Rotate Polygon to be Parallel with the X Axis

Question

By default, PostGIS calculates the Envelope or Extent of a polygon based on the bounding box of the polygon ((MINX, MINY), (MINX, MAXY), (MAXX, MAXY), (MAXX, MINY), (MINX, MINY)).

This gives a result like so:

However, I'm looking for a result more like this:

As far as I know, the best algorithm that I can come up with is:

1. Determine angle a to rotate the polygon X to be parallel with the x-axis
2. Rotate X a degrees, calculate the envelope Y of rotated polygon X
3. Rotate Y -a degrees

How could you calculate step #1 in PostGIS?

Answer 1

Here's an implementation. It doesn't handle degenerate cases and you may not like the rectangle chose if the areas tie. I'm working on a submission for postgis, but this should be enough to work with until I get a patch into postgis.

create or replace function ST_MinimumRectangle(g geometry) returns geometry
language 'plpgsql' as $$
declare
        hull geometry;
begin
        hull = ST_ExteriorRing(ST_ConvexHull(g));
        -- one side must lie along the rectangle.
        -- so, for each side, rotate, bbox, counter-rotate bbox
        with sides as (
                select ST_PointN(hull, n) as a, ST_PointN(hull, n+1) as b,
                        n as side
                        from generate_series(1,ST_NPoints(hull)-1) n
                ),
        angles as (
                select side, a, b, st_azimuth(a, b) as angle from sides
        ),
        boxes as (
                select ST_Rotate(ST_Envelope(ST_Rotate(hull, -angle)),angle) as rect, side, angle from angles
        )
        select rect into hull from boxes order by ST_Area(rect), side limit 1
        ;
        return hull;
end;
$$ immutable strict;

Answer 2

Small fix.

Somehow it worked for me as:

select ST_Rotate(ST_Envelope(ST_Rotate(hull, **angle**)),**-angle**)