intersections.m

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function [x0,y0,iout,jout] = intersections(x1,y1,x2,y2,robust)
% Theory of operation:
%
% Given two line segments, L1 and L2,
%
%   L1 endpoints:  (x1(1),y1(1)) and (x1(2),y1(2))
%   L2 endpoints:  (x2(1),y2(1)) and (x2(2),y2(2))
%
% we can write four equations with four unknowns and then solve them.  The
% four unknowns are t1, t2, x0 and y0, where (x0,y0) is the intersection of
% L1 and L2, t1 is the distance from the starting point of L1 to the
% intersection relative to the length of L1 and t2 is the distance from the
% starting point of L2 to the intersection relative to the length of L2.
%
% So, the four equations are
%
%    (x1(2) - x1(1))*t1 = x0 - x1(1)
%    (x2(2) - x2(1))*t2 = x0 - x2(1)
%    (y1(2) - y1(1))*t1 = y0 - y1(1)
%    (y2(2) - y2(1))*t2 = y0 - y2(1)
%
% Rearranging and writing in matrix form,
%
%  [x1(2)-x1(1)       0       -1   0;      [t1;      [-x1(1);
%        0       x2(2)-x2(1)  -1   0;   *   t2;   =   -x2(1);
%   y1(2)-y1(1)       0        0  -1;       x0;       -y1(1);
%        0       y2(2)-y2(1)   0  -1]       y0]       -y2(1)]
%
% Let's call that A*T = B.  We can solve for T with T = A\B.
%
% Once we have our solution we just have to look at t1 and t2 to determine
% whether L1 and L2 intersect.  If 0 <= t1 < 1 and 0 <= t2 < 1 then the two
% line segments cross and we can include (x0,y0) in the output.
%
% In principle, we have to perform this computation on every pair of line
% segments in the input data.  This can be quite a large number of pairs so
% we will reduce it by doing a simple preliminary check to eliminate line
% segment pairs that could not possibly cross.  The check is to look at the
% smallest enclosing rectangles (with sides parallel to the axes) for each
% line segment pair and see if they overlap.  If they do then we have to
% compute t1 and t2 (via the A\B computation) to see if the line segments
% cross, but if they don't then the line segments cannot cross.  In a
% typical application, this technique will eliminate most of the potential
% line segment pairs.

% Input checks.
narginchk(2,5)

% Adjustments when fewer than five arguments are supplied.
switch nargin
    case 2
        robust = true;
        x2 = x1;
        y2 = y1;
        self_intersect = true;
    case 3
        robust = x2;
        x2 = x1;
        y2 = y1;
        self_intersect = true;
    case 4
        robust = true;
        self_intersect = false;
    case 5
        self_intersect = false;
end

% x1 and y1 must be vectors with same number of points (at least 2).
if sum(size(x1) > 1) ~= 1 || sum(size(y1) > 1) ~= 1 || ...
        length(x1) ~= length(y1)
    error('X1 and Y1 must be equal-length vectors of at least 2 points.')
end
% x2 and y2 must be vectors with same number of points (at least 2).
if sum(size(x2) > 1) ~= 1 || sum(size(y2) > 1) ~= 1 || ...
        length(x2) ~= length(y2)
    error('X2 and Y2 must be equal-length vectors of at least 2 points.')
end


% Force all inputs to be column vectors.
x1 = x1(:);
y1 = y1(:);
x2 = x2(:);
y2 = y2(:);

% Compute number of line segments in each curve and some differences we'll
% need later.
n1 = length(x1) - 1;
n2 = length(x2) - 1;
xy1 = [x1 y1];
xy2 = [x2 y2];
dxy1 = diff(xy1);
dxy2 = diff(xy2);

% Determine the combinations of i and j where the rectangle enclosing the
% i'th line segment of curve 1 overlaps with the rectangle enclosing the
% j'th line segment of curve 2.
[i,j] = find(repmat(min(x1(1:end-1),x1(2:end)),1,n2) <= ...
    repmat(max(x2(1:end-1),x2(2:end)).',n1,1) & ...
    repmat(max(x1(1:end-1),x1(2:end)),1,n2) >= ...
    repmat(min(x2(1:end-1),x2(2:end)).',n1,1) & ...
    repmat(min(y1(1:end-1),y1(2:end)),1,n2) <= ...
    repmat(max(y2(1:end-1),y2(2:end)).',n1,1) & ...
    repmat(max(y1(1:end-1),y1(2:end)),1,n2) >= ...
    repmat(min(y2(1:end-1),y2(2:end)).',n1,1));

% Force i and j to be column vectors, even when their length is zero, i.e.,
% we want them to be 0-by-1 instead of 0-by-0.
i = reshape(i,[],1);
j = reshape(j,[],1);

