Geospatial Support for Circular Data in SQL Server 2012

SQL Server 2012 adds many significant improvements to the spatial support that was first introduced with SQL Server 2008. In this blog post, I’ll explore one of the more notable enhancements: support for curves and arcs (circular data). SQL Server 2008 only supported straight lines, or polygons composed of straight lines. The three new shapes in SQL Server 2012 are circular strings, compound curves, and curve polygons. All three are supported in Well-Known Text (WKT), Well-Known Binary (WKB), and Geometry Markup Language (GML) by both the geometry (planar, or “flat-earth” model) and geography (ellipsoidal sphere, or geodetic) data types, and all of the existing methods work on the new shapes.

Circular Strings

A circular string defines a basic curved line, similar to how a line string defines a straight line. It takes a minimum of three coordinates to define a circular string; the first and third coordinates define the end points of the line, and the second coordinate (the “anchor” point, which lies somewhere between the end points) determines the arc of the line. Here is the shape represented by CIRCULARSTRING(0 1, .25 0, 0 -1):

The following code produces four circular strings. All of them have the same start and end points, but different anchor points. The lines are buffered slightly to make them easier to see in the spatial viewer.

-- Create a "straight" circular line
SELECT geometry::Parse('CIRCULARSTRING(0 8, 4 0, 8 -8)').STBuffer(.1)
UNION ALL  -- Curve it
SELECT geometry::Parse('CIRCULARSTRING(0 8, 4 4, 8 -8)').STBuffer(.1)
UNION ALL  -- Curve it some more
SELECT geometry::Parse('CIRCULARSTRING(0 8, 4 6, 8 -8)').STBuffer(.1)
UNION ALL  -- Curve it in the other direction
SELECT geometry::Parse('CIRCULARSTRING(0 8, 4 -6, 8 -8)').STBuffer(.1)

The spatial viewer in SQL Server Management Studio shows the generated shapes:

The first shape is a “straight circular” line, because the anchor point is position directly between the start and end points. The next two shapes use the same end points with the anchor out to the right (4), one a bit further than the other (6). The last shape also uses the same end points, but specifies an anchor point that curves the line to the left rather than the right (-6).

You can extend circular strings with as many curve segments as you want. Do this by defining another two coordinates for each additional segment. The last point of the previous curve serves as the first end point of the next curve segment, so the two additional coordinates respectively specify the next segment’s anchor and second end point. Thus, valid circular strings will always have an odd number of points. You can extend a circular string indefinitely to form curves and arcs of any kind.

It’s easy to form a perfect circle by connecting two semi-circle segments. For example, the following illustration shows the circle produced by CIRCULARSTRING(0 4, 4 0, 8 4, 4 8, 0 4).

This particular example connects the end of the second segment to the beginning of the first segment to form a closed shape. Note that this is certainly not required of circular strings (or line strings), and that closing the shape by connecting the last segment to the first still does not result in a polygon, which is a two dimensional shape that has area. Despite being closed, this circle is still considered a one-dimensional shape with no area. As you’ll soon see, the curve polygon can be used to convert closed line shapes into true polygons.

Compound Curves

A compound curve is a set of circular strings, or circular strings combined with line strings, that form a desired curved shape. The end point of each element in the collection must match the starting point of the following element, so that compound curves are defined in a “connect-the-dots” fashion. The following code produces a compound curve and compares it with the equivalent geometry collection shape.

-- Compound curve
 DECLARE @CC geometry = '
  COMPOUNDCURVE(
   (4 4, 4 8),
   CIRCULARSTRING(4 8, 6 10, 8 8),
   (8 8, 8 4),
   CIRCULARSTRING(8 4, 2 3, 4 4)
  )'

-- Equivalent geometry collection
 DECLARE @GC geometry = '
  GEOMETRYCOLLECTION(
   LINESTRING(4 4, 4 8),
   CIRCULARSTRING(4 8, 6 10, 8 8),
   LINESTRING(8 8, 8 4),
   CIRCULARSTRING(8 4, 2 3, 4 4)
  )'

-- They both render the same shape in the spatial viewer
 SELECT @CC.STBuffer(.5)
 UNION ALL
 SELECT @GC.STBuffer(1.5)

This code creates a keyhole shape using a compound curve, and also creates an identical shape as a geometry collection (though notice that the LINESTRING keyword is not—and cannot—be specified when defining a compound curve). It then buffers both of them with different padding, so that the spatial viewer clearly shows the two identical shapes on top of one another, as shown:

Both the compound curve and the geometry collection yield identical shapes. In fact, the expression @CC.STEquals(@GC) which compares the two instances for equality returns 1 (for true). The STEquals method tests for “spatial equality,” meaning it returns true if two instances produce the same shape even if they are being rendered using different spatial data classes. Furthermore, recall that segments of a circular string can be made perfectly straight by positioning the anchor directly between the end points, meaning that the circular string offers yet a third option for producing the very same shape. So which one should you use? Comparing these spatial data classes will help you determine which one is best to use in different scenarios.

A geometry collection (which was already supported in SQL Server 2008) is the most accommodating, but carries the most storage overhead. Geometry collections can hold instances of any spatial data class, and the instances don’t need to be connected to (or intersected with) each other in any way. The collection simply holds a bunch of different shapes as a set, which in this example just happens to be several line strings and circular strings connected at their start and end points.

In contrast, the new compound curve class in SQL Server 2012 has the most constraints but is the most lightweight in terms of storage. It can only contain line strings or circular strings, and each segment’s start point must be connected to the previous segment’s end point (although it is most certainly not necessary to connect the first and last segments to form a closed shape as in this example). The DATALENGTH function shows the difference in storage requirements; DATALENGTH(@CC) returns 152 and DATALENGTH(@GC) returns 243. In our current example, DATALENGTH(@CC) returns 152 and DATALENGTH(@GC) returns 243. This means that the same shape requires 38% less storage space by using a compound curve instead of a geometry collection. A compound curve is also more storage-efficient than a multi-segment circular line string when straight lines are involved. This is because there is overhead for the mere potential of a curve, since the anchor point requires storage even when it’s position produces straight lines, whereas compound curves are optimized specifically to connect circular strings and (always straight) line strings.

Curve Polygons

A curve polygon is very similar to an ordinary polygon; like an ordinary polygon, a curve polygon specifies a “ring” that defines a closed shape, and can also specify additional inner rings to define “holes” inside the shape. The only fundamental difference between a polygon and a curve polygon is that the rings of a curve polygon can include circular shapes, whereas an ordinary polygon is composed exclusively with straight lines. Specifically, each ring in a curve polygon can consist of any combination of line strings, circular strings, and compound curves that collectively define the closed shape. For example, the following code produces a curve polygon with the same keyhole outline that I just demonstrated for the compound curve.

-- Curve polygon
 SELECT geometry::Parse('
  CURVEPOLYGON(
   COMPOUNDCURVE(
    (4 4, 4 8),
    CIRCULARSTRING(4 8, 6 10, 8 8),
    (8 8, 8 4),
    CIRCULARSTRING(8 4, 2 3, 4 4)
   )
  )')

This code has simply specified the same compound curve as the closed shape of a curve polygon. Although the shape is the same, the curve polygon is a two-dimensional object, whereas the compound curve version of the same shape is a one-dimensional object. This can be seen visually by the spatial viewer results, which shades the interior of the curve polygon as shown here:

Conclusion

Circular data support is an important new capability added to the spatial support in SQL Server 2012. In this blog post, I demonstrated the three new spatial data classes for curves and arcs: circular strings, compound curves, and curve polygons. Stay tuned for my next post (coming soon), for more powerful and fun spatial enhancements coming soon in SQL Server 2012!

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