A polygon is defined by a collection of rings. Each ring is a collection of contiguous line segments such that the start point and the end point are the same.

Ring

Boundary and Rings

The boundary of a polygon is the collection of rings by which the polygon is defined. The boundary contains one or more outer rings and zero or more inner rings. An outer ring is oriented clockwise while an inner ring is oriented counterclockwise. Imagine walking clockwise along an outer ring. The area to your immediate right is the interior of the polygon and to your left is the exterior.

Outer Ring

Similarly, if you were to walk counter-clockwise along an inner ring, the area to your immediate right is the interior of the polygon and to your left is the exterior.

Outer & Inner Ring

It is important to understand the boundary, interior and exterior of a geometry when using the various operators. The relational operators, for example, rely heavily on these concepts.

If a polygon has an inner ring, the inner ring looks like a hole. If the hole contains another outer ring, that outer ring looks like an island.

1 outer ring, no inner rings 4 outer rings, no inner rings 1 outer ring, 1 inner ring 2 outer rings, 1 inner ring

Valid polygons

A valid polygon has the following main property:

  • for every polygon segment, the polygon exterior is unambiguously to the left of the segment and the polygon interior is unambiguously to the right of the segment.
  • As a result of the above property the following is also true:
  • segments can touch other segments only at the end points,
  • segments have non-zero length,
  • outer rings are clockwise and holes are counterclockwise,
  • each polygon ring has non-zero area.
  • order of the rings does not matter,
  • rings can be self-tangent,
  • rings can not overlap.
  • A valid polygon is said to be simple. See the Simplify operator for a more detailed specification of simple polygons. Note that a simple polygon may not be OGC simple. See the Simplify operator with OGC restrictions.

    Examples

    Let's look at some examples of non-simple vs. simple polygons. The green circles are the vertices of the polygon, and the lavender colored area represents the interior of the polygon.

    Non-simple polygons
    Self-intersection Self-intersection Dangling segment Dangling segment Overlapping rings


    Simple polygons
    No self-intersection Self-intersection at vertex No dangling segment No dangling segment No overlapping rings

    When drawing a polygon, use the even-odd fill rule. The even-odd fill rule will guarantee that the polygon will draw correctly even if the ring orientation is not as described above for simple polygons.

    JSON format

    A polygon can be represented as a JSON string. A polygon in JSON format contains an array of rings and an optional spatialReference. A polygon can also have boolean-valued hasM and hasZ fields. The default value is false for both the hasM and hasZ fields.

    Each ring is represented by an array of points. Exterior rings are oriented clockwise, while interior rings are oriented counter-clockwise. The order of the rings is irrelevant. The first point of each ring is always the same as the last point. Each point in a ring is represented as an array of numbers. See the description of JSON multipoints for details on the interpretation of the point arrays.

    An empty polygon is represented with an empty array for the rings field.

    Syntax

    {
      "hasZ" : true | false,
      "hasM" : true | false,
      "rings": [
                 [[<x11>,<y11>,<z11>,<m11>],[<x12>,<y12>,<z12>,<m12>], ... ,[<x1j>,<y1j>,<z1j>,<m1j>]],
                 ... ,
                 [[<xn1>,<yn1>,<zn1>,<mn1>],[<xn2>,<yn2>,<zn2>,<mn2>], ... ,[<xnk>,<ynk>,<znk>,<mnk>]]
               ],
      "spatialReference" : {"wkid" : <wkid>}
    }

    2D Polygon

    {
      "rings": [
                 [[6453,16815],[10653,16423],[14549,5204],[-7003,6939],[6453,16815]],
                 [[914,7992],[3140,11429],[1510,10525],[914,7992]]
               ],
      "spatialReference" : {"wkid" : 54004}
    }
    

    3D Polygon with Ms

    Note that the third point does not have a z-value, and the second ring does not have any m-values.

    {
      "hasZ" : true,
      "hasM" : true,
      "rings": [
                 [[6453,16815,35,1],[10653,16423,36,2],[14549,5204,null,3],[-7003,6939,37,4],[6453,16815,35,1]],
                 [[914,7992,30],[3140,11429,29],[1510,10525,28],[914,7992,30]]
               ],
      "spatialReference" : {"wkid" : 54004}
    }
    

    Empty Polygon

    {"rings": []}

    Creating a polygon

    To create a polygon, we can use the Polygon class methods or one of the import operators.

    Each of the code samples below creates a polygon with two outer rings and one inner ring.
    The polygon looks like this:

    Create this polygon

    Polygon class methods

    When using the Polygon class methods to create a polygon, the order in which the rings are created doesn't matter. What matters is the ring orientation. Remember, clockwise implies an outer ring whereas counter-clockwise implies an inner ring.

    To begin each ring, we call the startPath method and then successive calls to the lineTo method. We don't need to repeat the start point as the end point.

    public static Polygon createPolygon1()
    {
        Polygon poly = new Polygon();
    
        // clockwise => outer ring
        poly.startPath(0, 0);
        poly.lineTo(-0.5, 0.5);
        poly.lineTo(0, 1);
        poly.lineTo(0.5, 1);
        poly.lineTo(1, 0.5);
        poly.lineTo(0.5, 0);
    
        // hole
        poly.startPath(0.5, 0.2);
        poly.lineTo(0.6, 0.5);
        poly.lineTo(0.2, 0.9);
        poly.lineTo(-0.2, 0.5);
        poly.lineTo(0.1, 0.2);
        poly.lineTo(0.2, 0.3);
    
        // island
        poly.startPath(0.1, 0.7);
        poly.lineTo(0.3, 0.7);
        poly.lineTo(0.3, 0.4);
        poly.lineTo(0.1, 0.4);
    
        return poly;
    }
    

    Import from JSON

    As with the Polygon class methods, when using OperatorImportFromJson to create a polygon the order in which the rings are created doesn't matter. What matters is the ring orientation. Unlike the Polygon class methods, the start point of each ring must be repeated to specify the end point.

