Object Rotation with polar angles python

Question

I would like to know how to rotate an object around its center point,that the angle from the z axis is theta and the XY plane rotates in angle phi via python scripting.

Answer 1

To avoid any misunderstanding let me reiterate with this illustration.

^{Click to enlarge.}

$\theta$ will control the orientation on the red axis.
$\phi$ will control the orientation on the blue axis.
If $\theta$ is a multiple of $\pi$, $\phi$ won't change the final orientation.

We can construct a quaternion from the $\theta$ and $\phi$ rotation and chain them. The mathutils.Quaternion class provides a rotate(other) function for this purpose.

import bpy
import math
import mathutils

θ = math.pi / 3
ϕ = math.pi / 2

q = mathutils.Quaternion((0, -1, 0), θ)
qϕ = mathutils.Quaternion((0, 0, 1), ϕ)

q.rotate(qϕ)

bpy.context.object.rotation_mode = 'QUATERNION'
bpy.context.object.rotation_quaternion = q

While this works, it doesn't really produce polar coordinates. Certain polar coordinates should be the same, and they currently aren't.

Calculation the spherical position

You can directly calculate the x, y and z values.

$x = \sin{\theta} \cos{\phi}$
$y = \sin{\theta} \sin{\phi}$
$z = \cos{\theta}$

Then you can generate a quaternion pointing to that position with the to_track_quat function.

position = mathutils.Vector((math.sin(θ) * math.cos(ϕ),
                             math.sin(θ) * math.sin(ϕ),
                             math.cos(θ)                ))

q = position.to_track_quat('-Z', 'X')

This will give you good control over which axis is oriented where.

Answer 2

Latitude and longitude.

It appears you are describing a system akin to latitude and longitude. Can pretty much emulate this by changing the Euler order to 'ZYX' and setting Z rotation for longitude and X for latitude.

Below is a sphere such that -Y with no rotation points to lat, long = (0, 0) A camera is globally position above this point.

Test script

With the setup above, can set the absolute rotation of the sphere to latitude and longitude.

Set the rotation of the sphere such that the cities using latitude and longitude are randomly presented under camera.

import bpy
from math import radians
from mathutils import Euler
from random import choice

context = bpy.context
ob = context.object

data = (
        (40, -73., "New York"),
        (-33, 151, "Sydney"),
        (51, 0, "London"),
        (-22, -42, "Rio"),
        (-1, 37, "Nairobi"),
        )

lat, long, name = choice(data)

R = Euler((radians(lat), 0, radians(-long)), 'ZYX').to_matrix()

ob.matrix_world = R.to_4x4()

Equirectangular world image used as map texture.

By Strebe - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=16115228