def setUp(self):
self.n_samples = 10
self.SO3_GROUP = SpecialOrthogonal(n=3, point_type='vector')
self.SE3_GROUP = SpecialEuclidean(n=3, point_type='vector')
self.S1 = Hypersphere(dim=1)
self.S2 = Hypersphere(dim=2)
self.H2 = Hyperbolic(dim=2)
self.H2_half_plane = PoincareHalfSpace(dim=2)
self.M32 = Matrices(m=3, n=2)
self.S32 = PreShapeSpace(k_landmarks=3, m_ambient=2)
self.KS = visualization.KendallSphere()
self.M33 = Matrices(m=3, n=3)
self.S33 = PreShapeSpace(k_landmarks=3, m_ambient=3)
self.KD = visualization.KendallDisk()
plt.figure()
def setup_method(self):
gs.random.seed(1234)
self.k_landmarks = 4
self.m_ambient = 3
self.space = PreShapeSpace(self.k_landmarks, self.m_ambient)
self.matrices = self.space.embedding_space
self.n_samples = 10
self.shape_metric = KendallShapeMetric(self.k_landmarks,
self.m_ambient)
self.base_point = self.space.random_point()
vector = gs.random.rand(11, self.k_landmarks, self.m_ambient)
tg_vec_0 = self.space.to_tangent(vector[0], self.base_point)
self.hor_x = self.space.horizontal_projection(tg_vec_0,
self.base_point)
tg_vec_1 = self.space.to_tangent(vector[1], self.base_point)
self.hor_y = self.space.horizontal_projection(tg_vec_1,
self.base_point)
tg_vec_2 = self.space.to_tangent(vector[2], self.base_point)
self.hor_z = self.space.horizontal_projection(tg_vec_2,
self.base_point)
tg_vec_3 = self.space.to_tangent(vector[3], self.base_point)
self.hor_h = self.space.horizontal_projection(tg_vec_3,
self.base_point)
tg_vec_4 = self.space.to_tangent(vector[4], self.base_point)
self.ver_v = self.space.vertical_projection(tg_vec_4, self.base_point)
tg_vec_5 = self.space.to_tangent(vector[5], self.base_point)
self.ver_w = self.space.vertical_projection(tg_vec_5, self.base_point)
tg_vec_6 = self.space.to_tangent(vector[6], self.base_point)
hor_dy = self.space.horizontal_projection(tg_vec_6, self.base_point)
tg_vec_7 = self.space.to_tangent(vector[7], self.base_point)
hor_dz = self.space.horizontal_projection(tg_vec_7, self.base_point)
tg_vec_8 = self.space.to_tangent(vector[8], self.base_point)
ver_dv = self.space.vertical_projection(tg_vec_8, self.base_point)
tg_vec_9 = self.space.to_tangent(vector[9], self.base_point)
ver_dw = self.space.vertical_projection(tg_vec_9, self.base_point)
tg_vec_10 = self.space.to_tangent(vector[10], self.base_point)
hor_dh = self.space.horizontal_projection(tg_vec_10, self.base_point)
