An algorithm for inflating/deflating (offsetting, buffering) polygons

Question

How would I "inflate" a polygon? That is, I want to do something similar to this:

<

Answer 1

For these types of things I usually use JTS. For demonstration purposes I created this jsFiddle that uses JSTS (JavaScript port of JTS). You just need to convert the coordinates you have to JSTS coordinates:

function vectorCoordinates2JTS (polygon) {
  var coordinates = [];
  for (var i = 0; i < polygon.length; i++) {
    coordinates.push(new jsts.geom.Coordinate(polygon[i].x, polygon[i].y));
  }
  return coordinates;
}

The result is something like this:

Additional info: I usually use this type of inflating/deflating (a little modified for my purposes) for setting boundaries with radius on polygons that are drawn on a map (with Leaflet or Google maps). You just convert (lat,lng) pairs to JSTS coordinates and everything else is the same. Example:

Answer 2

Here is an alternative solution, see if you like this better.

1. Do a triangulation, it don't have to be delaunay -- any triangulation would do.

2. Inflate each triangle -- this should be trivial. if you store the triangle in the anti-clockwise order, just move the lines to right-hand-side and do intersection.

3. Merge them using a modified Weiler-Atherton clipping algorithm

Answer 3

I thought I might briefly mention my own polygon clipping and offsetting library - Clipper.

While Clipper is primarily designed for polygon clipping operations, it does polygon offsetting too. The library is open source freeware written in Delphi, C++ and C#. It has a very unencumbered Boost license allowing it to be used in both freeware and commercial applications without charge.

Polygon offsetting can be performed using one of three offset styles - squared, round and mitered.

Answer 4

Based on advice from @JoshO'Brian, it appears the rGeos package in the R language implements this algorithm. See rGeos::gBuffer .

Answer 5

Each line should split the plane to "inside" and "outline"; you can find this out using the usual inner-product method.

Move all lines outward by some distance.

Consider all pair of neighbor lines (lines, not line segment), find the intersection. These are the new vertex.

Cleanup the new vertex by removing any intersecting parts. -- we have a few case here

(a) Case 1:

 0--7  4--3
 |  |  |  |
 |  6--5  |
 |        |
 1--------2

if you expend it by one, you got this:

0----a----3
|    |    |
|    |    |
|    b    |
|         |
|         |
1---------2

7 and 4 overlap.. if you see this, you remove this point and all points in between.

(b) case 2

 0--7  4--3
 |  |  |  |
 |  6--5  |
 |        |
 1--------2

if you expend it by two, you got this:

0----47----3
|    ||    |
|    ||    |
|    ||    |
|    56    |
|          |
|          |
|          |
1----------2

to resolve this, for each segment of line, you have to check if it overlap with latter segments.

(c) case 3

       4--3
 0--X9 |  |
 |  78 |  |
 |  6--5  |
 |        |
 1--------2

expend by 1. this is a more general case for case 1.

(d) case 4

same as case3, but expend by two.

Actually, if you can handle case 4. All other cases are just special case of it with some line or vertex overlapping.

To do case 4, you keep a stack of vertex.. you push when you find lines overlapping with latter line, pop it when you get the latter line. -- just like what you do in convex-hull.

Answer 6

The polygon you are looking for is called inward/outward offset polygon in computational geometry and it is closely related to the straight skeleton.

These are several offset polygons for a complicated polygon:

And this is the straight skeleton for another polygon:

As pointed out in other comments, as well, depending on how far you plan to "inflate/deflate" your polygon you can end up with different connectivity for the output.

From computation point of view: once you have the straight skeleton one should be able to construct the offset polygons relatively easily. The open source and (free for non-commercial) CGAL library has a package implementing these structures. See this code example to compute offset polygons using CGAL.

The package manual should give you a good starting point on how to construct these structures even if you are not going to use CGAL, and contains references to the papers with the mathematical definitions and properties:

CGAL manual: 2D Straight Skeleton and Polygon Offsetting