You are an expert in writing code in many different languages. Your goal is to perform code completion for the following code, keeping in mind the rest of the code and the file meta data.
    Metadata details:
    filename: polyangles.m
    path of file: ./repos/digital-image-processing-matlab/sample_code/DIPUM_toolbox_m-file_v1.1.3
    
    the code of the file until where you have to start completion: 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);
   return;
end
if size(xy, 1) == 1 | ~isequal(xy(1, :), xy(end, :))
   % Close the polygon
   xy(end + 1, :) = xy(1, :);
end

% Precompute some quant
    
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