Let us assume I have a set of planes
$$f_i(a_i,b_i,c_i,d_i) = a_ix+b_iy+c_iz + d_i = 0,\hspace{1cm} i \in \{0,\cdots,N\}$$
Then for each such plane $i$ I define a subset defined by selecting one of the following constraints for every $j$ :
- $f_j > 0$
- $f_j < 0$
- $0 = 0$ ("Don't care")
- $f_j = 0$ (If for some reason we will want to define lines or points)
The constraints are combined using logical and.
For a minimal example $z = 0$ & $x<1$ & $x>-1$ & $y<1$ & $y>-1$ shall define a square of side 2 centered in the x-y plane.
Now to my question :
Will this construction if we allow $N$ to be large enough be enough to define any piecewisely linear surface in $\mathbb R^3$ ? My intuition tells me that yes, it will be possible.
Own work: Once upon a time I tried creating a similar construction but without ability to select constraint of the forms 3 and 4, and then my conclusion (yet again without proof) that it probably will be enough for any convex shape in $\mathbb R^3$.
Part of my reasoning is that since we do not have any limit considering the uniqueness for the quadruple (a,b,c,d). We can define any number of convex shapes lying in any same plane and then just select the 0=0 don't care condition between those. So we can construct non-convex shapes as the union of such convex shapes.
As a follow up question, if the answer to the previous one is affirmative, will we be able to "compress" the representation?
I mean, any linear "piece" of an object constructed this way will be limited by it's closest surrounding planes. This gives me the intuition that we will be able to get away selecting constraints from 1 and 2 only for those closest planes and for all the rest condition 3 shall suffice. The importance of being able to assure this will be that we can escape the otherwise quadratic complexity of constraints by only coding the sparse subset which are not of type 3 or 4. Sure it is only a couple of bits per combination, but square complexity can nevertheless become a beast as $N$ grows and we can not ensure sparsity.