# Polygon

**Applies to:**

A **Polygon** is a two-dimensional surface stored as a sequence of points defining an exterior bounding ring and zero or more interior rings.

## Polygon instances

A **Polygon** instance can be formed from a ring that has at least three distinct points. A **Polygon** instance can also be empty.

The exterior and any interior rings of a **Polygon** define its boundary. The space within the rings defines the interior of the **Polygon**.

The illustration below shows examples of **Polygon** instances.

As shown in the illustration:

Figure 1 is a

**Polygon**instance whose boundary is defined by an exterior ring.Figure 2 is a

**Polygon**instance whose boundary is defined by an exterior ring and two interior rings. The area inside the interior rings is part of the exterior of the**Polygon**instance.Figure 3 is a valid

**Polygon**instance because its interior rings intersect at a single tangent point.

### Accepted instances

Accepted **Polygon** instances are instances that can be stored in a **geometry** or **geography** variable without throwing an exception. The following are accepted **Polygon** instances:

- An Empty
**Polygon**instance - A
**Polygon**instance that has an acceptable exterior ring (**LineString**) and zero or more acceptable interior rings (**LineString**s)

The following criteria are needed for a ring (**LineString**) to be acceptable.

- The
**LineString**instance must be accepted. - The
**LineString**instance must have at least four points. - The starting and ending points of the
**LineString**instance must be the same.

The following example shows accepted **Polygon** instances.

```
DECLARE @g1 geometry = 'POLYGON EMPTY';
DECLARE @g2 geometry = 'POLYGON((1 1, 3 3, 3 1, 1 1))';
DECLARE @g3 geometry = 'POLYGON((-5 -5, -5 5, 5 5, 5 -5, -5 -5),(0 0, 3 0, 3 3, 0 3, 0 0))';
DECLARE @g4 geometry = 'POLYGON((-5 -5, -5 5, 5 5, 5 -5, -5 -5),(3 0, 6 0, 6 3, 3 3, 3 0))';
DECLARE @g5 geometry = 'POLYGON((1 1, 1 1, 1 1, 1 1))';
```

As `@g4`

and `@g5`

show an accepted **Polygon** instance may not be a valid **Polygon** instance. `@g5`

also shows that a Polygon instance needs to only contain a ring with any four points to be accepted.

The following examples throw a `System.FormatException`

because the **Polygon** instances are not accepted.

```
DECLARE @g1 geometry = 'POLYGON((1 1, 3 3, 1 1))';
DECLARE @g2 geometry = 'POLYGON((1 1, 3 3, 3 1, 1 5))';
```

`@g1`

is not accepted because the **LineString** instance for the exterior ring does not contain enough points. `@g2`

is not accepted because the starting point of the exterior ring **LineString** instance is not the same as the ending point. The following example has an acceptable exterior ring, but the interior ring is not acceptable. This also throws a `System.FormatException`

.

```
DECLARE @g geometry = 'POLYGON((-5 -5, -5 5, 5 5, 5 -5, -5 -5),(0 0, 3 0, 0 0))';
```

### Valid instances

The interior rings of a **Polygon** can touch both themselves and each other at single tangent points, but if the interior rings of a **Polygon** cross, the instance is not valid.

The following example shows valid **Polygon** instances.

```
DECLARE @g1 geometry = 'POLYGON((-20 -20, -20 20, 20 20, 20 -20, -20 -20))';
DECLARE @g2 geometry = 'POLYGON((-20 -20, -20 20, 20 20, 20 -20, -20 -20), (10 0, 0 10, 0 -10, 10 0))';
DECLARE @g3 geometry = 'POLYGON((-20 -20, -20 20, 20 20, 20 -20, -20 -20), (10 0, 0 10, 0 -10, 10 0), (-10 0, 0 10, -5 -10, -10 0))';
SELECT @g1.STIsValid(), @g2.STIsValid(), @g3.STIsValid();
```