    The code shown below creates the same polygon as before, but notice that the inner ring that forms the hole is given before the outer ring. This was done just to drive home the point that the order of the rings doesn't matter when the polygon is in JSON format.

    static Polygon createPolygonFromJson() throws JsonParseException, IOException
    {
        String jsonString = "{\"rings\":[[[0.5,0.2],[0.6,0.5],[0.2,0.9],[-0.2,0.5],[0.1,0.2],[0.2,0.3],[0.5,0.2]],"
                          + "[[0.0,0.0],[-0.5,0.5],[0.0,1.0],[0.5,1.0],[1.0,0.5],[0.5,0.0],[0.0,0.0]],"
                          + "[[0.1,0.7],[0.3,0.7],[0.3,0.4],[0.1,0.4],[0.1,0.7]]],"
                          + " \"spatialReference\":{\"wkid\":4326}}";
    
        MapGeometry mapGeom = OperatorImportFromJson.local().execute(Geometry.Type.Polygon, jsonString);
    
        return (Polygon)mapGeom.getGeometry();
    }
    

    Import from GeoJSON

    Unlike the Polygon class methods and OperatorImportFromJson, when using OperatorImportFromGeoJson to create a polygon the order in which the rings are given does matter. Within an array of rings, the outer ring is always first followed by zero or more inner rings. However, the order of the arrays of rings doesn't matter. The start point of each ring must be repeated to specify the end point.

    The code shown below creates the same polygon as in the previous examples.

    static Polygon createPolygonFromGeoJson() throws JsonParseException, IOException
    {
        String geoJsonString = "{\"type\":\"MultiPolygon\","
                             + "\"coordinates\":[[[[0.0,0.0],[-0.5,0.5],[0.0,1.0],[0.5,1.0],[1.0,0.5],[0.5,0.0],[0.0,0.0]],"
                             + "[[0.5,0.2],[0.6,0.5],[0.2,0.9],[-0.2,0.5],[0.1,0.2],[0.2,0.3],[0.5,0.2]]],"
                             + "[[[0.1,0.7],[0.3,0.7],[0.3,0.4],[0.1,0.4],[0.1,0.7]]]],"
                             + "\"crs\":\"EPSG:4326\"}";
    
        MapGeometry mapGeom = OperatorImportFromGeoJson.local().execute(GeoJsonImportFlags.geoJsonImportDefaults, Geometry.Type.Polygon, geoJsonString, null);
    
        return (Polygon)mapGeom.getGeometry();
    }
    

    Import from WKT

    As with OperatorImportFromGeoJson, when using OperatorImportFromWkt to create a polygon the order in which the rings are given does matter. Within an array of rings, the outer ring is always first followed by zero or more inner rings. However, the order of the arrays of rings doesn't matter. The start point of each ring must be repeated to specify the end point.

    The code shown below creates the same polygon as in the previous examples, but notice that the outer ring that forms the island is given first. This was done for illustrative purposes. It could have been given last, which probably would be more understandable to the reader.

    static Polygon createPolygonFromWKT() throws JsonParseException, IOException
    {
        String wktString = "MULTIPOLYGON (((0.1 0.7, 0.1 0.4, 0.3 0.4, 0.3 0.7, 0.1 0.7)),"
                         + "((0 0, 0.5 0, 1 0.5, 0.5 1, 0 1, -0.5 0.5, 0 0),"
                         + "(0.5 0.2, 0.2 0.3, 0.1 0.2, -0.2 0.5, 0.2 0.9, 0.6 0.5, 0.5 0.2)))";
    
        Geometry geom = OperatorImportFromWkt.local().execute(WktImportFlags.wktImportDefaults, Geometry.Type.Polygon, wktString, null);
    
        return (Polygon)geom;
    }
    

    Extracting Outer Rings

    Many workflows require getting only the outer rings of a polygon.

    The method getAllOuterRings extracts all the outer rings of a polygon. The call to OperatorSimplifyOGC makes sure there are no self-intersections, all outer rings are clockwise, and holes are counterclockwise. You can skip the call to OperatorSimplifyOGC if you trust your input polygon.

    static List<Polygon> getAllOuterRings(Polygon polygon)
    {
      Polygon simplePolygon = (Polygon)OperatorSimplifyOGC.local().execute(polygon, null, true, null);
      ArrayList<Polygon> rings = new ArrayList<Polygon>();
      int n = simplePolygon.getPathCount();
      for (int i = 0; i < n; i++)
      {
        if (simplePolygon.calculateRingArea2D(i) <= 0)
          continue;
    
        Polygon ring = new Polygon();
        ring.addPath(simplePolygon, i, true);
        rings.add(ring);
      }
    
      return rings;
    }
    

    The method getOutermostRings extracts the outermost rings, that is, only rings that are not contained inside of any hole.

    static List<Polygon> getOutermostRings(Polygon polygon)
    {
      List<Polygon> allOuterRings = getAllOuterRings(polygon);
      GeometryCursor outerRingsCursor = new SimpleGeometryCursor(allOuterRings);
      GeometryCursor unionCursor = OperatorUnion.local().execute(outerRingsCursor, null, null);
      Geometry unionPoly = unionCursor.next();
    
      // Holes could have been produced during the union, so rerun just in case.
      return getAllOuterRings((Polygon)unionPoly);
    }