How to find the center and radius of an any-dimensional sphere given dims+1 points

Given a vector of N-dimensional points. The vector will be of size N+1.

Is there a generalized algorithm to find the center and radius of the ND sphere using those points where the sphere intersects every single one of those points?

The same question has been asked on the mathematics stackexchange and has received a constructive answer:

• Does a set of n+1 points that affinely span R^n lie on a unique (n-1)-sphere?

Here is an implementation in python/numpy of the algorithm described at that answer.

import numpy as np

def find_sphere_through_points(points):
    n_points, n_dim = points.shape
    if (n_points != n_dim + 1):
        raise ValueError('Number of points must be equal to 1 + dimension')
    a = np.concatenate((points, np.ones((n_points, 1))), axis=1)
    b = (points**2).sum(axis=1)
    x = np.linalg.solve(a, b)
    center = x[:-1] / 2
    radius = x[-1] + center@center
    return center, radius

To test this method, we can generate random points on the surface of a sphere, using the method described in this related question:

• Generate a random sample of points distributed on the surface of a unit spher

import numpy as np

def sample_spherical(npoints, ndim=3, center=None):
    vec = np.random.randn(npoints, ndim)
    vec /= np.linalg.norm(vec, axis=1).reshape(npoints,1)
    if center is None:
        return vec
    else:
        return vec + center

n = 5
center = np.random.rand(n)
points = sample_spherical(n+1, ndim=n, center=center)

guessed_center, guessed_radius = find_sphere_through_points(points)

print('True center:\n    ', center)
print('Calc center:\n    ', guessed_center)
print('True radius:\n    ', 1.0)
print('Calc radius:\n    ', guessed_radius)

# True center:
#      [0.18150032 0.94979547 0.07719378 0.26561175 0.37509931]
# Calc center:
#      [0.18150032 0.94979547 0.07719378 0.26561175 0.37509931]
# True radius:
#      1.0
# Calc radius:
#      0.9999999999999997