## Thursday, July 12, 2007

### The ultimate Point-In-Polygon implementation

The classic stabbing line/ray-crossing point-in-polygon algorithm has been around for longer than computational geometry itself. (See William Franklin and SoftSurfer for background and description. ) The algorithm as usually published is short, effective and clever. But it does have some limitations:
1. It is implemented as an iteration over a data structure representing a sequence of line segments. This ties the algorithm to a fixed data structure, which leads to code duplication
2. It handles simple rings, but not fully general polygons (which may contain holes and/or multiple shells)
3. It does not detect the case where the query point lies on one of the segments in the polygon
4. It runs in O(n) time. This is reasonable for a single query point, but is not the most efficient way to perform many point queries against a single fixed polygon
5. It uses conventional floating-point arithmetic. This makes it non-robust (i.e. potentially incorrect for some inputs)
This predicate forms a key component of many spatial processing algorithms. In JTS it's used all over the place. The more places it's used, the more the above limitations have become apparent. Over time I've been slowly chipping away at removing them.

Most recently, I've revisited P-I-P in the course of some work on improving the performance of spatial predicates. And I think I've finally arrived at the ultimate Point-In-Polygon implementation. It removes all of the above limitations with the following design features:
1. It uses an incremental design, which frees the algorithm from being tied to any particular segment data structure
2. It handles all polygonal geometry types defined in JTS. (Implementing this was made much cleaner by the pleasant realization that the PIP algorithm really doesn't care where in a polygonal geometry the segments come from. The only thing that matters is the parity of the segment crossings. This holds true for any number of holes and shells.)
3. It detcts the point-on-segment case, and reports it via a trivalent return value (in the set {INTERIOR, BOUNDARY, EXTERIOR}).
4. It is straightforward to use with a spatial index on the segments of the polygon (such as STRtree or BinTree), thus providing O(log n) performance
5. Thanks to the RobustDeterminant class in JTS, the implementation is much more robust. (Due to some remaining floating-point arithmetic it may still not be 100% robust - but I hope to address that sometime as well.)
Even with all these changes, it's not much more complex than the original algorithm (at least, not given various supporting components which are already available in JTS.)

Coming soon to a JTS version near you...

#### 1 comment:

moovida said...

Hi Martin, I think I will come more often back to this blog, you should have told at the conference :)

I already used some of your tricks from the foss4g presentation. Now I can't wait to try the 1.9 whenever it comes.

Great job man!