find coordinates of point of intersection of 3 planes

Question

i have normal to a plane and its distance from origin i.e. 4 components for each plane. if i have given 3 such planes and know that they are intersecting at a single point. how do i calculate coordinates of that point in my C program?

Answer 1

If your normal is (a, b, c), then your plane is given by the equation ax+by+cz=d, where d is chosen to match the distance from the origin.

Then you have three equations to solve, say:

$a_1x+b_1y+c_1z=d$

$a_2x+b_2y+c_2z=e$

$a_3x+b_3y+c_3z=f$

This is easiest done by writing them in the form of $M \cdot \left\{\begin{array}{c}x\\y\\z\end{array}\right\} = \left\{\begin{array}{c}d\\e\\f\end{array}\right\}$, where $M=\left\{\begin{array}{rcl}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{array}\right\}$, and multiplying on the left by $M^{-1}$.

I can give more details if needed; just let me know which part is not clear.