function dist = segments_dist_2d ( p1, p2, q1, q2 )
%*****************************************************************************80
%
%% segments_dist_2d() computes the distance between two line segments in 2D.
%
% Discussion:
%
% A line segment is the finite portion of a line that lies between
% two points.
%
% If the lines through [P1,P2] and [Q1,Q2] intersect, and both
% line segments include the point of intersection, then the distance
% is zero and we are done.
%
% Therefore, we compute the intersection of the two lines, and
% find the coordinates of that intersection point on each line.
% This will tell us if the zero distance case has occurred.
%
% Otherwise, let PN and QN be points in [P1,P2] and [Q1,Q2] for which
% the distance is minimal. If the lines do not intersect, then it
% cannot be the case that both PN and QN are strictly interior to their
% line segments, aside from the exceptional singular case when
% the line segments overlap or are parallel. Even then, one of PN
% and QN may be taken to be a segment endpoint.
%
% Therefore, our second computation finds the minimum of:
%
% Distance ( P1, [Q1,Q2] );
% Distance ( P2, [Q1,Q2] );
% Distance ( Q1, [P1,P2] );
% Distance ( Q2, [P1,P2] );
%
% Licensing:
%
% This code is distributed under the MIT license.
%
% Modified:
%
% 04 December 2010
%
% Author:
%
% John Burkardt
%
% Input:
%
% real P1(2,1), P2(2,1), the endpoints of the first
% segment.
%
% real Q1(2,1), Q2(2,1), the endpoints of the second
% segment.
%
% Output:
%
% real DIST, the distance between the line segments.
%
%
% Determine whether and where the underlying lines intersect.
%
[ ival, r ] = lines_exp_int_2d ( p1, p2, q1, q2 );
%
% If there is exactly one intersection point part of both lines,
% check that it is part of both line segments.
%
if ( ival == 1 )
[ rps, rpt ] = segment_point_coords_2d ( p1, p2, r );
[ rq2, rqt ] = segment_point_coords_2d ( q1, q2, r );
if ( 0.0 <= rpt && rpt <= 1.0 && 0.0 <= rqt && rqt <= 1.0 )
dist = 0.0;
return
end
end
%
% If there is no intersection, or the intersection point is
% not part of both line segments, then an endpoint of one
% line segment achieves the minimum distance.
%
dist2 = segment_point_dist ( q1, q2, p1 );
dist = dist2;
dist2 = segment_point_dist ( q1, q2, p2 );
dist = min ( dist, dist2 );
dist2 = segment_point_dist ( p1, p2, q1 );
dist = min ( dist, dist2 );
dist2 = segment_point_dist ( p1, p2, q2 );
dist = min ( dist, dist2 );
return
end