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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 $n_G$.

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Unless I'm missing something, I see only two equations in three unknowns. We expect to find a solution set that is "generically" one-dimensional. In other words, we won't be able to find a finite set of solutions. – Shaun Ault Oct 19 '11 at 20:56
Please let me know if I copied any of the equations incorrectly. – Brian M. Scott Oct 19 '11 at 21:09
Sorry for not explaining my notation- it's set of four equations with four unknowns: the equations (1) have index i (i=1 to 3) for each of three circles. In the first (vector) representation $n_G$ and $D$ are unknowns, but $n_G$ consists of three components ($x$, $y$ and $z$ in second representation)... edit: Thanks Brian, they're picture perfect and accurate. – Tomislav Petričević Oct 19 '11 at 21:12

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