function [olon, olat] = geo_simplify(lon, lat, nmax, tolerance)

% geo_simplify -- Simplify a (lon, lat) contour.
%  [olon, olat] = geo_simplify(lon, lat, nmax, tolerance)
%   simplifies the (lon, lat) contour, returning nmax points
%   if (nmax > 0) or those points that are more significant
%   than the given tolerance.  The gs_sort() routine is used,
%   in which the deviation, not the curvature, is the key
%   principle.  NaNs are ignored.
%  [out] = geo_simplity(...) returns [olon olat] in one array.
%  geo_simplify(nPoints) demonstrates itself with nPoints
%   (random); default = 24.
 
% Copyright (C) 1997 Dr. Charles R. Denham, ZYDECO.
%  All Rights Reserved.
%   Disclosure without explicit written consent from the
%    copyright owner does not constitute publication.
 
% Version of 06-Jul-1998 06:40:26.
% Revised    04-Sep-1998 15:29:19.

% Reference: Douglas, D.H. and T.K. Peucker, Algorithms for
%  the reduction of the number of points required to represent
%  a digitized line of (sic?) its caricature, Can. Cartogr.,
%  10, 112-122, 1973.  We have adopted a similar scheme for
%  selecting "significant" points.  Unlike the D-P algorithm,
%  we sort all the points first, then isolate those that satisfy
%  our criteria for tolerance or total number of points retained.

if nargin < 1, help(mfilename), lon = 'demo'; end

if isequal(lon, 'demo'), lon = 24; end

if ischar(lon), lon = eval(lon); end

if length(lon) == 1
	n = lon;
	lon = 1:n;
	lat = cumsum(rand(size(lon))-0.5) .* 10;
 	lat(ceil(n/2)) = NaN;
	nmax = fix((n+2) ./ 3);
	tolerance = 0;
	tic
	[lon1, lat1] = geo_simplify(lon, lat, nmax, tolerance);
	toc
	plot(lon1, lat1, '-*', lon, lat, 'o')
	xlabel('lon (degrees)'), ylabel('lat (degrees)')
	title('geo\_simplify')
	axis equal
	set(gcf, 'Name', 'geo_simplify')
	figure(gcf)
	return
end

if nargin < 3, nmax = length(lon); end
if nargin < 4
	tolerance = 0;
	if nmax == 0, nmax = length(lon); end
end

if nmax > 0 & nmax <= 1
	nmax = round(nmax*length(lon));
end

RCF = 180 ./ pi;
x = cos(lat/RCF).*cos(lon/RCF);
y = cos(lat/RCF).*sin(lon/RCF);
z = sin(lat/RCF);

% Save NaNs.

xx = x(:);
yy = y(:);
zz = z(:);

i = find(isfinite(xx) & isfinite(yy) & isfinite(zz));
j = find(~isfinite(xx) | ~isfinite(yy) | ~isfinite(zz));
if any(i), xx = xx(i); yy = yy(i); zz = zz(i); end

% Sort.

[ind, dev] = gs_sort(xx, yy, zz);

% Restore NaNs.

ind = i(ind);
ind = [ind; j(:)];
dev = [dev; NaN*j];

% Deviation angle: always < 90 degrees.

dev = asin(dev) .* 180 ./ pi;

% Trim by count or tolerance.

if nmax > 0
	if nmax <= 1, nmax = round(nmax .* length(lon)); end
	nmax = max(3, min(nmax, length(lon)));
	ind = ind(1:nmax);
	dev = dev(1:nmax);
else
	f = find(dev >= tolerance);
	if any(f)
		ind = ind(f);
		dev = dev(f);
	end
end

% Unsort.

[ind, s] = sort(ind);
dev = dev(s);

lon1 = lon(ind);
lat1 = lat(ind);

if nargout == 1
	olon = [lon1 lat1];
elseif nargout > 1
	olon = lon1;
	olat = lat1;
end

function [theIndices, theDeviations] = gs_sort(x, y, z)

% gs_sort -- Sort (x, y, z) data for line-simplification.
%  [theIndices, theDeviations] = gs_sort(x, y, z) returns
%   indices and deviations of the given (x, y, z) data,
%   in order of decreasing importance.  The (x, y, z)
%   data are geographic, expressed as direction-cosines.
%   The deviations are the sines of the angles of
%   deviation.
 
% Copyright (C) 1997 Dr. Charles R. Denham, ZYDECO.
%  All Rights Reserved.
%   Disclosure without explicit written consent from the
%    copyright owner does not constitute publication.
 
