GenerateRegularPolygon.m

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Creates a 2D polygon by sampling points on a Ellipse.
%
% Parameters:
% Inputs: aveRadius   - The average radius of the ellipse inside which the 
%                       polygon will be plotted
%                       This roughly controls how large the polygon will be
%         numVerts    - Number of vertices of the polygon 
%         AspectRatio - VerticalAxesLength/HorizontalAxesLength (b/a) of 
%                       the Ellipse
%
% Output: [x_vertices, y_vertices] = Returns a list of vertices as (x,y)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [x_vertices, y_vertices] = GenerateRegularPolygon ( aveRadius, numVerts, AspectRatio )

% generate n angle steps
% Make number of angles equal to number of vertices
angleSteps(1:numVerts) = (2*pi / numVerts);

% now generate the points
angle = unifrnd(0, 1);
angle(2:numVerts+1) = angleSteps(1:numVerts);
angle = cumsum(angle); % Incrementing angle per vertex
angle(end) = []; % We wouldnt need the last angle since it is a closed polygon with a flat base

%% Using polar cordinate equation for an ellipse
% The AspectRatio variables gives the ratio of a/b where a and b are vertical axis and horizontal radius lengths.
if AspectRatio > 1
    x_vertices = round((1/AspectRatio)*(aveRadius*cos(angle)));
    y_vertices = round(aveRadius*sin(angle));
else
    x_vertices = round((aveRadius*cos(angle)));
    y_vertices = round(AspectRatio*aveRadius*sin(angle));
end
    
% Close the polygon by making its end vertex = start vertex    
x_vertices(end+1) = x_vertices(1);
y_vertices(end+1) = y_vertices(1);

% Make the base of the polygon flat so that it can stand upright on a surface
index = find(y_vertices==min(y_vertices), 1 );
if index == numVerts
    y_vertices(index-1) = y_vertices(index);
else
    y_vertices(index+1) = y_vertices(index);
end

%% Check for center of gravity
% Offset the polygon to have a vertex at (0,0)
if min(y_vertices) < 0
y_vertices = y_vertices + abs(min(y_vertices));
else
y_vertices = y_vertices - abs(min(y_vertices));    
end

if min(x_vertices) < 0
x_vertices = x_vertices + abs(min(x_vertices));
else
x_vertices = x_vertices - abs(min(x_vertices));    
end

% Find the centroid
[x_centroid, ~] = polygoncentroid(x_vertices,y_vertices);

CenterOfMass_balanced = nnz(x_centroid >= min(x_vertices(y_vertices == min(y_vertices))) & x_centroid <= max(x_vertices(y_vertices == min(y_vertices))));

% Re-run of the structure's center of gravity does not fall within the base
if CenterOfMass_balanced ~= 1
    clearvars -except aveRadius numVerts AspectRatio;
%     disp("Not balanced, rerunning \n");
    [x_vertices, y_vertices] = GenerateRegularPolygon ( aveRadius, numVerts, AspectRatio );
end

end