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## Copyright (C) 2024 David Legland
## All rights reserved.
##
## Redistribution and use in source and binary forms, with or without
## modification, are permitted provided that the following conditions are met:
##
## 1 Redistributions of source code must retain the above copyright notice,
## this list of conditions and the following disclaimer.
## 2 Redistributions in binary form must reproduce the above copyright
## notice, this list of conditions and the following disclaimer in the
## documentation and/or other materials provided with the distribution.
##
## THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS ''AS IS''
## AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
## IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
## ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE FOR
## ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
## DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
## SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
## CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
## OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
## OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
##
## The views and conclusions contained in the software and documentation are
## those of the authors and should not be interpreted as representing official
## policies, either expressed or implied, of the copyright holders.
function [nodes, edges] = medialAxisConvex(points)
%MEDIALAXISCONVEX Compute medial axis of a convex polygon.
%
% [N, E] = medialAxisConvex(POLYGON);
% where POLYGON is given as a set of points [x1 y1;x2 y2 ...], returns
% the medial axis of the polygon as a graph.
% N is a set of nodes. The first elements of N are the vertices of the
% original polygon.
% E is a set of edges, containing indices of source and target nodes.
% Edges are sorted according to order of creation. Index of first vertex
% is lower than index of last vertex, i.e. edges always point to newly
% created nodes.
%
% Notes:
% - Is not fully implemented, need more development (usually crashes for
% polygons with more than 6-7 points...)
% - Works only for convex polygons.
% - Complexity is not optimal: this algorithm is O(n*log n), but linear
% algorithms exist.
%
% See also
% polygons2d, bisector
% ------
% Author: David Legland
% E-mail: david.legland@inrae.fr
% Created: 2005-07-07
% Copyright 2005-2023 INRA - TPV URPOI - BIA IMASTE
% eventually remove the last point if it is the same as the first one
if points(1,:) == points(end, :)
nodes = points(1:end-1, :);
else
nodes = points;
end
% special case of triangles:
% compute directly the gravity center, and simplify computation.
if size(nodes, 1)==3
nodes = [nodes; mean(nodes, 1)];
edges = [1 4;2 4;3 4];
return
end
% number of nodes, and also of initial rays
N = size(nodes, 1);
% create ray of each vertex
rays = zeros(N, 4);
rays(1, 1:4) = bisector(nodes([2 1 N], :));
rays(N, 1:4) = bisector(nodes([1 N N-1], :));
for i=2:N-1
rays(i, 1:4) = bisector(nodes([i+1, i, i-1], :));
end
% add indices of edges producing rays (indices of first vertex, second
% vertex is obtained by adding one modulo N).
rayEdges = [[N (1:N-1)]' (1:N)'];
pint = intersectLines(rays, rays([2:N 1], :));
% compute the distance between each intersection point and the closest
% edge. This distance is used as marker to propagate processing front.
ti = zeros(N, 1);
for i = 1:N
line = createLine(points(mod(i-2, N)+1, :), points(i, :));
ti(i) = abs(distancePointLine(pint(i,:), line));
end
% create list of events.
% terms are : R1 R2 X Y t0
% R1 and R2 are indices of involved rays
% X and Y is coordinate of intersection point
% t0 is position of point on rays
events = sortrows([ (1:N)' [2:N 1]' pint ti], 5);
% initialize edges
edges = zeros(0, 2);
%% process each event until there is no more
% start after index of last vertex, and process N-3 intermediate rays
for i = N+1:2*N-3
% add new node at the rays intersection
nodes(i,:) = events(1, 3:4);
% add new couple of edges
edges = [edges; events(1,1) i; events(1,2) i]; %#ok<AGROW>
% find the two edges creating the new emanating ray
n1 = rayEdges(events(1, 1), 1);
n2 = rayEdges(events(1, 2), 2);
% create the new ray
line1 = createLine(nodes(n1, :), nodes(mod(n1,N)+1, :));
line2 = createLine(nodes(mod(n2,N)+1, :), nodes(n2, :));
ray0 = bisector(line1, line2);
% set its origin to emanating point
ray0(1:2) = nodes(i, :);
% add the new ray to the list
rays = [rays; ray0]; %#ok<AGROW>
rayEdges(size(rayEdges, 1)+1, 1:2) = [n1 n2];
% find the two neighbour rays
ind = sum(ismember(events(:,1:2), events(1, 1:2)), 2) ~= 0;
ir = unique(events(ind, 1:2));
ir = ir(~ismember(ir, events(1,1:2)));
% create new intersections
pint = intersectLines(ray0, rays(ir, :));
ti = abs(distancePointLine(pint, line1));
% remove all events involving old intersected rays
ind = sum(ismember(events(:,1:2), events(1, 1:2)), 2) == 0;
events = events(ind, :);
% add the newly formed events
events = [events; ir(1) i pint(1,:) ti(1); ir(2) i pint(2,:) ti(2)]; %#ok<AGROW>
% and sort them according to 'position' parameter
events = sortrows(events, 5);
end
% centroid computation for last 3 rays
nodes = [nodes; mean(events(:, 3:4))];
edges = [edges; [unique(events(:,1:2)) ones(3, 1)*(2*N-2)]];
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