# generate valid derivatives of horizontal / vertical vector fields.
a_x_y = self.space.integrability_tensor(self.hor_x, self.hor_y,
self.base_point)
self.nabla_x_y = hor_dy + a_x_y
a_x_z = self.space.integrability_tensor(self.hor_x, self.hor_z,
self.base_point)
self.nabla_x_z = hor_dz + a_x_z
a_x_v = self.space.integrability_tensor(self.hor_x, self.ver_v,
self.base_point)
self.nabla_x_v = ver_dv + a_x_v
a_x_w = self.space.integrability_tensor(self.hor_x, self.ver_w,
self.base_point)
self.nabla_x_w = ver_dw + a_x_w
a_x_h = self.space.integrability_tensor(self.hor_x, self.hor_h,
self.base_point)
self.nabla_x_h = hor_dh + a_x_h
def setUp(self):
gs.random.seed(1234)
self.k_landmarks = 4
self.m_ambient = 3
self.space = PreShapeSpace(self.k_landmarks, self.m_ambient)
self.matrices = self.space.embedding_manifold
self.n_samples = 10
self.shape_metric = KendallShapeMetric(self.k_landmarks,
self.m_ambient)
def fit(self):
"""
This is the main entry of fitting PNS to data
"""
## 0. make sure the data are distributed on a unit sphere
d, n = self.data.shape
if not is_on_unit_sphere(self.data):
print("Mapping data to preshape space")
data_in_3d = np.reshape(self.data, (-1, 3, n))
_, k_landmarks, _ = data_in_3d.shape
from geomstats.geometry.pre_shape import PreShapeSpace
preshape = PreShapeSpace(m_ambient=3, k_landmarks=k_landmarks)
data_preshape = preshape.projection(data_in_3d)
base_point = data_preshape[0]
data_shape = preshape.align(point=data_preshape,
base_point=base_point)
self.data = np.reshape(data_shape, (d, n))
## 1. rotate data to get a tight space, excluding the null space
eps = 1e-15
u, s, _ = np.linalg.svd(self.data, full_matrices=False)
small_singular_val = np.where(s < eps)[0]
maxd = len(small_singular_val)
if maxd == 0:
maxd = np.min([d, n]) + 1
## the dimension of null space
nullspdim = d - maxd + 1
## 2. intrinsic dimension of sphere is 1 dimension lower than extrinsic_dim
dm = maxd - 2
basisu = []
if nullspdim > 0:
basisu = u[:, :dm + 1]
## extract the signal by projecting to the kernel space (complementary of the null space)
currentSphere = np.matmul(u[:, :dm + 1].T, self.data)
else:
currentSphere = self.data
if self.itype == 9:
## Use hypothesis testing (Kurtosis test) to decide whether great or small circle for EACH subsphere
self.output = self.automatic_fit_subspheres(
currentSphere, dm, nullspdim, basisu)
else:
## Otherwise, always fit data with one particular circle type (great or small)
self.output = self.fit_with_subspheres(currentSphere, dm,
nullspdim, basisu)
def integrability_tensor_test_data(self):
space = PreShapeSpace(4, 3)
vector = gs.random.rand(2, 4, 3)
base_point = space.random_point()
random_data = [
dict(
k_landmarks=4,
m_ambient=3,
tangent_vec_a=space.to_tangent(vector[0], base_point),
tangent_vec_b=space.to_tangent(vector[1], base_point),
base_point=base_point,
)
]
return self.generate_tests(random_data)
def vertical_projection_test_data(self):
vector = gs.random.rand(10, 4, 3)
space = PreShapeSpace(4, 3)
point = space.random_point()
smoke_data = [
dict(
k_landmarks=4,
m_ambient=3,
tangent_vec=space.to_tangent(vector[0], point),
point=point,
),
dict(
k_landmarks=4,
m_ambient=3,
tangent_vec=space.to_tangent(vector, point),
point=point,
),
]
return self.generate_tests(smoke_data)
def parallel_transport_test_data(self):
k_landmarks = 4
m_ambient = 3
n_samples = 10
space = PreShapeSpace(4, 3)
base_point = space.projection(gs.eye(4)[:, :3])
vec_a = gs.random.rand(n_samples, k_landmarks, m_ambient)
tangent_vec_a = space.to_tangent(space.center(vec_a), base_point)
vec_b = gs.random.rand(n_samples, k_landmarks, m_ambient)
tangent_vec_b = space.to_tangent(space.center(vec_b), base_point)
