how to find the point in a country furthest from a boundary

Question

i'm trying to find the point in south africa furthest from the sea or a landward boundary.

my method thus far:

1. i obtained a first guess at the centre of the inscribed circle and the three segments of boundary this circle might touch using a compass and 1:2 500 000 map.

2. i then clipped these three segments of boundary from a data set of boundary coordinates supplied by national geo-spatial information (WGS84; decimal degrees; 9 decimal places; roughly one point for every 40 metres of boundary).

3. i converted the boundary coordinates to radians and used cartesian equations (distance between two points and centre of a circle through three points) to iteratively improve the centre of my circle and the three boundary points it touches. (obviously, i'm ignoring curvature; i'm effectively finding the centre of the base of the spherical cap defined by the inscribed circle -- i.e. a point approximately 16km underground.)

my questions:

1. is it reasonable to plug lat-long coordinates expressed in radians into cartesian equations to find the centre of the base of the spherical cap defined by the inscribed circle (given that the surface distance represented by one radian varies with latitude, and that the base of the spherical cap in question is roughly 800km in diameter and spans latitude 30S)? what order of error might be involved here?

2. is it reasonable to assume that the three boundary points tangential to the base of the spherical cap will likewise be the three boundary points closest to the actual point in the country furthest from a boundary?

3. i have the feeling i'm not even asking the right questions here. i guess there must be a better way to find the point in a country furthest from a boundary ...

cheers, william

Answer 1

To be honest, it sounds like you know what you're doing more than me, but my approach would be:

• Rasterize geopoltical/sea boundaries when the polylines are 1 and everything else is 0.

• Use the Euclidean Distance tool in Arc to produce raster of 'distance to the nearest 1' (i.e. nearest boundary/sea pixel)

• Follow the advice from this Stackexchange explaining how to get the location of the pixel with the highest value in a raster.

Answer 2

Here's another approach, if you're going to be doing this manually rather than algorithmically, which keeps things in a vector form:

I'm working out of QGIS.

• Extract the nodes of the country polygon.
• Create voronoi polygons from these nodes.
• Follow the acute angles until you find an intersection that has bordering polygons extending outwards to split the country roughly in three.

I'm not 100% sure how to perform this last step without human intervention (perhaps something with measuring the principal axis?). I can say that it will be something other than the centroid of the polygon in most cases.

Disclaimer: Determine very deliberately how far upstream into a bay/river you should go before you consider yourself "in the country". Also won't work for countries where a node-less line segment is one of the three nearest boundaries to the Farthest Point.

EDIT:

Instead of Voronoi polygons, use the related Delaunay Triangulation tool, and select the triangle with the largest area. Take the centroid of that triangle, and you will have your point.

EDIT2:

I'm not 100% certain if the 'largest area' criteria will always align with the farthest point, mathematically. Interesting problem... In the end, if you want a strong scriptable solution you may need to use iterative buffers, or resort to the aforementioned raster.