% Find segments pairs which have at least one vertex = NaN and remove them.
% This line is a fast way of finding such segment pairs.  We take
% advantage of the fact that NaNs propagate through calculations, in
% particular subtraction (in the calculation of dxy1 and dxy2, which we
% need anyway) and addition.
% At the same time we can remove redundant combinations of i and j in the
% case of finding intersections of a line with itself.
if self_intersect
    remove = isnan(sum(dxy1(i,:) + dxy2(j,:),2)) | j <= i + 1;
else
    remove = isnan(sum(dxy1(i,:) + dxy2(j,:),2));
end
i(remove) = [];
j(remove) = [];

% Initialize matrices.  We'll put the T's and B's in matrices and use them
% one column at a time.  AA is a 3-D extension of A where we'll use one
% plane at a time.
n = length(i);
T = zeros(4,n);
AA = zeros(4,4,n);
AA([1 2],3,:) = -1;
AA([3 4],4,:) = -1;
AA([1 3],1,:) = dxy1(i,:).';
AA([2 4],2,:) = dxy2(j,:).';
B = -[x1(i) x2(j) y1(i) y2(j)].';

% Loop through possibilities.  Trap singularity warning and then use
% lastwarn to see if that plane of AA is near singular.  Process any such
% segment pairs to determine if they are colinear (overlap) or merely
% parallel.  That test consists of checking to see if one of the endpoints
% of the curve 2 segment lies on the curve 1 segment.  This is done by
% checking the cross product
%
%   (x1(2),y1(2)) - (x1(1),y1(1)) x (x2(2),y2(2)) - (x1(1),y1(1)).
%
% If this is close to zero then the segments overlap.

% If the robust option is false then we assume no two segment pairs are
% parallel and just go ahead and do the computation.  If A is ever singular
% a warning will appear.  This is faster and obviously you should use it
% only when you know you will never have overlapping or parallel segment
% pairs.

if robust
    overlap = false(n,1);
    warning_state = warning('off','MATLAB:singularMatrix');
    % Use try-catch to guarantee original warning state is restored.
    try
        lastwarn('')
        for k = 1:n
            T(:,k) = AA(:,:,k)\B(:,k);
            [~,last_warn] = lastwarn;
            lastwarn('')
            if strcmp(last_warn,'MATLAB:singularMatrix')
                % Force in_range(k) to be false.
                T(1,k) = NaN;
                % Determine if these segments overlap or are just parallel.
                overlap(k) = rcond([dxy1(i(k),:);xy2(j(k),:) - xy1(i(k),:)]) < eps;
            end
        end
        warning(warning_state)
    catch err
        warning(warning_state)
        rethrow(err)
    end
    % Find where t1 and t2 are between 0 and 1 and return the corresponding
    % x0 and y0 values.
    in_range = (T(1,:) >= 0 & T(2,:) >= 0 & T(1,:) <= 1 & T(2,:) <= 1).';
    % For overlapping segment pairs the algorithm will return an
    % intersection point that is at the center of the overlapping region.
    if any(overlap)
        ia = i(overlap);
        ja = j(overlap);
        % set x0 and y0 to middle of overlapping region.
        T(3,overlap) = (max(min(x1(ia),x1(ia+1)),min(x2(ja),x2(ja+1))) + ...
            min(max(x1(ia),x1(ia+1)),max(x2(ja),x2(ja+1)))).'/2;
        T(4,overlap) = (max(min(y1(ia),y1(ia+1)),min(y2(ja),y2(ja+1))) + ...
            min(max(y1(ia),y1(ia+1)),max(y2(ja),y2(ja+1)))).'/2;
        selected = in_range | overlap;
    else
        selected = in_range;
    end
    xy0 = T(3:4,selected).';

    % Remove duplicate intersection points.
    [xy0,index] = unique(xy0,'rows');
    x0 = xy0(:,1);
    y0 = xy0(:,2);

    % Compute how far along each line segment the intersections are.
    if nargout > 2
        sel_index = find(selected);
        sel = sel_index(index);
        iout = i(sel) + T(1,sel).';
        jout = j(sel) + T(2,sel).';
    end
else % non-robust option
    for k = 1:n
        [L,U] = lu(AA(:,:,k));
        T(:,k) = U\(L\B(:,k));
    end

    % Find where t1 and t2 are between 0 and 1 and return the corresponding
    % x0 and y0 values.
    in_range = (T(1,:) >= 0 & T(2,:) >= 0 & T(1,:) < 1 & T(2,:) < 1).';
    x0 = T(3,in_range).';
    y0 = T(4,in_range).';

    % Compute how far along each line segment the intersections are.
    if nargout > 2
        iout = i(in_range) + T(1,in_range).';
        jout = j(in_range) + T(2,in_range).';
    end
end

% Plot the results (useful for debugging).
% plot(x1,y1,x2,y2,x0,y0,'ok');