Can one solve this system of qudratic equations unambigiously?

Question

Given the parameters $p_1, p_2, p_3$ I want to know if the following system can be solved:

$p_1 a + p_2 c + ef = 0\\ p_1 b + cd + p_3 f = 0\\ ab + p_2 d + p_3 f = 0\\ p_1^2 +a^2+b^2=1\\ c^2+p_2^2+d^2=1\\ e^2+f^2+p_3^2=1\\ cf-p_2 e=b\\ ae-p_1 f=d\\ p_1 p_2 -ac=p_3\\ $

Background:

This system actually discribes a Rotation Matrix

$p_1\ \ a\ \ \ \ b\\ c\ \ \ \ p_2\ \ d\\ e\ \ \ \ f\ \ \ \ p_3$

where only $p_1$, $p_2$ and $p_3$ are know. For example $p_1=p_2=p_3=1$ would result in $a=b=c=d=e=f=0$.

I would like to know two things:

• Why is this system (unabigiously) solvable / not solvable?
• If it is solvable, what is the solution? (Less important)

Answer 1

Let $$M = \left( \begin{array}{ccc} p_1 & a & b \\ c & p_2 & d \\ e & f & p_3 \end{array} \right).$$

If $M$ is a rotation matrix, then $M^{-1} = M^T.$ This implies that $M$ and $M^{-1}$ have the same entries on the main diagonal. But if $M$ is not the identity, $M \ne M^{-1},$ so the rotation matrix is not completely determined by the entries on its main diagonal.

On the other hand, if $[u_1\ u_2\ u_3]^T$ is a unit vector on the axis of rotation of $M$ and if $\theta$ is the angle of rotation about that axis, then $$ p_i = (1 - \cos\theta) u_i^2 + \cos\theta \tag{1}$$ for $i = 1, 2, 3.$ Moreover, $$p_1 + p_2 + p_3 = 1 + 2 \cos\theta.$$ Therefore we can express $\cos\theta$ in terms of $p_1,$ $p_2,$ and $p_3.$ Plug that value of $\cos\theta$ into equation $(1)$ for each $i;$ this either gives $u_1 = 0$ or gives two possible values of $u_1$ which differ only by a sign change.

We can safely assume that $0 \le \theta \le \pi,$ because the rotation described by angle $-\theta$ and unit vector $[u_1\ u_2\ u_3]^T$ is the same as the rotation described by angle $\theta$ and unit vector $[{-u_1}\ {-u_2}\ {-u_3}]^T.$ That means that in general there are eight possible ways to fill in the matrix $M$ (one for each choice of the signs of each of the $u_i),$ therefore eight possible solutions to the given set of equations. (For $0 < \theta < \pi,$ there are four solutions if exactly one of the $u_i$ is zero, two solutions if two of the $u_i$ are equal to zero. There are half as many solutions if $\theta = \pi,$ and of course only one solution if $\theta = 0.)$ Moreover, by computing the rotation matrix for the rotation by angle $\theta$ around the axis given by $[u_1\ u_2\ u_3]^T,$ we can compute all the unknown entries $a, b, c, d, e,$ and $f$ in the rotation matrix for a specific choice of $[u_1\ u_2\ u_3]^T.$

Answer 2

Yes. it can be solved. I need a case distinction. First assume that $p_1+p_2=0$. Then the system is obviously solved by $f=e=b=d=0$, $c=a$, and $a$ a solution of $a^2 + p_1^2 - 1=0$, i.e., with $a=\sqrt{1-p_1^2}$. In the second case we may assume that $p_1+p_2=1$ by rescaling. Then we may take $f=e$, and $$ a= -\frac{1}{p_1}(e^2 - (p_1 - 1)c), $$

$$ c=-\frac{1}{2e^2( 1- p_1)}(e^4 + 2e^2p_1 - e^2 - 2p_1 + 1); $$ with $e\neq 0$ and $p_1\neq 1$, and $e$ satisfying a monic polynomial equation of degree $8$, which has a solution over the complex numbers, but is rather complicated. Also, the subcases $p_1=1$ and $e=0$ can be solved.