Pi Day PostGIS Circles

What's your favourite infinite sequence of non-repeating digits? There are some people who make a case for e, but to my mind nothing beats the transcendental and curvy utility of π, the ratio of a circle's circumference to its diameter.

Drawing circles is a simple thing to do in PostGIS -- take a point, and buffer it. The result is circular, and we can calculate an estimate of pi just by measuring the perimeter of the unit circle.

SELECT ST_Buffer('POINT(0 0)', 1.0);

Except, look a little more closely -- this "circle" seems to be made up of short straight lines. What is the ratio of its circumference to its diameter?

SELECT ST_Perimeter(ST_Buffer('POINT(0 0)', 1.0)) / 2;

3.1365484905459406

That's close to pi, but it's not pi. Can we generate a better approximation? What if we make the edges even shorter? The third parameter to ST_Buffer() is the "quadsegs", the number of segments to build each quadrant of the circle.

SELECT ST_Perimeter(ST_Buffer('POINT(0 0)', 1.0, quadsegs => 128)) / 2;

3.1415729403671087

Much closer!

We can crank this process up a lot more, keep adding edges, but at some point the process becomes silly. We should just be able to say "this edge is a portion of a circle, not a straight line", and get an actual circular arc.

Good news, we can do exactly that! The CIRCULARSTRING is the curvy analogue to a LINESTRING wherein every connection is between three points that define a portion of a circle.

The circular arc above is the arc that starts at A and ends at C, passing through B. Any three points define a unique circular arc. A CIRCULARSTRING is a connected sequence of these arcs, just as a LINESTRING is a connected sequence of linear edges.

How does this help us get to pi though? Well, PostGIS has a moderate amount of support for circular arc geometry, so if we construct a circle using "natively curved" objects, we should get an exact representation of a circle rather than an approximation.

So, what is an arc that starts and ends at the same point? A circle! This is the unit circle -- a circle of radius one centered on the origin -- expressed as a CIRCULARSTRING.

SELECT ST_Length('CIRCULARSTRING(1 0, -1 0, 1 0)') / 2;

3.141592653589793

That looks a lot like pi!

Let's bust out the built-in pi() function from PostgreSQL and check to be sure.

SELECT pi() - ST_Length('CIRCULARSTRING(1 0, -1 0, 1 0)') / 2;

0

Yep, a perfect π to celebrate "Pi Day" with!