plane intersection in practical sense

Question

suppose i want to model a 3D plane intersections when they make a corner. that is three or more than three planes will be intersected at some corners of cubes or other volumetric objects. i can find this corner point by considering as a plane intersection problem. also, i can do this by considering as line intersections problem. here, lines can be found by intrersecting two planes at a time.so, i would like to know this plane intersection and line intersection is given almost same result or two different results. suppose, i fitted planes by least square and found the best planes from the given set of points. now, i am confused and think these two method would be given 2 results. any comment please.

Answer 1

Here is what I have done in a similar situation. Call your three planes $A$, $B$, $C$, and their lines of intersection $ab$, $bc$, $ca$. Each pair of lines should intersect in a point, but likely they slightly miss. A good candidate is the midpoint of the shortest segment connecting the pair of lines. So, you could write a routine to compute that shortest segment (you can find the computation in many locations on the web, e.g., this StackOverflow question); let's call it $m(L_1,L_2)$ for two lines $L_1$ and $L_2$. Then average the three points:
$$\frac{1}{3} \left[ m( ab, bc ) + m( bc, ca ) + m(ca, ab) \right] \;.$$ This is a bit of a hack, but it is computationally easy and recognizes the numerical realities.