Rotation of plane tangential to sphere

[Equanimity]

I'm using some software where the location of a plane with respect to a point is defined using spherical coordinates. I've attached a visual guide of the software parameters to this post, if someone is interested.

I'm not having problems defining the position of the plane with respect to a given origin [there are no problems calculating the polar angle theta using acos(k), the azimuthal angle phi using atan2(i, -j)* and the distance r], but I am having problems determining the rotation in the plane tangential to the sphere (positions in this plane a defined using x,y coordinates, with 0,0 being the point defined by the spherical coordinates).

Different theta/phi combinations effect a rotation in this plane. I understand why this is the case: a combination of pitch and yaw results in roll. I can observe the effect, for instance, when I open up Google Earth and rotate to a different latitude and longitude - I see that the camera rotates (the north and south directions change).

* this -j is required for the software to provide the correct answer; I believe it indicates that the plane is facing the origin of the spherical coordinate system

I'm a stuck on how to quantify this rotation. Is there a simple way to calculate the rotation in this tangential plane between two sets of i,j,k (or r, theta, phi) values?

 

[tiny-tim]

Welcome to PF!

Hi Equanimity! Welcome to PF!

(have a theta: θ and a phi: φ )

I can observe the effect, for instance, when I open up Google Earth and rotate to a different latitude and longitude - I see that the camera rotates (the north and south directions change).
…
I'm a stuck on how to quantify this rotation. Is there a simple way to calculate the rotation in this tangential plane between two sets of i,j,k (or r, theta, phi) values?

I think you're talking about the excess angle …

when you add all the interior angles of a polygon on a sphere, they come to more than they would on a plane, eg > 180° for a triangle or > 360° for a quadrilateral.

If you go opposite ways round the polygon, and meet up, the error is that excess angle.

The excess angle is proportional to the area of the polygon (area/radius²) …

there's some explanation and formulas at http://etc.usf.edu/lit2go/contents/3900/3934/3934.pdf"

 

[Equanimity]

Thanks for the suggestion, but I'm not sure that is the relevant concept. The plane in this case isn't on the surface of the sphere, it is tangential.

I think I can be more thorough in phrasing my question.

In this software, the 'default' parameters are polar angle θ = 90, azimuthal angle Φ = -90 and the plane rotation φ = -90. This results in a plane with an origin on the y-axis, facing the positive y-direction. The output of the software with these parameters is as expected. A φ = -90 produces expected output whenever θ = 90. But with different combinations of θ and Φ, it seems there is an effected rotation of the plane, that I need to quantify, in order to remove it.

Through guess and check, I have a whole lot of combinations of working sets:
θ=90, Φ=-90, φ≈-90
θ=90, Φ=0, φ≈-90
θ=90, Φ=90, φ≈-90
θ=110, Φ=-90, φ≈0
θ=125, Φ=-90, φ≈173
θ=123, Φ=-130, φ≈226
θ=65, Φ=-90, φ≈0
and so on ...

I know that the rotation of the plane φ is dependent on θ and Φ. But I cannot figure out the nature of that dependance.

I believe what is happening is the same thing that occurs when a camera rotates about a sphere, as in the picture I've attached. There are varying rotations in the orientation of the plane at different points about the sphere. This makes sense when considering the constant φ when θ=90: the x and y projections in the plane do not change as you rotate about along the meridian.

I first thought that if I gathered enough sets of working angles, I'd be able to see the relationship between φ and (θ,Φ); but it isn't apparent. I've read about rotation matrices, euler angles and quaternions, but I don't know how to apply them to get the relationship I'm after (or if they are even appropriate).