{
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        "author": "Yunseong Jeong",
        "author_email": "Yunseong14@naver.com",
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            "Programming Language :: Python :: 3",
            "Programming Language :: Python :: 3.2",
            "Programming Language :: Python :: 3.3",
            "Programming Language :: Python :: 3.4",
            "Programming Language :: Python :: 3.5",
            "Programming Language :: Python :: 3.6",
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        "description": "# n_sphere\n\nSpherical Coordinate system, Rectangular coordinate system and Stereographic Projection in N-sphere (Using Python)\n\nhttps://pypi.org/project/n-sphere/\n\n1. Spherical Coordinate ([Show Image](https://wikimedia.org/api/rest_v1/media/math/render/svg/fff301fb5050086e409e13519f82295f23b2987a))\n    ```\n    r = math.sqrt( (x_n)^2 + ((x_n-1)^2) + ... + (x_1)^2 )\n    Phi_1 = math.acos( x_1 / math.sqrt((x_n)^2 + (x_n-1)^2 + ... + (x_1)^2 ))\n    Phi_2 = math.acos( x_2 / math.sqrt((x_n)^2 + (x_n-1)^2 + ... + (x_2)^2 ))\n    ...\n    Phi_n-2 = math.acos( x_n-2 / math.sqrt((x_n)^2 + (x_n-1)^2 ))\n    Phi_n-1 = math.acos( x_n-1 / math.sqrt((x_n)^2 + (x_n-1)^2 )) (x_n >= 0)\n          = 2 * Pi - math.acos( x_n-1 / math.sqrt((x_n)^2 + (x_n-1)^2 )) (x_n < 0) \n    ```\n   - R = radius\n   - n - 1 angular coordinates Phi_1, Phi_2, ... , Phi_n-1  in n-dimensional Euclidean Space\n   - Phi_1 , Phi_2 , ... , Phi_n-2 range over **[0, Pi]** radians\n   - Phi_n-1 range over **[0, 2 * Pi)** radians\n2. Rectangular Coordinate ([Show Image](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c4349e9ce260f719ebf573067bc7b84305ae31c))\n   ```\n   x_1 = r * math.cos(Phi_1)\n   x_2 = r * math.sin(Phi_1) * math.cos(Phi_2)\n   x_3 = r * math.sin(Phi_1) * math.sin(Phi_2) * math.cos(Phi_3)\n   ...\n   x_n-1 = r * math.sin(Phi_1) * ... * math.sin(Phi_n-2) * math.cos(Phi_n-1)\n   x_n = r * math.sin(Phi_1) * ... * math.sin(Phi_n-2) * math.sin(Phi_n-1)\n   ``` \n\n3. Stereographic Projection (**incomplete**)\n   - In 3-dimension\n    ```\n   [x,y,z] -> [x/1-z, y/1-z]\n   ```\n   - In N-dimension\n   ```\n   [x_1 , x_2 , ... , x_n] -> [ x_1 / 1-x_n , x_2 / 1-x_n , ... , x_n-1 / 1-x_n] \n   ```\n\n4. Generating random points (Plan)\n\n\n# Project Object\nhttps://en.wikipedia.org/wiki/N-sphere\n\n# How to Use\n```\npip install n-sphere\n```\n- Tensor (Using Torch)\n``` shell script\n> python\nPython 3.7.2 (tags/v3.7.2:9a3ffc0492, Dec 23 2018, 23:09:28) [MSC v.1916 64 bit (AMD64)] on win32\nType \"help\", \"copyright\", \"credits\" or \"license\" for more information.\n>>> import torch\n>>> import n_sphere\n>>> x = torch.randint(10, size(5,3,))\nTraceback (most recent call last):\n  File \"<stdin>\", line 1, in <module>\nNameError: name 'size' is not defined\n>>> x = torch.randint(10, size=(5,3,))\n>>> x\ntensor([[1, 0, 4],\n        [2, 7, 9],\n        [1, 8, 6],\n        [4, 9, 3],\n        [0, 8, 1]])\n>>> c_x = n_sphere.convert_spherical(x)\n>>> c_x\ntensor([[ 4.1231,  1.3258,  1.5708],\n        [11.5758,  1.3972,  0.9098],\n        [10.0499,  1.4711,  0.6435],\n        [10.2956,  1.1718,  0.3218],\n        [ 8.0623,  1.5708,  0.1244]], dtype=torch.float64)\n>>> r_x = n_sphere.convert_rectangular(c_x)\n>>> r_x\ntensor([[1.0000e+00, 2.4493e-16, 4.0000e+00],\n        [2.0000e+00, 7.0000e+00, 9.0000e+00],\n        [1.0000e+00, 8.0000e+00, 6.0000e+00],\n        [4.0000e+00, 9.0000e+00, 3.0000e+00],\n        [2.6347e-06, 8.0000e+00, 1.0000e+00]], dtype=torch.float64)\n```\n\n- List\n```shell script\n>>> list = [\n... [2,3,4,5,6],\n... [4,1,3,6,7],\n... [2,2,2,2,2],\n... [4,9,1,2,8]]\n>>> c_list = n_sphere.convert_spherical(list)\n>>> c_list\narray([[ 9.48683298,  1.358384  ,  1.241372  ,  1.097478  ,  0.87605805],\n       [10.53565375,  1.181364  ,  1.468018  ,  1.256207  ,  0.86217005],\n       [ 4.47213595,  1.107149  ,  1.047198  ,  0.955317  ,  0.78539816],\n       [12.88409873,  1.255119  ,  0.745355  ,  1.450118  ,  1.32581766]])\n>>> r_list = n_sphere.convert_rectangular(c_list)\n>>> r_list\narray([[2.000001, 3.      , 4.      , 5.      , 6.      ],\n       [4.000001, 0.999996, 2.999996, 6.000001, 7.000001],\n       [1.999999, 1.999999, 2.      , 2.000001, 2.000001],\n       [4.000002, 9.000003, 0.999998, 1.999999, 7.999996]])\n>>>\n```\n- Number of digits can be changed in Round Function (default = 6)\n\n```sh\n>>> list = [\n... [2,3,4,5,6],\n... [4,1,3,6,7],\n... [2,2,2,2,2],\n... [4,9,1,2,8]]\n>>> c_list = n_sphere.convert_spherical(list,10)\n>>> c_list\narray([[ 9.48683298,  1.35838411,  1.24137205,  1.097478  ,  0.87605805],\n       [10.53565375,  1.18136412,  1.46801764,  1.25620658,  0.86217005],\n       [ 4.47213595,  1.10714872,  1.04719755,  0.95531662,  0.78539816],\n       [12.88409873,  1.2551192 ,  0.74535537,  1.45011777,  1.32581766]])\n>>> r_list = n_sphere.convert_rectangular(c_list,10)\n>>> r_list\narray([[2., 3., 4., 5., 6.],\n       [4., 1., 3., 6., 7.],\n       [2., 2., 2., 2., 2.],\n       [4., 9., 1., 2., 8.]])\n>>>\n```\n\n# Contact Us\nyunseong14@naver.com\n\n# Reference\n- [N-sphere](https://en.wikipedia.org/wiki/N-sphere)\n- [Stereographic projection](https://en.wikipedia.org/wiki/Stereographic_projection)\n\n\n\n",
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