CircularString

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)';

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();

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)).

This example shows how to create an empty CircularString instance:

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

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();

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));  

This produces the following output:

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));

This produces the following output:

Perimeter = 5.65685

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

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)';

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)';

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);