# CircularString

**Applies to:**

A **CircularString** is a collection of zero or more continuous circular arc segments. A circular arc segment is a curved segment defined by three points in a two-dimensional plane; the first point cannot be the same as the third point. If all three points of a circular arc segment are collinear, the arc segment is treated as a line segment.

## CircularString instances

The drawing below shows valid **CircularString** instances:

### Accepted instances

A **CircularString** instance is accepted if it is either empty or contains an odd number of points, n, where n > 1. The following **CircularString** instances are accepted.

```
DECLARE @g1 geometry = 'CIRCULARSTRING EMPTY';
DECLARE @g2 geometry = 'CIRCULARSTRING(1 1, 2 0, -1 1)';
DECLARE @g3 geometry = 'CIRCULARSTRING(1 1, 2 0, 2 0, 2 0, 1 1)';
```

`@g3`

shows that **CircularString** instance may be accepted, but not valid. The following CircularString instance declaration is not accepted. This declaration throws a `System.FormatException`

.

```
DECLARE @g geometry = 'CIRCULARSTRING(1 1, 2 0, 2 0, 1 1)';
```

### Valid instances

A valid **CircularString** instance must be empty or have the following attributes:

- It must contain at least one circular arc segment (that is, have a minimum of three points).
- The last endpoint for each circular arc segment in the sequence, except for the last segment, must be the first endpoint for the next segment in the sequence.
- It must have an odd number of points.
- It cannot overlap itself over an interval.
- Although
**CircularString**instances may contain line segments, these line segments must be defined by three collinear points.

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

```
DECLARE @g1 geometry = 'CIRCULARSTRING EMPTY';
DECLARE @g2 geometry = 'CIRCULARSTRING(1 1, 2 0, -1 1)';
DECLARE @g3 geometry = 'CIRCULARSTRING(1 1, 2 0, 2 0, 1 1, 0 1)';
DECLARE @g4 geometry = 'CIRCULARSTRING(1 1, 2 2, 2 2)';
SELECT @g1.STIsValid(), @g2.STIsValid(), @g3.STIsValid(),@g4.STIsValid();
```

A **CircularString** instance must contain at least two circular arc segments to define a complete circle. A **CircularString** instance cannot use a single circular arc segment (such as (1 1, 3 1, 1 1)) to define a complete circle. Use (1 1, 2 2, 3 1, 2 0, 1 1) to define the circle.

The following example shows CircularString instances that are not valid.

```
DECLARE @g1 geometry = 'CIRCULARSTRING(1 1, 2 0, 1 1)';
DECLARE @g2 geometry = 'CIRCULARSTRING(0 0, 0 0, 0 0)';
SELECT @g1.STIsValid(), @g2.STIsValid();
```

### Instances with collinear points

In the following cases a circular arc segment will be treated as a line segment:

- When all three points are collinear (for example, (1 3, 4 4, 7 5)).
- When the first and middle point are the same, but the third point is different (for example, (1 3, 1 3, 7 5)).
- When the middle and last point are the same, but the first point is different (for example, (1 3, 4 4, 4 4)).

## Examples

### A. Instantiating a Geometry Instance with an Empty CircularString

This example shows how to create an empty **CircularString** instance:

```
DECLARE @g geometry;
SET @g = geometry::Parse('CIRCULARSTRING EMPTY');
```

### B. Instantiating a Geometry Instance Using a CircularString with One Circular Arc Segment

The following example shows how to create a **CircularString** instance with a single circular arc segment (half-circle):

```
DECLARE @g geometry;
SET @g = geometry:: STGeomFromText('CIRCULARSTRING(2 0, 1 1, 0 0)', 0);
SELECT @g.ToString();
```

### C. Instantiating a Geometry Instance Using a CircularString with Multiple Circular Arc Segments

The following example shows how to create a **CircularString** instance with more than one circular arc segment (full circle):

```
DECLARE @g geometry;
SET @g = geometry::Parse('CIRCULARSTRING(2 1, 1 2, 0 1, 1 0, 2 1)');
SELECT 'Circumference = ' + CAST(@g.STLength() AS NVARCHAR(10));
```

Here is the result set.

```
Circumference = 6.28319
```

Compare the output when **LineString** is used instead of **CircularString**:

```
DECLARE @g geometry;
SET @g = geometry::STGeomFromText('LINESTRING(2 1, 1 2, 0 1, 1 0, 2 1)', 0);
SELECT 'Perimeter = ' + CAST(@g.STLength() AS NVARCHAR(10));
```

Here is the result set.

```
Perimeter = 5.65685
```

Notice that the value for the **CircularString** example is close to 2∏, which is the actual circumference of the circle.

### D. Declaring and Instantiating a Geometry Instance with a CircularString in the Same Statement

This snippet shows how to declare and instantiate a **geometry** instance with a **CircularString** in the same statement:

```
DECLARE @g geometry = 'CIRCULARSTRING(0 0, 1 2.1082, 3 6.3246, 0 7, -3 6.3246, -1 2.1082, 0 0)';
```

### E. Instantiating a Geography Instance with a CircularString

The following example shows how to declare and instantiate a **geography** instance with a **CircularString**:

```
DECLARE @g geography = 'CIRCULARSTRING(-122.358 47.653, -122.348 47.649, -122.348 47.658, -122.358 47.658, -122.358 47.653)';
```

### F. Instantiating a Geometry Instance with a CircularString that is a Straight Line

The following example shows how to create a **CircularString** instance that is a straight line:

```
DECLARE @g geometry;
SET @g = geometry::STGeomFromText('CIRCULARSTRING(0 0, 1 2, 2 4)', 0);
```

## See Also

Spatial Data Types Overview

CompoundCurve

MakeValid (geography Data Type)

MakeValid (geometry Data Type)

STIsValid (geometry Data Type)

STIsValid (geography Data Type)

STLength (geometry Data Type)

STStartPoint (geometry Data Type)

STEndpoint (geometry Data Type)

STPointN (geometry Data Type)

STNumPoints (geometry Data Type)

STIsRing (geometry Data Type)

STIsClosed (geometry Data Type)

STPointOnSurface (geometry Data Type)

LineString