What Möbius Map Rotates the Riemann Sphere About the x-Axis?

A rotation of the Riemann sphere is represented by a Möbius transformation. The exact signs depend on how you identify points on the sphere with complex numbers, so first fix a stereographic-projection convention.

Use Cartesian coordinates $(X,Y,Z)$ on the unit sphere and define

$$z = \frac{X+iY}{1-Z}.$$

This convention projects from the north pole $(0,0,1)$ onto the complex plane, so the north pole corresponds to $z=\infty$. With a right-handed rotation by angle $\theta$ about the positive $X$-axis, the transformation is

$$z' = \frac{z\cos(\theta/2)+i\sin(\theta/2)}{iz\sin(\theta/2)+\cos(\theta/2)}.$$

Here the sphere’s $X$-axis is the axis of rotation; its two intersections with the sphere correspond to the fixed complex points $z=1$ and $z=-1$.

Deriving the transformation

The inverse stereographic formulas for this convention are

$$X=\frac{2\operatorname{Re}z}{1+|z|^2},\qquad Y=\frac{2\operatorname{Im}z}{1+|z|^2},\qquad Z=\frac{|z|^2-1}{1+|z|^2}.$$

A right-handed rotation about the positive $X$-axis leaves $X$ unchanged and rotates the $YZ$-plane:

$$X'=X,\qquad Y'=Y\cos\theta-Z\sin\theta,\qquad Z'=Y\sin\theta+Z\cos\theta.$$

Convert the rotated point back to the complex coordinate using $z'=(X'+iY')/(1-Z')$. Substituting the inverse stereographic formulas and simplifying gives the stated Möbius map. The half-angle appears because a rotation of the sphere through $\theta$ is represented by coefficients involving $\cos(\theta/2)$ and $\sin(\theta/2)$.

The formula is Möbius because it has the form $(az+b)/(cz+d)$. Its coefficient matrix can be taken as

$$\begin{pmatrix} \cos(\theta/2) & i\sin(\theta/2)\\ i\sin(\theta/2) & \cos(\theta/2) \end{pmatrix},$$

whose determinant is $1$. As usual, multiplying all four coefficients by the same nonzero number does not change the map.

Worked example: a quarter-turn

Take $\theta=\pi/2$. Then $\cos(\theta/2)=\sin(\theta/2)=1/\sqrt{2}$, so

$$z'=\frac{z+i}{iz+1}.$$

For example, let $z=1/2$. Its sphere coordinates are

$$(X,Y,Z)=\left(\frac{4}{5},0,-\frac{3}{5}\right).$$

A positive quarter-turn about the $X$-axis sends this point to $(4/5,3/5,0)$. The complex coordinate of the rotated point is therefore $4/5+3i/5$. The Möbius formula gives the same result:

$$\frac{1/2+i}{i/2+1}=\frac{4}{5}+\frac{3}{5}i.$$

This check is useful because reversing the rotation direction would change the signs of the imaginary terms.

What the formula does to special points

The points $z=1$ and $z=-1$ are fixed for every angle. They represent $(1,0,0)$ and $(-1,0,0)$, the two points where the rotation axis meets the sphere. The north pole is $z=\infty$; the formula sends it to $-i\cot(\theta/2)$ when the denominator’s leading coefficient is nonzero, consistent with rotating the north pole around the $X$-axis.

For $\theta=\pi$, the map reduces to $z'=1/z$. This is the half-turn about the $X$-axis: it fixes the two axial points and reverses the other two Cartesian coordinates. For $\theta=2\pi$, the formula is the identity, as a full rotation of the sphere should be.

Common mistakes

• Using the wrong stereographic convention. If you instead define the coordinate using $1+Z$ in the denominator, the north and south poles trade roles and the formula’s signs can differ. State the projection convention before comparing formulas.
• Confusing active and passive rotations. The formula above moves each point by a right-handed rotation about positive $X$. Describing the coordinate frame as rotating in the opposite direction gives the inverse map, obtained by replacing $\theta$ with $-\theta$.
• Using the full angle in the coefficients. The coefficients use $\theta/2$, not $\theta$. Substituting $\cos\theta$ and $\sin\theta$ generally gives a different rotation.
• Treating infinity as an exception. Möbius transformations act on the extended complex plane. Evaluate the map at $z=\infty$ by taking its limit, rather than concluding the formula is undefined there.

If your convention projects from the south pole, or if your positive rotation direction differs, derive the map from that convention’s inverse stereographic formulas and rotation matrix; do not change signs by guesswork.