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I would like to rotate an array about the axis that is normal to a given location on the surface of a sphere, but unsure how to go about it.

As an example, the below code creates a series of points (a dipole field line), shifts it away from the centre of a sphere and rotates to some angle in the xy plane. At this location, I would like to rotate the field line to any angle around the axis that is normal to the sphere surface.

I think it could be done with Euler angles or Rodriguez' formula, but I'm not familiar enough to implement this.

Here is an attempt to perform the required rotation at the sphere centre, and then displace and rotate to the desired location, however the resulting field line does not seem to retain any rotation around angle gamma.

**What I have tried:**

Performing the rotating at (0,0) but it doesn't seem to translate and rotate correctly to where the above code displays the field line.

As an example, the below code creates a series of points (a dipole field line), shifts it away from the centre of a sphere and rotates to some angle in the xy plane. At this location, I would like to rotate the field line to any angle around the axis that is normal to the sphere surface.

import numpy as np import matplotlib.pyplot as plt field=np.linspace(-np.pi/2,np.pi/2,100) circle=np.linspace(0,2*np.pi,100) theta=60*np.pi/180 r_shift=0.9 r=1.5*np.sin(field+np.pi/2)**2 x0=r*np.cos(field)+r_shift y0=r*np.sin(field) # rotate around y-axis x=x0*np.cos(theta) y=y0 fig,ax=plt.subplots() ax.set_box_aspect(1) ax.plot(np.cos(circle),np.sin(circle),color='k') ax.plot(x,y) ax.set_xlabel('X') ax.set_ylabel('Y') ax.set_xlim(-2,2) ax.set_ylim(-2,2) plt.show()

I think it could be done with Euler angles or Rodriguez' formula, but I'm not familiar enough to implement this.

Here is an attempt to perform the required rotation at the sphere centre, and then displace and rotate to the desired location, however the resulting field line does not seem to retain any rotation around angle gamma.

<pre>import numpy as np import matplotlib.pyplot as plt field=np.linspace(-np.pi/2,np.pi/2,100) circle=np.linspace(0,2*np.pi,100) theta=60*np.pi/180 gamma=45*np.pi/180 r_shift=0.9 r=1.5*np.sin(field+np.pi/2)**2 x0=r*np.cos(field) y0=r*np.sin(field) # rotate around x-axis to angle gamma # [1 0 0 ] # [0 cos(gamma) -sin(gamma)] # [0 sin(gamma) cos(gamma)] y=y0*np.cos(gamma) # displace in x to r_shift x=x0+r_shift # rotate around y-axis to angle theta # [cos(theta) 0 sin(theta)] # [0 1 0] # [sin(theta) 0 cos(theta)] x=x*np.cos(theta) fig,ax=plt.subplots() ax.set_box_aspect(1) ax.plot(np.cos(circle),np.sin(circle),color='k') ax.plot(x0,y0) ax.plot(x,y) ax.set_xlabel('X') ax.set_ylabel('Y') ax.set_xlim(-2,2) ax.set_ylim(-2,2) plt.show()

Performing the rotating at (0,0) but it doesn't seem to translate and rotate correctly to where the above code displays the field line.

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a.) Translate the point to your 'rotation origin'

b.) Rotate

c.) Translate back

https://python3-9-code.trinket.io/python3-generated/5ciydimu/trinket_plot.png[^]

where orange is the not rotated dipol and blue is rotatet?

This had been done by:

`x= x0*np.cos(theta) - y0*np.sin(theta)`

y= x0*np.sin(theta) + y0*np.cos(theta)

i.e. for whatever location I place the origin of the field line, the axis that is normal to the sphere surface at that location is the what I want to rotate around.

Especally you mentioned 'At this location', baut what you mean exactly with 'this location'?

So if you imagine any position on the surface of a sphere, and the axis that would be normal to that point of the sphere (i.e. pointing straight out of the sphere surface) - this is the axis I want to rotate around.

It should look like this before (blue) and after (orange) rotation:

https://imgur.com/a/VAV2DCe