n_sphere
Spherical Coordinate system, Rectangular coordinate system and Stereographic Projection in N-sphere (Using Python)
https://pypi.org/project/n-sphere/
-
Spherical Coordinate (Show Image)
r = math.sqrt( (x_n)^2 + ((x_n-1)^2) + ... + (x_1)^2 ) Phi_1 = math.acos( x_1 / math.sqrt((x_n)^2 + (x_n-1)^2 + ... + (x_1)^2 )) Phi_2 = math.acos( x_2 / math.sqrt((x_n)^2 + (x_n-1)^2 + ... + (x_2)^2 )) ... Phi_n-2 = math.acos( x_n-2 / math.sqrt((x_n)^2 + (x_n-1)^2 )) Phi_n-1 = math.acos( x_n-1 / math.sqrt((x_n)^2 + (x_n-1)^2 )) (x_n >= 0) = 2 * Pi - math.acos( x_n-1 / math.sqrt((x_n)^2 + (x_n-1)^2 )) (x_n < 0)- R = radius
- n - 1 angular coordinates Phi_1, Phi_2, ... , Phi_n-1 in n-dimensional Euclidean Space
- Phi_1 , Phi_2 , ... , Phi_n-2 range over [0, Pi] radians
- Phi_n-1 range over [0, 2 * Pi) radians
-
Rectangular Coordinate (Show Image)
x_1 = r * math.cos(Phi_1) x_2 = r * math.sin(Phi_1) * math.cos(Phi_2) x_3 = r * math.sin(Phi_1) * math.sin(Phi_2) * math.cos(Phi_3) ... x_n-1 = r * math.sin(Phi_1) * ... * math.sin(Phi_n-2) * math.cos(Phi_n-1) x_n = r * math.sin(Phi_1) * ... * math.sin(Phi_n-2) * math.sin(Phi_n-1) -
Stereographic Projection (incomplete)
- In 3-dimension
[x,y,z] -> [x/1-z, y/1-z]- In N-dimension
[x_1 , x_2 , ... , x_n] -> [ x_1 / 1-x_n , x_2 / 1-x_n , ... , x_n-1 / 1-x_n] -
Generating random points (Plan)
Project Object
https://en.wikipedia.org/wiki/N-sphere
How to Use
pip install n-sphere
- Tensor (Using Torch)
> python
Python 3.7.2 (tags/v3.7.2:9a3ffc0492, Dec 23 2018, 23:09:28) [MSC v.1916 64 bit (AMD64)] on win32
Type "help", "copyright", "credits" or "license" for more information.
>>> import torch
>>> import n_sphere
>>> x = torch.randint(10, size(5,3,))
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
NameError: name 'size' is not defined
>>> x = torch.randint(10, size=(5,3,))
>>> x
tensor([[1, 0, 4],
[2, 7, 9],
[1, 8, 6],
[4, 9, 3],
[0, 8, 1]])
>>> c_x = n_sphere.convert_spherical(x)
>>> c_x
tensor([[ 4.1231, 1.3258, 1.5708],
[11.5758, 1.3972, 0.9098],
[10.0499, 1.4711, 0.6435],
[10.2956, 1.1718, 0.3218],
[ 8.0623, 1.5708, 0.1244]], dtype=torch.float64)
>>> r_x = n_sphere.convert_rectangular(c_x)
>>> r_x
tensor([[1.0000e+00, 2.4493e-16, 4.0000e+00],
[2.0000e+00, 7.0000e+00, 9.0000e+00],
[1.0000e+00, 8.0000e+00, 6.0000e+00],
[4.0000e+00, 9.0000e+00, 3.0000e+00],
[2.6347e-06, 8.0000e+00, 1.0000e+00]], dtype=torch.float64)- List
>>> list = [
... [2,3,4,5,6],
... [4,1,3,6,7],
... [2,2,2,2,2],
... [4,9,1,2,8]]
>>> c_list = n_sphere.convert_spherical(list)
>>> c_list
array([[ 9.48683298, 1.358384 , 1.241372 , 1.097478 , 0.87605805],
[10.53565375, 1.181364 , 1.468018 , 1.256207 , 0.86217005],
[ 4.47213595, 1.107149 , 1.047198 , 0.955317 , 0.78539816],
[12.88409873, 1.255119 , 0.745355 , 1.450118 , 1.32581766]])
>>> r_list = n_sphere.convert_rectangular(c_list)
>>> r_list
array([[2.000001, 3. , 4. , 5. , 6. ],
[4.000001, 0.999996, 2.999996, 6.000001, 7.000001],
[1.999999, 1.999999, 2. , 2.000001, 2.000001],
[4.000002, 9.000003, 0.999998, 1.999999, 7.999996]])
>>>- Number of digits can be changed in Round Function (default = 6)
>>> list = [
... [2,3,4,5,6],
... [4,1,3,6,7],
... [2,2,2,2,2],
... [4,9,1,2,8]]
>>> c_list = n_sphere.convert_spherical(list,10)
>>> c_list
array([[ 9.48683298, 1.35838411, 1.24137205, 1.097478 , 0.87605805],
[10.53565375, 1.18136412, 1.46801764, 1.25620658, 0.86217005],
[ 4.47213595, 1.10714872, 1.04719755, 0.95531662, 0.78539816],
[12.88409873, 1.2551192 , 0.74535537, 1.45011777, 1.32581766]])
>>> r_list = n_sphere.convert_rectangular(c_list,10)
>>> r_list
array([[2., 3., 4., 5., 6.],
[4., 1., 3., 6., 7.],
[2., 2., 2., 2., 2.],
[4., 9., 1., 2., 8.]])
>>>
