Rotate the surface of a sphere using altitude

Question

I have a unit sphere with points on its surface described by their azimuth and elevation. Azimuth is given as a number in $[0, 2 \pi]$, while elevation is a between $[-\pi/2, \pi/2]$.

I would like to rotate the sphere on the elevation axis (rotating by azimuth is trivial). For example, I'd rotate the sphere by $\pi / 2$ at azimuth 0, meaning that any point at azimuth 0 will be "elevated" by $\pi / 2$, while points on other azimuths will be elevated by a different amount depending on how far they are from the pivot point.

On the other side of the azimuth, at $\pi$, each point would be pushed down by $\pi / 2$.

What I'm after is the transformation formula for any point on the surface, for a given elevation angle and azimuth pivot point.

I'm having trouble working out the math for this. I have a version but I get false results. I'd really appreciate some help.

Answer 1

It is very easy to compute this rotation in a 3D cartesian coordinate system on the 2-sphere $\Bbb S^2$, simply by applying a certain rotation matrix. But it is tricky to do this in your azimuth/elevation system on $[0,2\pi]\times[-\pi/2,\pi/2]$.

I therefore recommend convert between these systems. How this can be done in detail is well-explained on the Wikipedia article on sphereical coordinate systems.

So I assume we have

• A conversion from spherical into cartesian: $f:(\phi,\theta)\mapsto(x,y,z)$.
• A conversion from cartesian into spherical: $g:(x,y,z)\mapsto(\phi,\theta)$.
• A rotation around azimuth by $\Delta\phi$: $h_{\Delta\phi}(\phi,\theta)=(\phi+\Delta\phi,\theta)$.
• A rotation matrix for rotation around the the $y$-axis by an angle of $\theta$ in cartesian coordinates: $$R_\theta=\begin{pmatrix}\cos(\theta)&0&-\sin(\theta)\\0&0&0\\\sin(\theta)&0&\cos(\theta)\end{pmatrix}.$$

I assume that $f(0,0)=(1,0,0)$ is mapped to the positive $x$-axis, and $f(\pi/2,0)=(0,1,0)$ is mapped to the positive $y$-axis. This means that elevation describes the rotation out of the $xy$-plane.

Now, let's say you want to rotatate around elevation by an angle of $\Delta\theta$ at azimuth $\phi_0$. This can be done by the following transformation:

$$h_{\phi_0}\circ g\circ R_{\Delta\theta}\circ f\circ h_{-\phi_0}\quad:\quad (\phi,\theta)\quad\mapsto\quad h_{\phi_0}(g(R_{\Delta\theta}f(h_{-\phi_0}(\phi,\theta)))).$$

It is not a nice formula, but it works. One might write this out as a single formula, but it will probably be long and ugly.

Answer 2

I should think quaternions would work well for your problem. As you see on the wiki, given a vector $\mathbf{p}$ and axis of rotation $\mathbf{u}$, a unit vector, and angle $\theta$ through which to rotate, you form the rotation quaternion as $$q=\cos(\theta/2)+(u_x\mathbf{i}+u_y\mathbf{j}+u_z\mathbf{k})\sin(\theta/2),$$ and the vector extension into quaternions as $$p=0+p_x\mathbf{i}+p_y\mathbf{j}+p_z\mathbf{k}.$$ You do the rotation by $p'=qpq^{-1}$, and the inverse $q^{-1}$ you can find simply by "conjugating" the $q$ quaternion. The quaternion multiplication is what William Rowan Hamilton discovered, and I would totally do that in a computer software package, as it's a lot of multiplication, addition, and subtraction.

The advantage of quaternions is that you avoid gymbal lock, and the representation of a quaternion is relatively efficient. It also allows you to rotate about any axis whatever, unlike Euler rotation matrices. You have to extend Euler matrices as illustrated here to do that.