% Version of 02-Jul-1998 09:36:15.

if nargin < 1, help(mfilename), x = 'demo'; end

if isequal(x, 'demo'), x = 24; end

if ischar(x), x = eval(x); end

if length(x) == 1
	n = x;
	a = cumsum(rand(n, 3)-0.5);
	for i = 1:n
		a(i, :) = a(i, :) ./ norm(a(i, :));
	end
	x = a(:, 1); y = a(:, 2); z = a(:, 3);
	tic
	[i, d] = gs_sort(x, y, z);
	toc
	reduce = 3;   % Reduction-factor.
	k = fix((n+reduce-1)./reduce);
	ii = i(1:k);
	dd = d(1:k);
	xx = x(ii);
	yy = y(ii);
	zz = z(ii);
	[ii, s] = sort(ii);
	dd = dd(s);
	xx = xx(s);
	yy = yy(s);
	zz = zz(s);
	subplot(2, 1, 1)
	plot3(x, y, z, 'b+', xx, yy, zz, 'go-')
	title('Data and Subset')
	xlabel('x'), ylabel('y'), zlabel('z')
	view(2)
	subplot(2, 1, 2)
	plot(i, d, '.', xx, dd, 'o')
	title('Deviation')
	xlabel('x'), ylabel('y'), zlabel('z')
	view(2)
	set(gcf, 'Name', 'gs_sort demo')
	figure(gcf)
	return
end

ind = zeros(size(x));
dev = zeros(size(x));

nstack = ceil(1 + 2*log(length(x))/log(2));
stack = zeros(nstack, 2);
nstack = 1;
stack(nstack, :) = [1 length(x)];

ind(1) = 1;
ind(2) = length(x);
nout = 2;

% Call gs_pivot() repeatedly and stack the
%  longer interval before the shorter.

while nstack > 0
	a = stack(nstack, 1);
	b = stack(nstack, 2);
	nstack = nstack - 1;
	if b-a > 1
		[p, d] = gs_pivot(x([a:b]), y([a:b]), z([a:b]));
		if any(p)
			p = a+p-1;
			nout = nout+1;
			ind(nout) = p;
			dev(nout) = d;
			if b-p >= p-a
				if b-p > 1
					nstack = nstack + 1;
					stack(nstack, :) = [p b];
				end
			end
			if p-a > 1
				nstack = nstack + 1;
				stack(nstack, :) = [a p];
			end
			if b-p < p-a
				if b-p > 1
					nstack = nstack + 1;
					stack(nstack, :) = [p b];
				end
			end
		end
	end
end

% Sort in descending order of importance.

dev(1:2) = max(dev);
[ignore, s] = sort(-dev);
ind = ind(s);
dev = dev(s);

% Output.

if nargout > 0
	theIndices = ind;
	theDeviations = dev;
else
	ind, dev
end

function [thePivot, theDeviation] = gs_pivot(x, y, z)

% gs_pivot -- Geographic line-simplification pivot.
%  [thePivot, theDeviation] = gs_pivot(x, y, z) returns
%   the index of the (x, y, z) point that deviates most
%   from the great-circle path between the first and last
%   of the geographic points, given as direction-cosines.
%   Only finite points are allowed.  The deviation is
%   returned as the sine of the deviation-angle.
 
% Copyright (C) 1997 Dr. Charles R. Denham, ZYDECO.
%  All Rights Reserved.
%   Disclosure without explicit written consent from the
%    copyright owner does not constitute publication.
 
% Version of 02-Jul-1998 08:32:40.

if nargin < 1, help(mfilename), x = 'demo'; end

if isequal(x, 'demo'), x = 10; end

if ischar(x), x = eval(x); end

if length(x) == 1
	n = x;
	a = cumsum(rand(n, 3)-0.5);
	for i = 1:n
		a(i, :) = a(i, :) ./ norm(a(i, :));
	end
	x = a(:, 1); y = a(:, 2); z = a(:, 3);
	tic
	[piv, dev] = gs_pivot(x, y, z)
	toc
	plot3(x, y, z, 'b-', ...
			x([1 n]), y([1 n]), z([1, n]), 'g-', ...
			x(piv), y(piv), z(piv), 'r*')
	title(['gs\_pivot(' int2str(n) ')'])
	xlabel('x'), ylabel('y'), zlabel('z')
	set(gca, 'Box', 'on')
	figure(gcf)
	return
end

thePivot = 0;
theDeviation = -inf;

n = length(x);
if n < 3, return, end

% Pole of the spanning great-circle.
%  WARNING: no such pole will exist if the two
%  end-points are the same.  In that case, use
%  point farthest from the ends.

if any([x(1) y(1) z(1)] ~= [x(n) y(n) z(n)])
	c(1) = y(1)*z(n)-y(n)*z(1);
	c(2) = z(1)*x(n)-z(n)*x(1);
	c(3) = x(1)*y(n)-x(n)*y(1);
	c = c.' ./ norm(c);
	d = [x y z] * c;
else
	c = [x(1) y(1) z(1)].';
	d = [x y z] * c;
	d = sqrt(1 - d.*d);
end

% Find the pivot and deviation.

temp = abs(d(2:n-1));
f = find(temp == max(temp));

piv = f(1) + 1;
dev = abs(d(piv));

% Output.

if nargout > 0
	thePivot = piv;
	theDeviation = dev;
else
	piv, dev
end