PostGIS has supported curved geometry types — CIRCULARSTRING, COMPOUNDCURVE< CURVEPOLYGON — since version 1.4, but the number of functions that directly calculate against the curved features has remained pretty small. You can generate a bounding box, or calculate a length, but that’s about it.

A COMPOUNDCURVE made up of a line string, a circular arc, and another line string.

In order to do more complex calculations like area calculation or intersections, you have to first convert the curved object into a linearized approximation, using the ST_CurveToLine function. This is fine for functions that return numbers (like ST_Area) or booleans (like ST_Intersects), but what about functions that return derived geometries?

Linearized version of the compound curve. The arc has been replaced by a regular collection of lines.

For derived geometries, the result will be linearized like the inputs. But portions of the geometry will be linearized versions of the original curves: wouldn’t it be nice to have those curves back for storage?

Yes, it would which is why ST_LineToCurve exists. The line to curve logic works on the premise that a linearized version of a curve will have a certain amount of regularity in it.

The version of the code from PostGIS 1.4 and 1.5 works by looking at the angles between successive segments. Segments that share an angle of deflection with neighbours are probably components of a circular arc. This worked OK, but the code involved a fair amount of trigonometry.

By looking at angles between edges, you can find edges that are former components of an arc.

A simpler approach used for 2.0 turned out to be looking at the circle the arc is inscribed on. Any circular arc in PostGIS is defined by a start point, mid point and end point. Between them, they imply a circle, and the center of the circle can be calculated. Any successive point which is the same distance from that center point as the arc points can be considered part of the arc.

By using the circle as a basis for comparison, each successive point needs a simple distance check, instead of a trig check.

The new code is a lot simpler, and can deal with derived segments of more variable length that the old code.

The simplest way to prove that it works is to wrap a curved geometry in multiple nests of ST_LineToCurve and ST_CurveToLine, pushing the geometry back and forth between representations. While the functions are not perfect inverses (the segmentization routine doesn’t necessarily include the middle control points of the input arcs) you can see that the space bounded by the geometries does not change.

Happy curving!

Tags: arcs, development, postgis