```import math
import numpy as np

"""
compute_delta_theta() and compute_delta_phi().
"""
K = 2.0

"""
The 3D coordinate structure
"""

class Point4d(object):

"""
The constructor for Point4d

:param float x: the x coordinate for a node in cartesian space
:param float y: the y coordinate for a node in cartesian space
:param float z: the z coordinate for a node in cartesian space
"""

def __init__(self, x=0.0, y=0.0, z=0.0, w=1.0):
self.x = x
self.y = y
self.z = z
self.w = w

"""
Transform coodinate from spherical space to cartesian space

:param float theta: the theta coordinate for a node in spherical space, polar angle
:param float phi: the phi coordinate for a node in spherical space, elevation angle
:param float r: the radius for a node in spherical space, radial distance
"""

def sph_to_cart(self, theta, phi, r):
self.x = r * math.sin(phi) * math.cos(theta)
self.y = r * math.sin(phi) * math.sin(theta)
self.z = r * math.cos(phi)

"""
Add an offset to a node's coordinate in cartesian space

:param Point4d offset: the offset to be added to current node's coordinate
"""

def cart_offset(self, offset):
self.x += offset.x
self.y += offset.y
self.z += offset.z

"""
Translate node's coordinate in cartesian space by translation matrix

:param Point4d offset: the offset to be added to current node's coordinate
"""

def translate(self, offset):
translation_matrix = np.array([[1, 0, 0, offset.x],
[0, 1, 0, offset.y],
[0, 0, 1, offset.z],
[0, 0, 0, 1]])
target = translation_matrix.dot(np.array([self.x, self.y, self.z, self.w]))
self.x, self.y, self.z, self.w = target[0], target[1], target[2], target[3]

"""
Transform node's coordinate in cartesian space by transformation matrix by
spherical space coordinate values

:param float theta: the theta coordinate for a node in spherical space, polar angle
:param float phi: the phi coordinate for a node in spherical space, elevation angle
"""

def coordinate_transformation(self, theta, phi):
rotation_matrix = rotation_matrix_z(theta).dot(rotation_matrix_y(phi))
target = rotation_matrix.dot(np.array([self.x, self.y, self.z, self.w]))
self.x, self.y, self.z, self.w = target[0], target[1], target[2], target[3]

"""
Compute the node's hemisphere radius by its hemisphere space reserved

:returns: return the node's radius as a float
"""

return K * math.asinh(math.sqrt(H_p / (2 * math.pi * K * K)))

"""
:returns: return the node's hemisphere space reservation as a float
"""

beta = 1.00
return 2 * math.pi * (math.cosh(radius / K) - 1.0) * beta

"""
Compute the proper delta variant value for node's hemisphere placement, similar to
the space reservation the size of the node's hemisphere radius

:returns: return the node's half hemisphere space reservation as angle delta_theta
in spherical space as a float number
"""

def compute_delta_theta(r, rp, phi):
return math.atan(math.tanh(r / K) /
(math.sinh(rp / K) * math.sinh(phi)))

"""
Compute the proper phi variant value for a band of nodes' hemisphere placement,
similar to the space reservation the size of the largest node's hemisphere radius
of the whole band

:returns: return the nodes' max half hemisphere space reservation in the same band
as angle delta_phi in spherical space as a float number
"""

def compute_delta_phi(r, rp):
return math.atan(math.tanh(r / K) / math.sinh(rp / K))

"""
The Klein metric for visualizing hyperbolic space: unusual uses of 4x4 matrices
by Phillips and Gunn
"""

def minkowski(x, y):
return (x.x * y.x + x.y * y.y + x.z * y.z - x.w * y.w)

"""
Calculate 3D hyperbolic distance as a Klein metric
"""

def hyperbolic_distance(x, y):
t1 = minkowski(x, y)
t2 = minkowski(x, x)
t3 = minkowski(y, y)
return (2 * math.acosh(((t1 * t1) / (t2 * t3))**2))

"""
Rotation matrix around X axis

:params: the angle for rotating around X axis
:returns: return the rotation matrix around X axis by the given angle
"""

def rotation_matrix_x(angle):
return np.array([[1, 0, 0, 0],
[0, math.cos(angle), -1 * math.sin(angle), 0],
[0, math.sin(angle), math.cos(angle), 0],
[0, 0, 0, 1]])

"""
Rotation matrix around Y axis

:params: the angle for rotating around Y axis
:returns: return the rotation matrix around Y axis by the given angle
"""

def rotation_matrix_y(angle):
return np.array([[math.cos(angle), 0, math.sin(angle), 0],
[0, 1, 0, 0],
[-1 * math.sin(angle), 0, math.cos(angle), 0],
[0, 0, 0, 1]])

"""
Rotation matrix around Z axis

:params: the angle for rotating around Z axis
:returns: return the rotation matrix around X axis by the given angle
"""

def rotation_matrix_z(angle):
return np.array([[math.cos(angle), -1 * math.sin(angle), 0, 0],
[math.sin(angle), math.cos(angle), 0, 0],
[0, 0, 1, 0],
[0, 0, 0, 1]])
```