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My question is about a labeling scheme for the cells (convex polygon regions) resulting from the division of a 2D plane by a set of lines.

I am seeking answer for the Euclidean (flat) 2D plane, although I don't know whether the answers might be equally applicable to the 2D Euclidean plane and the 2D Projective plane.

I already know the maximum number of cells from such divisions:

$1 + {n \over 2}(n + 1)$ , or $\binom{n+1}{2} + 1$

Now, using the ideas from the comments from this question, one can derive a scheme for labeling the cells with a bitstring.

  1. Enumerate the lines with natural numbers, starting with 0, 1, 2, ...
  2. For each line, obtain its line equation in the form of $ax+by+c = 0$. Take the parameters (a, b, c).
  3. Notice that given a line $(a, b, c)$ and a point $(x, y)$ on the plane, we can decide if the point falls on one side of the line (arbitrarily chosen when the (a, b, c) were taken) or the other side, by testing the sign of $ax+by+c$ to be positive or negative.
  4. The $k$-th bit of the bitstring that labels a cell is thus the sign of the points interior to the cell tested against the $k$-th line, with positive mapped to a bit value of 1 and negative mapped to 0.

Sorry for the verbose and confusing description. Please refer to the comments in the linked question for an alternative description. The Wikipedia article of Convex polytope also has a section on intersection of half-spaces which describes the same.

Here is my question:

It is obvious that for a given 2D plane division by $n$ distinct lines, the vast majority of bitstrings of length $n$ do not correspond to any cells, as is evident when one compares $2^n$ against $\binom{n+1}{2} + 1$.

Suppose the set of $n$ lines, and their parameters $(a_k, b_k, c_k)$ for the $k$-th line are given. How do we compute the set of valid bitstrings of length $n$ that correspond to a valid cell in the 2D plane division?

My goal is to seek an algorithm for computing this set of valid bitstrings, but I also appreciate any heuristics that can be obtained by mathematical means.

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rwong
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1 Answers1

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You should investigate oriented matroids. Since it's really late, I'm not perfectly sure which term in that world corresponds to cells of the arrangement, but I guess “tope” is the one to look for. In any case, deciding whether a given sign vector (your bitstring) is valid for an oriented matroid is easy, but deciding whether that oriented matroid is realizable, i.e. corresponds to a line configuration in the plane, is really hard.

MvG
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