`@g3`

is valid because the two interior rings touch at a single point and do not cross each other. The following example shows `Polygon`

instances that are not valid.

```
DECLARE @g1 geometry = 'POLYGON((-20 -20, -20 20, 20 20, 20 -20, -20 -20), (20 0, 0 10, 0 -20, 20 0))';
DECLARE @g2 geometry = 'POLYGON((-20 -20, -20 20, 20 20, 20 -20, -20 -20), (10 0, 0 10, 0 -10, 10 0), (5 0, 1 5, 1 -5, 5 0))';
DECLARE @g3 geometry = 'POLYGON((-20 -20, -20 20, 20 20, 20 -20, -20 -20), (10 0, 0 10, 0 -10, 10 0), (-10 0, 0 10, 0 -10, -10 0))';
DECLARE @g4 geometry = 'POLYGON((-20 -20, -20 20, 20 20, 20 -20, -20 -20), (10 0, 0 10, 0 -10, 10 0), (-10 0, 1 5, 0 -10, -10 0))';
DECLARE @g5 geometry = 'POLYGON((10 0, 0 10, 0 -10, 10 0), (-20 -20, -20 20, 20 20, 20 -20, -20 -20) )';
DECLARE @g6 geometry = 'POLYGON((1 1, 1 1, 1 1, 1 1))';
SELECT @g1.STIsValid(), @g2.STIsValid(), @g3.STIsValid(), @g4.STIsValid(), @g5.STIsValid(), @g6.STIsValid();
```

`@g1`

is not valid because the inner ring touches the exterior ring in two places. `@g2`

is not valid because the second inner ring in within the interior of the first inner ring. `@g3`

is not valid because the two inner rings touch at multiple consecutive points. `@g4`

is not valid because the interiors of the two inner rings overlap. `@g5`

is not valid because the exterior ring is not the first ring. `@g6`

is not valid because the ring does not have at least three distinct points.

## Examples

### Example A.

The following example creates a simple `geometry`

`Polygon`

instance with a gap and SRID 10.

```
DECLARE @g geometry;
SET @g = geometry::STPolyFromText(
'POLYGON((0 0, 0 3, 3 3, 3 0, 0 0), (1 1, 1 2, 2 1, 1 1))',
10);
```

### Example B.

An instance that is not valid may be entered and converted to a valid `geometry`

instance. In the following example of a `Polygon`

, the interior and exterior rings overlap and the instance is not valid.

```
DECLARE @g geometry;
SET @g = geometry::Parse(
'POLYGON((1 0, 0 1, 1 2, 2 1, 1 0), (2 0, 1 1, 2 2, 3 1, 2 0))'
);
```

### Example C.

In the following example, the invalid instance is made valid with `MakeValid()`

.

```
SET @g = @g.MakeValid();
SELECT @g.ToString();
```

The `geometry`

instance returned from the above example is a `MultiPolygon`

.

```
MULTIPOLYGON (((2 0, 3 1, 2 2, 1.5 1.5, 2 1, 1.5 0.5, 2 0)),
((1 0, 1.5 0.5, 1 1, 1.5 1.5, 1 2, 0 1, 1 0)))
```

### Example D.

This is another example of converting an invalid instance to a valid geometry instance. In the following example the `Polygon`

instance has been created using three points that are exactly the same:

```
DECLARE @g geometry
SET @g = geometry::Parse('POLYGON((1 3, 1 3, 1 3, 1 3))');
SET @g = @g.MakeValid();
SELECT @g.ToString()
```

The geometry instance returned above is a `Point(1 3)`

. If the `Polygon`

given is `POLYGON((1 3, 1 5, 1 3, 1 3))`

then `MakeValid()`

would return `LINESTRING(1 3, 1 5)`

.

## See Also

STArea (geometry Data Type)

STExteriorRing (geometry Data Type)

STNumInteriorRing (geometry Data Type)

STInteriorRingN (geometry Data Type)

STCentroid (geometry Data Type)

STPointOnSurface (geometry Data Type)

MultiPolygon

Spatial Data (SQL Server)

STIsValid (geography Data Type)

STIsValid (geometry Data Type)