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: eq_caps_symm.m
    path of file: ./repos/libDirectional/lib/external/eq_sphere_partitions/eq_partitions
    
    the code of the file until where you have to start completion: function [s_cap,n_regions] = eq_caps_symm(dim,N)
%EQ_CAPS_SYMM Partition a sphere into to nested spherical caps
%
%Syntax
% [s_cap,n_regions] = eq_caps_symm(N);
%
%Description
% [S_CAP,N_REGIONS] = EQ_CAPS_symm(N) does the following:
% 1) partitions the unit sphere S^dim into a list of spherical caps of
%    increasing colatitude and thus increasing area,
% 2) sets S_CAP to be an array of size (1 by N_COLLARS+2),
%    containing increasing colatitudes of caps, and
% 3) sets N_REGIONS to be an array of size (1 by N_COLLARS+2),
%    containing the intger number of regions in each corresponding zone of 
%    S^dim.
%
% The argument N is assumed to be a positive integer.
%
%Notes
% The value N_COLLARS is a positive integer function of dim and N.
%
% S_CAP[1] is C_POLAR, the colatitude of the North polar cap.
% S_CAP[N_COLLARS+1] is pi-C_POLAR.
% S_CAP[N_COLLARS+2] is pi.
%
% N_REGIONS[1] is 1.
% N_REGIONS[N_COLLARS+2] is 1.
% The sum of N_REGIONS is N.
%
%Examples
% > [s_cap,n_regions] = eq_caps_symm(2,10)
% s_cap =
%     0.6435    1.5708    2.4981    3.1416
% n_regions =
%      1     4     4     1
%  
% > [s_cap,n_regions] = eq_caps_symm(3,6)
% s_cap =
%     0.9845    2.1571    3.1416
% n_regions =
%      1     4     1
%
%See also
% EQ_REGIONS, EQ_POINT_SET_POLAR

    
function [s_cap,n_regions] = eq_caps_symm(dim,N)
%EQ_CAPS_SYMM Partition a sphere into to nested spherical caps
%
%Syntax
% [s_cap,n_regions] = eq_caps_symm(N);
%
%Description
% [S_CAP,N_REGIONS] = EQ_CAPS_symm(N) does the following:
% 1) partitions the unit sphere S^dim into a list of spherical caps of
%    increasing colatitude and thus increasing area,
% 2) sets S_CAP to be an array of size (1 by N_COLLARS+2),
%    containing increasing colatitudes of caps, and
% 3) sets N_REGIONS to be an array of size (1 by N_COLLARS+2),
%    containing the intger number of regions in each corresponding zone of 
%    S^dim.
%
% The argument N is assumed to be a positive integer.
%
%Notes
% The value N_COLLARS is a positive integer function of dim and N.
%
% S_CAP[1] is C_POLAR, the colatitude of the North polar cap.
% S_CAP[N_COLLARS+1] is pi-C_POLAR.
% S_CAP[N_COLLARS+2] is pi.
%
% N_REGIONS[1] is 1.
% N_REGIONS[N_COLLARS+2] is 1.
% The sum of N_REGIONS is N.
%
%Examples
% > [s_cap,n_regions] = eq_caps_symm(2,10)
% s_cap =
%     0.6435    1.5708    2.4981    3.1416
% n_regions =
%      1     4     4     1
%  
% > [s_cap,n_regions] = eq_caps_symm(3,6)
% s_cap =
%     0.9845    2.1571    3.1416
% n_regions =
%      1     4     1
%
%See also
% EQ_REGIONS, EQ_POINT_SET_POLAR

% Adapted version by Florian Pfaff (pfaff@kit.edu) for libDirectional
% 2020-03-28
% Copyright 2004-2005 Paul Leopardi for the University of New South Wales.
% $Revision 1.10 $ $Date 2005-06-01 $
% Documentation files renamed
% $Revision 1.00 $ $Date 2005-02-12 $
%
% For licensing, see COPYING.
% For references, see AUTHORS.
% For revision history, see CHANGELOG.

%
% Check number of arguments
%
error(nargchk(2,2,nargin));
error(nargoutchk(2,2,nargout));
%
% dim is the number of dimensions
% N is the number of regions
%
if dim == 1
    %
    % We have a circle. Return the angles of N equal sectors.
    %
    sector = 1:N;
    %
    % Make dim==1 consistent with dim>1 by
    % returning the longitude of a sector enclosing the
    % cumulative sum of arc lengths given by summing n_regions.
    %
    s_cap = sector*2*pi/N;
    n_regions = ones(size(sector));
    %
elseif N == 1
    %
    % We have only one region, which must be the whole sphere.
    %
    s_cap = [pi];
    n_regions = [1];
    %
else
    %
    % Given dim and N, determine c_polar,
    % the colatitude of the North polar spherical cap.
    %
    c_polar = polar_colat(dim,N);
    %
    % Given dim and N, determine the ideal angle for spherical collars.
    % Based on N, this ideal angle, and c_polar,
    % determine n_collars, the number of collars between the polar caps.
    %
    % for quaternion
    a_ideal = ideal_collar_angle(dim,N);
    n_collars = zeros(size(N));
    enough = (N > 2) & (a_ideal > 0);
    n_collars(enough) = 2 * max(1/2,round(0.5*(pi-2*c_polar(enough))./a_ideal(enough)));
    %
    % Given dim, N, c_polar and n_collars, determine r_regions,
    % a list of the ideal real number of regions in each collar,
    % plus the polar caps.
    % The number of elements is n_collars+2.
    % r_regions[1] is 1.
    % r_regions[n_collars+2] is 1.
    % The sum of r_regions is N.
    %
    r_regions = ideal_region_list(dim,N,c_polar,n_collars);
    %
    % Given N and r_regions, determine n_regions,
    % a list of the natural number of regions in each collar and
    % the polar caps.
    % This list is as close as possible to r_regions.
    % The number of elements is n_collars+2.
    % n_regions[1] is 1.
    % n_regions[n_collars+2] is 1.
    % The sum of n_regions is N.
    %
    n_regions = round_to_naturals(N,r_regions);
    %
    % Given dim, N, c_polar and n_regions, determine s_cap,
    % an increasing list of colatitudes of spherical caps which enclose the same area
    % as that given by the cumulative sum of regions.
    % The number of elements is n_collars+2.
    % s_cap[1] is c_polar.
    % s_cap[n_collars+1] is Pi-c_polar.
    % s_cap[n_collars+2] is Pi.
    %
    s_cap = cap_colats(dim,N,c_polar,n_regions);
    %
end