smoke_data = [
dict(
k_landmarks=k_landmarks,
m_ambient=m_ambient,
tangent_vec_a=tangent_vec_a,
tangent_vec_b=tangent_vec_b,
base_point=base_point,
)
]
return self.generate_tests(smoke_data)
def alignment_is_symmetric_test_data(self):
space = PreShapeSpace(4, 3)
random_data = [
dict(
k_landmarks=4,
m_ambient=3,
point=space.random_point(),
base_point=space.random_point(),
),
dict(
k_landmarks=4,
m_ambient=3,
point=space.random_point(),
base_point=space.random_point(2),
),
dict(
k_landmarks=4,
m_ambient=3,
point=space.random_point(2),
base_point=space.random_point(2),
),
]
return self.generate_tests([], random_data)
from geomstats.geometry.poincare_half_space import PoincareHalfSpace
from geomstats.geometry.pre_shape import KendallShapeMetric, PreShapeSpace
from geomstats.geometry.special_euclidean import SpecialEuclidean
from geomstats.geometry.special_orthogonal import SpecialOrthogonal
from mpl_toolkits.mplot3d import Axes3D # NOQA
SE3_GROUP = SpecialEuclidean(n=3, point_type='vector')
SE2_GROUP = SpecialEuclidean(n=2, point_type='matrix')
SE2_VECT = SpecialEuclidean(n=2, point_type='vector')
SO3_GROUP = SpecialOrthogonal(n=3, point_type='vector')
S1 = Hypersphere(dim=1)
S2 = Hypersphere(dim=2)
H2 = Hyperboloid(dim=2)
POINCARE_HALF_PLANE = PoincareHalfSpace(dim=2)
M32 = Matrices(m=3, n=2)
S32 = PreShapeSpace(k_landmarks=3, m_ambient=2)
METRIC_S32 = KendallShapeMetric(k_landmarks=3, m_ambient=2)
M33 = Matrices(m=3, n=3)
S33 = PreShapeSpace(k_landmarks=3, m_ambient=3)
METRIC_S33 = KendallShapeMetric(k_landmarks=3, m_ambient=3)
AX_SCALE = 1.2
IMPLEMENTED = [
'SO3_GROUP', 'SE3_GROUP', 'SE2_GROUP', 'S1', 'S2', 'H2_poincare_disk',
'H2_poincare_half_plane', 'H2_klein_disk', 'poincare_polydisk', 'S32',
'M32', 'S33', 'M33'
]
def tutorial_matplotlib():
class PreShapeMetricTestData(_RiemannianMetricTestData):
k_landmarks_list = random.sample(range(3, 6), 2)
m_ambient_list = [random.sample(range(2, n), 1)[0] for n in k_landmarks_list]
metric_args_list = list(zip(k_landmarks_list, m_ambient_list))
shape_list = metric_args_list
space_list = [PreShapeSpace(k, m) for k, m in metric_args_list]
n_points_list = random.sample(range(1, 7), 2)
n_samples_list = random.sample(range(1, 7), 2)
n_points_a_list = random.sample(range(1, 7), 2)
n_points_b_list = [1]
batch_size_list = random.sample(range(2, 7), 2)
n_tangent_vecs_list = random.sample(range(2, 7), 2)
alpha_list = [1] * 2
n_rungs_list = [1] * 2
scheme_list = ["pole"] * 2
def exp_shape_test_data(self):
return self._exp_shape_test_data(
self.metric_args_list, self.space_list, self.shape_list
)
def log_shape_test_data(self):
return self._log_shape_test_data(self.metric_args_list, self.space_list)
def squared_dist_is_symmetric_test_data(self):
return self._squared_dist_is_symmetric_test_data(
self.metric_args_list,
self.space_list,
self.n_points_a_list,
self.n_points_b_list,
atol=gs.atol * 1000,
)
def exp_belongs_test_data(self):
return self._exp_belongs_test_data(
self.metric_args_list,
self.space_list,
self.shape_list,
self.n_samples_list,
belongs_atol=gs.atol * 1000,
)
def log_is_tangent_test_data(self):
return self._log_is_tangent_test_data(
self.metric_args_list,
self.space_list,
self.n_samples_list,
is_tangent_atol=gs.atol * 1000,
)
def geodesic_ivp_belongs_test_data(self):
return self._geodesic_ivp_belongs_test_data(
self.metric_args_list,
self.space_list,
self.shape_list,
self.n_points_list,
belongs_atol=gs.atol * 1000,
)
def geodesic_bvp_belongs_test_data(self):
return self._geodesic_bvp_belongs_test_data(
self.metric_args_list,
self.space_list,
self.n_points_list,
belongs_atol=gs.atol * 1000,
)
def exp_after_log_test_data(self):
