Rotating one 3d-vector to another

I know this is a long-dead and well-answered question, but I woke this morning thinking "All of the answers involve an "if" statement, i.e., all are discontinuous functions of the inputs. Is there a continuous answer, i.e., a continuous function $$ R: S^2 \times S^2 \to SO(3) $$ that takes a pair of vectors $u, v$ to a rotation matrix $R(u, v)$ with the property that $R(u, v) u = v$?"

The paper that Tomas Moller and I wrote back in 1999, for instance, uses "the coordinate vector corresponding to the smallest entry of $w$" for some vector $w$, which doesn't vary continuously as a function of $w$. And I wondered, "Did we really do as well as possible, or might there have been a continuous solution?"

The answer is "no." But the proof uses a bit of topology.

Fix the vector $u$ (set it to be $e_1$, for instance), and look at the map $$ K : S^2 \to SO(3) : v \mapsto R(u, v). $$ Then compose this with the map $$ H : SO(3) \to S^2 : M \mapsto Mu. $$ The composite map $$ H\circ K: S^2 \to S^2 $$ is $v \mapsto R(u, v)u = v$, i.e., the identity map on $S^2$. But that means that $$ (H\circ K)_{*} : H_2(S^2) \to H_2(S^2), $$ the induced map on second homology, must be the identify from $\mathbb Z $ to $\mathbb Z$. But since $$ (H\circ K)_{*} = H_{*} \circ K_{*}$$ this map must factor through $H_2(SO(3)) = 0$, which is impossible.

Thus: There's no continuous solution to the "rotate one vector to another" problem, a fact that I should have mentioned back in our original paper. Sigh. Hindshight is 20-20.

This is the right general approach, but the corner case $\|a\times b\| \approx 0$ must be handled.

If $\theta < \epsilon,$ $R=I$.

If $\pi-\theta < \epsilon$, you can choose for $\mathbf{x}$ any vector orthogonal to $\mathbf{a}$, for instance $\mathbf{x} = \frac{\mathbf{a} \times e_i}{\|\mathbf{a}\times e_i\|}$, where $i$ is the index of the component of $\mathbf{a}$ with least magnitude.

Let $F_n(x) = x - {2 \over n . n} n (n \ . \ x)$ be the transformation that reflects $x$ through the plane that is perpendicular to $n$. Composing two reflections gives a rotation: if the angle from $a$ to $b$ is $\phi$ then $F_b(F_a(x))$ rotates $x$ on the plane spanned by $a$ and $b$ by $2 \phi$.

Given the normalized vectors $\hat a=a/|a|$, $\hat b=b/|b|$, $\hat c=c/|c|$, where $c = \hat a + \hat b$, the angle from $\hat a$ to $\hat c$ is $\phi$, half the angle from $\hat a$ to $\hat b$.

In fact, $F_{\hat c}(F_{\hat a}(x))$ rotates $x$ by $\phi$ on the plane spanned by $a$ and $b$. This is valid even if $a$ and $b$ are parallel. This avoids computing a cross product, inverse cosine, sine and cosine, or division by a magnitude that can be arbitrarily close to zero.