If you are serious about sampling uniformly within a distance of a
particular point then you need account for the curvature of the
ellipsoid. This is pretty easy to do using a rejection technique. The
following Matlab code implements this.
function [lat, lon] = geosample(lat0, lon0, r0, n)
% [lat, lon] = geosample(lat0, lon0, r0, n)
%
% Return n points on the WGS84 ellipsoid within a distance r0 of
% (lat0,lon0) and uniformly distributed on the surface. The returned
% lat and lon are n x 1 vectors.
%
% Requires Matlab package
% http://www.mathworks.com/matlabcentral/fileexchange/39108
todo = true(n,1); lat = zeros(n,1); lon = lat;
while any(todo)
n1 = sum(todo);
r = r0 * max(rand(n1,2), [], 2); % r = r0*sqrt(U) using cheap sqrt
azi = 180 * (2 * rand(n1,1) - 1); % sample azi uniformly
[lat(todo), lon(todo), ~, ~, m, ~, ~, sig] = ...
geodreckon(lat0, lon0, r, azi);
% Only count points with sig <= 180 (otherwise it's not a shortest
% path). Also because of the curvature of the ellipsoid, large r
% are sampled too frequently, by a factor r/m. This following
% accounts for this...
todo(todo) = ~(sig <= 180 & r .* rand(n1,1) <= m);
end
end
This code samples uniformly within a circle on the azimuthal equidistant
projection centered at lat0, lon0. The radial, resp. azimuthal,
scale for this projection is 1, resp. r/m. Hence the areal
distortion is r/m and this is accounted for by accepting such points
with a probability m/r.
This code also accounts for the situation where r0 is about half the
circumference of the earth and avoids double sampling nearly antipodal
points.
In ArcGIS, this can be done with Create random point if you have spatial analyst or 3D analyst or Advanced(ArcInfo) licence
select each shapefile as "constraining_feature_class" and set the "minimum_allowed_distance"to 250 m.
In QGIS, from what I know, you can use "Vector ‣ Research Tools ‣ Random points" with a constraining feature class, but not for distance. Again, you can use this tool in a loop to add points one by one until you reach the required number.
On the other hand, a solution for simple random sampling with distance contraint is to recursively add your random point and erase the buffer around your random point from the feature class until all your features are completely erased. As mentioned by @Whuber, this will not optimize the number of sample points, but you can control the number of points per strata.
Finally, the optimal number of points with a distance constraint can be achieved by systematic sampling : create a grid (with fishnet) with evenly spaced points (in your case 250 m) and randomly select a rotation angle and shift value betwen 0 and 250 in X and Y direction. This will optimize the number of point samples (and you could randomly subselect some points to reach a given number inside each startum). However, note that systematic sample could be biased because of the spatial autocorrelation.
Best Answer
With some Python:
or for a loop: