polyangles.m ( File view )

  • By yueqiaohuayuan 2014-11-17
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			function angles = polyangles(x, y)
%POLYANGLES Computes internal polygon angles.
%   ANGLES = POLYANGLES(X, Y) computes the interior angles (in
%   degrees) of an arbitrary polygon whose vertices are given in 
%   [X, Y], ordered in a clockwise manner.  The program eliminates
%   duplicate adjacent rows in [X Y], except that the first row may
%   equal the last, so that the polygon is closed.  

%   Copyright 2002-2004 R. C. Gonzalez, R. E. Woods, & S. L. Eddins
%   Digital Image Processing Using MATLAB, Prentice-Hall, 2004
%   $Revision: 1.6 $  $Date: 2003/11/21 14:44:06 $

% Preliminaries.
[x y] = dupgone(x, y); % Eliminate duplicate vertices.
xy = [x(:) y(:)];
if isempty(xy)
   % No vertices!
   angles = zeros(0, 1);
if size(xy, 1) == 1 | ~isequal(xy(1, :), xy(end, :))
   % Close the polygon
   xy(end + 1, :) = xy(1, :);

% Precompute some quantities.
d = diff(xy, 1);
v1 = -d(1:end, :);
v2 = [d(2:end, :); d(1, :)];
v1_dot_v2 = sum(v1 .* v2, 2);
mag_v1 = sqrt(sum(v1.^2, 2));
mag_v2 = sqrt(sum(v2.^2, 2));

% Protect against nearly duplicate vertices; output angle will be 90
% degrees for such cases. The "real" further protects against
% possible small imaginary angle components in those cases.
mag_v1(~mag_v1) = eps;
mag_v2(~mag_v2) = eps;
angles = real(acos(v1_dot_v2 ./ mag_v1 ./ mag_v2) * 180 / pi);

% The first angle computed was for the second vertex, and the
% last was for the first vertex. Scroll one position down to 
% make the last vertex be the first.
angles = circshift(angles, [1, 0]);

% Now determine if any vertices are concave and adjust the angles
% accordingly.
sgn = convex_angle_test(xy);

% Any element of sgn that's -1 indicates that the angle is
% concave. The corresponding angles have to be subtracted 
% from 360.
I = find(sgn == -1);
angles(I) = 360 - angles(I);

function sgn = convex_angle_test(xy)
%   The rows of array xy are ordered vertices of a polygon. If the
%   kth angle is convex (>0 and <= 180 degress) then sgn(k) =
%   1. Otherwise sgn(k) = -1. This function assumes that the first
%   vertex in the list is convex, and that no other vertex has a
%   smaller value of x-coordinate. These two conditions are true in
%   the first vertex generated by the MPP algorithm. Also the
%   vertices are assumed to be ordered in a clockwise sequence, and
%   there can be no duplicate vertices. 
%   The test is based on the fact that every convex vertex is on the
%   positive side of the line passing through the two vertices
%   immediately following each vertex being considered.  If a vertex
%   is concave then it lies on the negative side of the line joining
%   the next two vertices. This property is true also if positive and
%   negative are interchanged in the preceding two sentences.

% It is assumed that the polygon is closed.  If not, close it.
if size(xy, 1) == 1 | ~isequal(xy(1, :), xy(end, :))
   xy(end + 1, :) = xy(1, :);

% Sign convention: sgn = 1 for convex vertices (i.e, interior angle > 0 
% and <= 180 degrees), sgn = -1 for concave vertices.

% Extreme points to be used in the following loop.  A 1 is appended 
% to perform the inner (dot) product with w, which is 1-by-3 (see 
% below).
L = 10^25;
top_left = [-L, -L, 1];
top_right = [-L, L, 1];
bottom_left = [L, -L, 1];
bottom_right = [L, L, 1];

sgn = 1; % The first vertex is known to be convex.

% Start following the vertices. 
for k = 2:length(xy) - 1
   pfirst= xy(k - 1, :);
   psecond = xy(k, :); % This is the point tested for convexity.
   pthird = xy(k + 1, :);
   % Get the coefficients of the line (polygon edge) passing 
   % through pfirst and psecond. 
   w = polyedge(pfirst, psecond);

   % Establish the positive side of the line w1x + w2y + w3 = 0.
   % The positive side of the line should be in the right side of the
   % vector (psecond - pfirst).  deltax and deltay of this vector
   % give the direction of travel. This establishes which of the
   % extreme points (see above) should be on the + side. If that 
   % point is o
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