return self._exp_after_log_test_data(
self.metric_args_list,
self.space_list,
self.n_samples_list,
rtol=gs.rtol * 100,
atol=1e-4,
)
def log_after_exp_test_data(self):
return self._log_after_exp_test_data(
self.metric_args_list,
self.space_list,
self.shape_list,
self.n_samples_list,
rtol=gs.rtol * 100,
atol=1e-2,
)
def exp_ladder_parallel_transport_test_data(self):
return self._exp_ladder_parallel_transport_test_data(
self.metric_args_list,
self.space_list,
self.shape_list,
self.n_samples_list,
self.n_rungs_list,
self.alpha_list,
self.scheme_list,
)
def exp_geodesic_ivp_test_data(self):
return self._exp_geodesic_ivp_test_data(
self.metric_args_list,
self.space_list,
self.shape_list,
self.n_samples_list,
self.n_points_list,
rtol=gs.rtol * 10000,
atol=gs.atol * 10000,
)
def parallel_transport_ivp_is_isometry_test_data(self):
return self._parallel_transport_ivp_is_isometry_test_data(
self.metric_args_list,
self.space_list,
self.shape_list,
self.n_samples_list,
is_tangent_atol=gs.atol * 1000,
atol=gs.atol * 1000,
)
def parallel_transport_bvp_is_isometry_test_data(self):
return self._parallel_transport_bvp_is_isometry_test_data(
self.metric_args_list,
self.space_list,
self.shape_list,
self.n_samples_list,
is_tangent_atol=gs.atol * 1000,
atol=gs.atol * 1000,
)
def dist_is_symmetric_test_data(self):
return self._dist_is_symmetric_test_data(
self.metric_args_list,
self.space_list,
self.n_points_a_list,
self.n_points_b_list,
)
def dist_is_positive_test_data(self):
return self._dist_is_positive_test_data(
self.metric_args_list,
self.space_list,
self.n_points_a_list,
self.n_points_b_list,
)
def squared_dist_is_positive_test_data(self):
return self._squared_dist_is_positive_test_data(
self.metric_args_list,
self.space_list,
self.n_points_a_list,
self.n_points_b_list,
)
def dist_is_norm_of_log_test_data(self):
return self._dist_is_norm_of_log_test_data(
self.metric_args_list,
self.space_list,
self.n_points_a_list,
self.n_points_b_list,
)
def dist_point_to_itself_is_zero_test_data(self):
return self._dist_point_to_itself_is_zero_test_data(
self.metric_args_list, self.space_list, self.n_points_list
)
def inner_product_is_symmetric_test_data(self):
return self._inner_product_is_symmetric_test_data(
self.metric_args_list,
self.space_list,
self.shape_list,
self.n_tangent_vecs_list,
)
def triangle_inequality_of_dist_test_data(self):
return self._triangle_inequality_of_dist_test_data(
self.metric_args_list, self.space_list, self.n_points_list
)
class KendallShapeMetricTestData(_RiemannianMetricTestData):
k_landmarks_list = random.sample(range(3, 6), 2)
m_ambient_list = [random.sample(range(2, n), 1)[0] for n in k_landmarks_list]
metric_args_list = list(zip(k_landmarks_list, m_ambient_list))
shape_list = metric_args_list
space_list = [PreShapeSpace(k, m) for k, m in metric_args_list]
n_points_list = random.sample(range(1, 4), 2)
n_samples_list = random.sample(range(1, 4), 2)
n_points_a_list = random.sample(range(1, 4), 2)
n_points_b_list = [1]
n_tangent_vecs_list = random.sample(range(1, 4), 2)
batch_size_list = random.sample(range(2, 4), 2)
alpha_list = [1] * 2
n_rungs_list = [1] * 2
scheme_list = ["pole"] * 2
def curvature_is_skew_operator_test_data(self):
base_point = smoke_space.random_point(2)
vec = gs.random.rand(4, 4, 3)
smoke_data = [dict(k_landmarks=4, m_ambient=3, vec=vec, base_point=base_point)]
return self.generate_tests(smoke_data)
def curvature_bianchi_identity_test_data(self):
smoke_data = [
dict(
k_landmarks=4,
m_ambient=3,
tangent_vec_a=tg_vec_0,
tangent_vec_b=tg_vec_1,
tangent_vec_cs=tg_vec_2,
base_point=base_point,
)
]
return self.generate_tests(smoke_data)
def kendall_sectional_curvature_test_data(self):
