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### Re^3: Speeding up point-in-polygon -- take two

by roboticus (Chancellor)
 on Aug 29, 2006 at 15:19 UTC ( #570189=note: print w/replies, xml ) Need Help??

punkish:

What I really have to figure out is to reduce those tests.
That's why I made my original suggestion. There are two ways that breaking the polygons into simple shapes will help you:

• With known shapes, your _pointIsInPolygon function is simpler, so you can use quick exits and/or the compiler can optimize it, and
• you can increase your density of polygon vs. bounding boxes.
In your original _pointIsInPolygon, you use variable indices into your vertex arrays, as well as adding a loop with it's inherent termination condition. By breaking your polygons into triangles or quadrilaterals, you can use fixed indices and omit the loop. That may significantly increase the subroutines speed. Also, your algorithm requires that you examine every edge in the polygon. By enforcing an order to the edges, you may be able to return early from the routine by taking advantage of that knowledge.

Regarding the 'density of polygon vs. bounding boxes': A rectangle aligned with your X and Y axes has a 100% chance of containing a point returned by your SELECT statement. An arbitrary triangle has a 50% chance of containing a point in the bounding rectangle. So breaking your polygons into triangles and (aligned) rectangles will have a worst case coverage of 50%. And a higher coverage density directly correlates to a lower false-positive rate. (I.e., with a higher coverage, you'll need fewer tests because your SELECT will hit fewer bounding boxes.) With arbitrary polygons, you can have nearly *any* hit rate:

```*--------------------*  *--------------------*  *--------------------*
|                    |   \                  /   |*------------------*|
|                    |    \                /    ||                  ||
|                   **     \              /     ||                  ||
*-------------------*      *-------------*      **                  **
~99%                       ~90                      ~32%

While I still think my original suggestion would be interesting, you state:

Yes, the polys are relatively static, although the points are not so. Nevertheless, I do indeed unroll both polys and points, and move them into the SQLite db. That way I avoid going back to the files repeatedly.
By this, I'm assuming you mean that the points are in the same rough relative position in the polygons, meaning that (1) the points are connected in the same order, and (2) the edges will never cross. For example, the shape on the left could morph into the shape in the center, but not the one on the right:

```   a----g        g             g
/     |       / \           / \
/      |      /   \         /   \
b       | ==  /     \  !=   /     \
\  e---f    /       \     /       \
\  \      a-b       f   a  c     e
c---d      /       /     \/|    /
c---d   /      /\|   /
\ /      /  b  /
e      d-----f
In that case, breaking the polygons into X-aligned trapezoids (my original suggestion) may be a bit too restrictive. Perhaps using arbitrary quadrilaterals (or triangles) would give you (a) enough coverage density (i.e.just breaking the shape into quadrilaterals instead of trapezoids may simplify things enough. So you'd increase your coverage, and by breaking the polygons into triangles, you could still simplify your _pointIsInPolygon subroutine to eliminate the loops and variable indices.

--roboticus

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