space {}. The spaces available for visualization'
' are: {}.'.format(space, IMPLEMENTED))
if points is None:
raise ValueError("No points given for plotting.")
if points.ndim < 2:
points = gs.expand_dims(points, 0)
if space in ('SO3_GROUP', 'SE3_GROUP'):
if ax is None:
ax = plt.subplot(111, projection='3d')
if space == 'SE3_GROUP':
ax_s = AX_SCALE * gs.amax(gs.abs(points[:, 3:6]))
elif space == 'SO3_GROUP':
ax_s = AX_SCALE * gs.amax(gs.abs(points[:, :3]))
ax_s = float(ax_s)
bounds = (-ax_s, ax_s)
plt.setp(ax,
xlim=bounds,
ylim=bounds,
zlim=bounds,
xlabel='X',
ylabel='Y',
zlabel='Z')
trihedrons = convert_to_trihedron(points, space=space)
for t in trihedrons:
t.draw(ax, **point_draw_kwargs)
elif space == 'S1':
circle = Circle()
ax = circle.set_ax(ax=ax)
circle.add_points(points)
circle.draw(ax, **point_draw_kwargs)
elif space == 'S2':
sphere = Sphere()
ax = sphere.set_ax(ax=ax)
sphere.add_points(points)
sphere.draw(ax, **point_draw_kwargs)
elif space == 'H2_poincare_disk':
if point_type is None:
point_type = 'extrinsic'
poincare_disk = PoincareDisk(point_type=point_type)
ax = poincare_disk.set_ax(ax=ax)
poincare_disk.add_points(points)
poincare_disk.draw(ax, **point_draw_kwargs)
plt.axis('off')
elif space == 'poincare_polydisk':
if point_type is None:
point_type = 'extrinsic'
n_disks = points.shape[1]
poincare_poly_disk = PoincarePolyDisk(point_type=point_type,
n_disks=n_disks)
n_columns = int(gs.ceil(n_disks**0.5))
n_rows = int(gs.ceil(n_disks / n_columns))
axis_list = []
for i_disk in range(n_disks):
axis_list.append(ax.add_subplot(n_rows, n_columns, i_disk + 1))
for i_disk, one_ax in enumerate(axis_list):
ax = poincare_poly_disk.set_ax(ax=one_ax)
poincare_poly_disk.clear_points()
poincare_poly_disk.add_points(points[:, i_disk, ...])
poincare_poly_disk.draw(ax, **point_draw_kwargs)
elif space == 'H2_poincare_half_plane':
if point_type is None:
point_type = 'half-space'
poincare_half_plane = PoincareHalfPlane(point_type=point_type)
ax = poincare_half_plane.set_ax(points=points, ax=ax)
poincare_half_plane.add_points(points)
poincare_half_plane.draw(ax, **point_draw_kwargs)
elif space == 'H2_klein_disk':
klein_disk = KleinDisk()
ax = klein_disk.set_ax(ax=ax)
klein_disk.add_points(points)
klein_disk.draw(ax, **point_draw_kwargs)
elif space == 'SE2_GROUP':
plane = SpecialEuclidean2()
ax = plane.set_ax(ax=ax)
plane.add_points(points)
plane.draw(ax, **point_draw_kwargs)
elif space == 'S32':
sphere = KendallSphere()
sphere.add_points(points)
sphere.draw()
sphere.draw_points()
ax = sphere.ax
elif space == 'M32':
sphere = KendallSphere(point_type='extrinsic')
sphere.add_points(points)
sphere.draw()
sphere.draw_points()
ax = sphere.ax
elif space == 'S33':
disk = KendallDisk()
disk.add_points(points)
disk.draw()
disk.draw_points()
ax = disk.ax
elif space == 'M33':
disk = KendallDisk(point_type='extrinsic')
disk.add_points(points)
disk.draw()
disk.draw_points()
ax = disk.ax
return ax
def plot(points, ax=None, space=None, **point_draw_kwargs):
"""
Plot points in the 3D Special Euclidean Group,
by showing them as trihedrons.
"""
if space not in IMPLEMENTED:
raise NotImplementedError(
'The plot function is not implemented'
' for space {}. The spaces available for visualization'
' are: {}.'.format(space, IMPLEMENTED))
if points is None:
raise ValueError("No points given for plotting.")
points = gs.to_ndarray(points, to_ndim=2)
if space in ('SO3_GROUP', 'SE3_GROUP'):
if ax is None:
ax = plt.subplot(111, projection='3d')
if space == 'SE3_GROUP':
ax_s = AX_SCALE * gs.amax(gs.abs(points[:, 3:6]))
elif space == 'SO3_GROUP':
ax_s = AX_SCALE * gs.amax(gs.abs(points[:, :3]))
plt.setp(ax,
xlim=(-ax_s, ax_s),
ylim=(-ax_s, ax_s),
zlim=(-ax_s, ax_s),
xlabel='X',
ylabel='Y',
zlabel='Z')
trihedrons = convert_to_trihedron(points, space=space)
for t in trihedrons:
t.draw(ax, **point_draw_kwargs)
elif space == 'S1':
circle = Circle()
ax = circle.set_ax(ax=ax)
circle.add_points(points)
circle.draw(ax, **point_draw_kwargs)
elif space == 'S2':
sphere = Sphere()
ax = sphere.set_ax(ax=ax)
sphere.add_points(points)
sphere.draw(ax, **point_draw_kwargs)
elif space == 'H2_poincare_disk':
poincare_disk = PoincareDisk()
ax = poincare_disk.set_ax(ax=ax)
poincare_disk.add_points(points)
poincare_disk.draw(ax, **point_draw_kwargs)
elif space == 'H2_poincare_half_plane':
poincare_half_plane = PoincareHalfPlane()
ax = poincare_half_plane.set_ax(ax=ax)
poincare_half_plane.add_points(points)
poincare_half_plane.draw(ax, **point_draw_kwargs)
elif space == 'H2_klein_disk':
klein_disk = KleinDisk()
ax = klein_disk.set_ax(ax=ax)
klein_disk.add_points(points)
klein_disk.draw(ax, **point_draw_kwargs)
return ax
