Measuring 3D Rotation from Two Values

Question

Suppose there is an matrix $T$ with unknown elements, but it it known that $T$ is a linear transformation representing a series of rotations.

Without knowing anything about how this rotation was performed (there are a number of ways to represent it anyway), how much information can I obtain about this rotation from only the following information:

The projection of the transformed x-axis onto the original x-axis; that is: $$\begin{bmatrix}1 \\ 0\\ 0\end{bmatrix}\cdot T\begin{bmatrix}1 \\ 0\\ 0\end{bmatrix}$$ And the projection of the transformed z-axis onto the original z-axis, that is: $$\begin{bmatrix}0 \\ 0\\ 1\end{bmatrix}\cdot T\begin{bmatrix}0 \\ 0\\ 1\end{bmatrix}$$

From these "measurements", is it possible to determine enough information such that there is only one degree of freedom in constructing $T$ from a series of rotations?

Answer 1

In a sense, yes. For instance, in the Wikipedia page you linked to, there is a way to represent a rotation by a unit quaternion $q=w+ix+jy+kz$ (with $w^2+x^2+y^2+z^2=1$ and $w\ge0$): $$ T = \begin{bmatrix} 1 - 2 y^2 - 2 z^2 & 2 x y - 2 z w & 2 x z + 2 y w \\ 2 x y + 2 z w & 1 - 2 x^2 - 2 z^2 & 2 y z - 2 x w \\ 2 x z - 2 y w & 2 y z + 2 x w & 1 - 2 x^2 - 2 y^2 \end{bmatrix}. $$ So, if $t_{11}$ and $t_{33}$ are given, define $a=\frac12(1-t_{11})$ and $b=\frac12(1-t_{33})$. Then $y^2+z^2=a$ and $x^2+y^2=b$. Hence $$ \sqrt{\max\left(\min(a,\,b),\,a+b-1\right)}\le|y|\le\sqrt{\max(a,\,b)}. $$ Once you have chosen the free variable $|y|$ in the above range, you can determine $|x|,|z|$ and $w$ uniquely according to $x^2+y^2=b,\ y^2+z^2=a$ and $w^2+x^2+y^2+z^2=1$. Picking also the signs for $x,y,z$, you get eight different rotation matrices for each feasible $|y|$. In other words, there are eight loci of $T$, on each there is only one degree of freedom.

Answer 2

Each projection determines a circle on which the transformed unit vector must lie. For the transformed $x$-axis $\mathbf u$, we know $u_x$, so this circle is $x=u_x$, $y^2+z^2=1-u_x^2$. Similarly, the transformed $z$-axis, $\mathbf w$, lies somewhere on $z=w_z$, $x^2+y^2=1-w_z^2$. For any choice of $\mathbf u$, the corresponding $\mathbf w$ lies on the intersection of the latter circle and the plane $\mathbf u\cdot\mathbf p=0$, hence there can be zero, one or two corresponding vectors $\mathbf w$. These two vectors are the first and third columns of the rotation matrix. There are no intersections when $u_z^2\gt1-w_z^2$, so this constrains the possibilities for $\mathbf u$ somewhat. A symmetric constraint comes from considering the intersection of the plane $\mathbf w\cdot\mathbf p=0$ with the first circle: $w_x^2\le1-u_x^2$. The transformed $y$-axis, and hence the middle column of the rotation matrix is, of course, completely determined by $\mathbf u$ and $\mathbf w$.