k_landmarks = 4
m_ambient = 3
space = smoke_space
n_samples = 4 * k_landmarks * m_ambient
base_point = space.random_point(1)
vec_a = gs.random.rand(n_samples, k_landmarks, m_ambient)
tg_vec_a = space.to_tangent(space.center(vec_a), base_point)
vec_b = gs.random.rand(n_samples, k_landmarks, m_ambient)
tg_vec_b = space.to_tangent(space.center(vec_b), base_point)
smoke_data = [
dict(
k_landmarks=4,
m_ambient=3,
tangent_vec_a=tg_vec_a,
tangent_vec_b=tg_vec_b,
base_point=base_point,
)
]
return self.generate_tests(smoke_data)
def kendall_curvature_derivative_bianchi_identity_test_data(self):
smoke_data = [
dict(
k_landmarks=4,
m_ambient=3,
hor_x=hor_x,
hor_y=hor_y,
hor_z=hor_z,
hor_h=hor_h,
base_point=base_point,
)
]
return self.generate_tests(smoke_data)
def curvature_derivative_is_skew_operator_test_data(self):
smoke_data = [
dict(
k_landmarks=4,
m_ambient=3,
hor_x=hor_x,
hor_y=hor_y,
hor_z=hor_z,
base_point=base_point,
)
]
return self.generate_tests(smoke_data)
def directional_curvature_derivative_test_data(self):
smoke_data = [
dict(
k_landmarks=4,
m_ambient=3,
hor_x=hor_x,
hor_y=hor_y,
base_point=base_point,
)
]
return self.generate_tests(smoke_data)
def directional_curvature_derivative_is_quadratic_test_data(self):
coef_x = -2.5
coef_y = 1.5
smoke_data = [
dict(
k_landmarks=4,
m_ambient=3,
coef_x=coef_x,
coef_y=coef_y,
hor_x=hor_x,
hor_y=hor_y,
base_point=base_point,
)
]
return self.generate_tests(smoke_data)
def parallel_transport_test_data(self):
k_landmarks = 4
m_ambient = 3
n_samples = 10
space = PreShapeSpace(4, 3)
base_point = space.projection(gs.eye(4)[:, :3])
vec_a = gs.random.rand(n_samples, k_landmarks, m_ambient)
tangent_vec_a = space.to_tangent(space.center(vec_a), base_point)
vec_b = gs.random.rand(n_samples, k_landmarks, m_ambient)
tangent_vec_b = space.to_tangent(space.center(vec_b), base_point)
smoke_data = [
dict(
k_landmarks=k_landmarks,
m_ambient=m_ambient,
tangent_vec_a=tangent_vec_a,
tangent_vec_b=tangent_vec_b,
base_point=base_point,
)
]
return self.generate_tests(smoke_data)
def exp_shape_test_data(self):
return self._exp_shape_test_data(
self.metric_args_list, self.space_list, self.shape_list
)
def log_shape_test_data(self):
return self._log_shape_test_data(self.metric_args_list, self.space_list)
def squared_dist_is_symmetric_test_data(self):
return self._squared_dist_is_symmetric_test_data(
self.metric_args_list,
self.space_list,
self.n_points_a_list,
self.n_points_b_list,
atol=gs.atol * 1000,
)
def exp_belongs_test_data(self):
return self._exp_belongs_test_data(
self.metric_args_list,
self.space_list,
self.shape_list,
self.n_samples_list,
belongs_atol=gs.atol * 1000,
)
def log_is_tangent_test_data(self):
return self._log_is_tangent_test_data(
self.metric_args_list,
self.space_list,
self.n_samples_list,
is_tangent_atol=gs.atol * 1000,
)
def geodesic_ivp_belongs_test_data(self):
return self._geodesic_ivp_belongs_test_data(
self.metric_args_list,
self.space_list,
self.shape_list,
self.n_points_list,
belongs_atol=gs.atol * 1000,
)
def geodesic_bvp_belongs_test_data(self):
return self._geodesic_bvp_belongs_test_data(
self.metric_args_list,
self.space_list,
self.n_points_list,
belongs_atol=gs.atol * 1000,
)
def exp_after_log_test_data(self):
return self._exp_after_log_test_data(
self.metric_args_list,
self.space_list,
self.n_samples_list,
rtol=gs.rtol * 100,
atol=gs.atol * 10000,
)
def log_after_exp_test_data(self):
return self._log_after_exp_test_data(
self.metric_args_list,
self.space_list,
self.shape_list,
self.n_samples_list,
rtol=gs.rtol * 100,
atol=gs.atol * 10000,
)
def exp_ladder_parallel_transport_test_data(self):
return self._exp_ladder_parallel_transport_test_data(
