I'm facing a rather trivial problem which I seem unable to solve... Not being a mathematician (but an engineer with a bit of knack for math), I managed to formulate it in a way that seemed solvable to me, but when I try to solve it seemingly I chase my own tail.
Here are the equations:
$\begin{align*} &r_{T_i}\sqrt{1-(\overrightarrow{n_G}\cdot\overrightarrow{n_{W_i}})^2}=\overrightarrow{n_G}\cdot\overrightarrow{W_{C_i}}+D, \qquad i=1,2,3\tag{1}\\ &|\overrightarrow{n_G}|=1\tag{2}\\ (1)\qquad\dots\qquad&aa_i\cdot x^2+bb_i\cdot y^2+cc_i\cdot z^2+ab_i\cdot x\cdot y+bc_i\cdot y\cdot z+ca_i\cdot z\cdot x+\\ &\qquad+a_i\cdot x\cdot D+b_i\cdot y\cdot D+c_i\cdot z\cdot D+D^2-(r_i)^2 = 0\\ (2)\qquad\dots\qquad&x^2+y^2+z^2=1 \end{align*}$
If the origin of the problem is of any consequence (in case anyone knows a simpler way to solve it) - I'm trying to get the plane tangent to three circles (ground plane defined by being tangential to three wheels). In those equations $n_G$ is the vector normal to the ground plane and $D$ it's distance from origin; $W_C$ are center points of wheels and $n_W$ are their axes (perpendicular to the wheel plane). In 2nd set of equations $x$, $y$ and $z$ are components of vector $n_G$.