What is the width of the beach of a little island polygon?

Question

This is a "classic" problem, that have no practical utility (see note-1 for applications), but illustrate another GIS problems...

We can do some approximations to modeling the situation,

• the real center (coconut) is not at the ST_Centroid point, but can be approximated to;
• the real shape (a island) is not a circle, neither a square, they are optional references.

So, translating the question to an equation, and using OGC standard functions... "What is the value of w?" at the equation

ST_Area(ST_Buffer(ST_Centroid(littleIsland), w, quad_segs)) = ST_Area(littleIsland)

When quad_segs=1, the reference is a square, when quad_segs>=8 the reference is near to a circle (see examples of generated polygons).

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NOTE-1: For understand this problem into a context of applications, see "Is there a st_buffer inverse function, that returns a width estimation?" and see "How to fit a polygon to a “best buffer” about a reference geometry in it?".

NOTE-2: the "inverse function" is so general and complex, because can operates with point, line and polygon inputs. Here we can discuss the simplest case, when the input is like a point, with no area or length.

Answer 1

I am using this function to solve the problem, but not have any source/reference (do you have? please inform posting a comment)... So, I posted also demonstrations and tests.

CREATE FUNCTION pointbuffer_width(g geometry, N integer DEFAULT 8) RETURNS float AS $$
   -- Returns the value of W in the approximation "g ~ ST_Buffer(ST_Centroid(g),W,'quad_segs=N')"
   -- where N>0 indicates the reference-shape, 1=square, 2=heptagon, ... circle.
   --
   SELECT sqrt(  area/(0.5*n*sin(2*pi()/n))  )
   FROM ( SELECT ST_Area($1) as area, 
                 (CASE WHEN $2<=1 THEN 4 ELSE 4*$2 END)::float as n
   ) as t;
$$ LANGUAGE SQL immutable;

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DEMONSTRATING

The main shapes, circle and square, have areas

A = pi*w^2   # circle with radius w
A = 2*w^2    # square circumscribed into a circle of radius w 

The intermediary shapes are the regular polygons of n sides. The square have n=4, the octagon n=8, etc. So, we can substitute the factor of w^2 by a function f(n) that supply the adequate value for the n sides. The formula of the area of a regular polygon of n sides with a circumradius w, is:

 A = 0.5*n*sin(2*pi/n)*w^2

Where we can check: for n=4 the factor 0.5*n*sin(2*pi/n) is 2; for n=400 (infinite) is 3.14 (~pi).

For the standard ST_Buffer(g,'quad_segs=N') polygon generator we can check that this N is 4*n.

Confirmation of N=4*n with PostGIS:

SELECT ST_AsEWKT((p_geom).geom) As geom_ewkt
  FROM (SELECT ST_DumpPoints( 
      ST_Buffer(ST_GeomFromText('POINT(100 90)'),50,'quad_segs=5') 
 ) AS p_geom )  AS t;

It returns 4+1 rows when N=1, 20+1 rows when N=5, etc. The dumped last row repeat the first, to close the polygon.

TESTING

Tests simulating buffers (need more tests with irregular ones)

-- FORMULA TESTER FOR POINTS AND ITS BUFFERS:
SELECT w, 
       (array['square','octagon','','','','','','~circle'])[shape] as shape, 
       round(pointbuffer_width(circle8,shape),1) as circle8,
       round(pointbuffer_width(octagon,shape),1) as octagon,
       round(pointbuffer_width(square,shape),1) as square
FROM (
 SELECT
    w, shape, -- 1, 2, 3 or 10 
    ST_Buffer(g, w) as circle8, -- default 8, use 20 for perfect circle
    ST_Buffer(g, w, 'quad_segs=2') as octagon,
    ST_Buffer(g, w, 'quad_segs=1') as square
 FROM ( SELECT ST_GeomFromText('POINT(100 90)') as g,  
        unnest(array[1.0,150.0]) as w,
        unnest(array[1,2,8]) as shape    
      ) as t
 ) as t2;

RESULTS:

   w   |  shape  | circle8 | octagon | square 
-------+---------+---------+---------+--------
   1.0 | square  |     1.2 |     1.2 |     *1
 150.0 | octagon |   157.6 |    *150 |  126.1
   1.0 | ~circle |      *1 |       1 |    0.8
 150.0 | square  |   187.4 |   178.4 |   *150
   1.0 | octagon |     1.1 |      *1 |    0.8
 150.0 | ~circle |    *150 |   142.8 |  120.1

Conclusion: it is stable for any w value (!); it is exact when the modeled shape fits the geometry shape (*).