How to rotate points on spherical cap with specific theta and phi?

I generate N=1000 points in spherical cap {rho = 10, 0 deg<theta<30 deg, 0<phi<2pi} as shown in the figure below.

enter image description here

And now I want to rotate these points together such that the center of the cap goes to the specific theta and phi with the distances to other points unchanged. For example, if we move the center of the cap from (theta=0, phi=0) to (theta=90 deg, phi=90 deg), the cap should be shown as

enter image description here

I first try to add the specific theta and phi to the positions of all points, but with the spherical coordinates, it doesn’t work as shown in the figure below. (It becomes a ring T_T)

enter image description here

Any one can help me to rotate all points together to such that the center of the cap goes to the specific theta and phi with the distances to other points unchanged. Or please give me some advice. Thanks in advance.

Here’s my code in python

from math import * import numpy as np  def polarToxyz(polar_cord):  # theta, phi radian     r = polar_cord[0]     theta = polar_cord[1]     phi = polar_cord[2]     return [r * sin(theta) * cos(phi), r * sin(theta) * sin(phi), r * cos(theta)]  theta_prime_deg = 30 theta_prime_rad = theta_prime_deg * np.pi / 180  delta_theta_deg = 0 delta_theta_rad = delta_theta_deg * np.pi / 180  delta_phi_deg = 0 delta_phi_rad = delta_phi_deg * np.pi / 180  list_x = [] list_y = [] list_z = [] for i in range(numbPoints):     rho = 10     phi = 2 * np.pi * np.random.uniform(0,1)     theta = acos(1 - np.random.uniform(0,1)*(1-cos(theta_prime_rad)))      theta += delta_theta_rad     phi += delta_phi_rad      [x,y,z] = polarToxyz([rho, theta, phi])     list_x.append(x)     list_y.append(y)     list_z.append(z)  drawFig3D(list_x, list_y, list_z)