self.metric_args_list,
self.space_list,
self.shape_list,
self.n_samples_list,
self.n_rungs_list,
self.alpha_list,
self.scheme_list,
)
def exp_geodesic_ivp_test_data(self):
return self._exp_geodesic_ivp_test_data(
self.metric_args_list,
self.space_list,
self.shape_list,
self.n_samples_list,
self.n_points_list,
rtol=gs.rtol * 10000,
atol=gs.atol * 10000,
)
def parallel_transport_ivp_is_isometry_test_data(self):
return self._parallel_transport_ivp_is_isometry_test_data(
self.metric_args_list,
self.space_list,
self.shape_list,
self.n_samples_list,
is_tangent_atol=gs.atol * 1000,
atol=gs.atol * 1000,
)
def parallel_transport_bvp_is_isometry_test_data(self):
return self._parallel_transport_bvp_is_isometry_test_data(
self.metric_args_list,
self.space_list,
self.shape_list,
self.n_samples_list,
is_tangent_atol=gs.atol * 1000,
atol=gs.atol * 1000,
)
def dist_is_symmetric_test_data(self):
return self._dist_is_symmetric_test_data(
self.metric_args_list,
self.space_list,
self.n_points_a_list,
self.n_points_b_list,
)
def dist_is_positive_test_data(self):
return self._dist_is_positive_test_data(
self.metric_args_list,
self.space_list,
self.n_points_a_list,
self.n_points_b_list,
)
def squared_dist_is_positive_test_data(self):
return self._squared_dist_is_positive_test_data(
self.metric_args_list,
self.space_list,
self.n_points_a_list,
self.n_points_b_list,
)
def dist_is_norm_of_log_test_data(self):
return self._dist_is_norm_of_log_test_data(
self.metric_args_list,
self.space_list,
self.n_points_a_list,
self.n_points_b_list,
)
def dist_point_to_itself_is_zero_test_data(self):
return self._dist_point_to_itself_is_zero_test_data(
self.metric_args_list, self.space_list, self.n_points_list
)
def inner_product_is_symmetric_test_data(self):
return self._inner_product_is_symmetric_test_data(
self.metric_args_list,
self.space_list,
self.shape_list,
self.n_tangent_vecs_list,
)
def triangle_inequality_of_dist_test_data(self):
return self._triangle_inequality_of_dist_test_data(
self.metric_args_list, self.space_list, self.n_points_list
)
import random import geomstats.backend as gs from geomstats.geometry.pre_shape import PreShapeSpace from tests.data_generation import _LevelSetTestData, _RiemannianMetricTestData smoke_space = PreShapeSpace(4, 3) vector = gs.random.rand(11, 4, 3) base_point = smoke_space.random_point() tg_vec_0 = smoke_space.to_tangent(vector[0], base_point) hor_x = smoke_space.horizontal_projection(tg_vec_0, base_point) tg_vec_1 = smoke_space.to_tangent(vector[1], base_point) hor_y = smoke_space.horizontal_projection(tg_vec_1, base_point) tg_vec_2 = smoke_space.to_tangent(vector[2], base_point) hor_z = smoke_space.horizontal_projection(tg_vec_2, base_point) tg_vec_3 = smoke_space.to_tangent(vector[3], base_point) hor_h = smoke_space.horizontal_projection(tg_vec_3, base_point) tg_vec_4 = smoke_space.to_tangent(vector[4], base_point) ver_v = smoke_space.vertical_projection(tg_vec_4, base_point) tg_vec_5 = smoke_space.to_tangent(vector[5], base_point) ver_w = smoke_space.vertical_projection(tg_vec_5, base_point) tg_vec_6 = smoke_space.to_tangent(vector[6], base_point) hor_dy = smoke_space.horizontal_projection(tg_vec_6, base_point) tg_vec_7 = smoke_space.to_tangent(vector[7], base_point) hor_dz = smoke_space.horizontal_projection(tg_vec_7, base_point) tg_vec_8 = smoke_space.to_tangent(vector[8], base_point) ver_dv = smoke_space.vertical_projection(tg_vec_8, base_point) tg_vec_9 = smoke_space.to_tangent(vector[9], base_point) ver_dw = smoke_space.vertical_projection(tg_vec_9, base_point) tg_vec_10 = smoke_space.to_tangent(vector[10], base_point) hor_dh = smoke_space.horizontal_projection(tg_vec_10, base_point)