Below is a demonstration of the features of the quadSphere function

## Contents

```clear; close all; clc;
```

Plot settings

```fontSize=15;
faceAlpha=0.75;
edgeColor=0.3*ones(1,3);
edgeWidth=1.5;
```

The function inputs are n and r which define the mesh refinement and radius respectively. The mesh refinement number n defines the number of subquadrangulation (see function subQuad) iterations performed on an initial solid. The function outputs the faces (F) and vertices (V). The default initial solid is the cube. However this can be altered by varying the optional 3rd input oriOpt. Its value sets the initial solid as: oriOpt=1 -> Tetrahedron oriOpt=2 -> Cube oriOpt=3 -> Octahedron oriOpt=4 -> Icosahedron oriOpt=5 -> Rhombic dodecahedron oriOpt=6 -> (Buckminster-Fuller) geoSphere For option 6, geoSphere the subdevisions are based on subtriangulations of the icosahedron (see geoSphere) after which all faces are converted to quadrilateral faces (see tri2quad). Below is a visualisation for n=0:1:3 for the cube as initial solid.

```% Open figure for plotting
hf=cFigure;

%Defining geodesic dome
n=0:1:3; %Refinements
pColors=autumn(numel(n));
for q=1:1:numel(n);

subplot(2,2,q); hold on;
title([num2str(n(q)),' refinement iterations'],'FontSize',fontSize);
xlabel('X','FontSize',fontSize); ylabel('Y','FontSize',fontSize); zlabel('Z','FontSize',fontSize);
hp=patch('Faces',F,'Vertices',V);
set(hp,'FaceColor',pColors(q,:),'FaceAlpha',faceAlpha,'lineWidth',edgeWidth,'edgeColor',edgeColor);
set(gca,'FontSize',fontSize);
view(3); axis tight;  axis equal;  grid on;
end
```

## Demo showing initial shapes and subquadrangulated results

```hf=cFigure;

subplot(2,3,1); hold on;
title(['Cube 0 ref. it.'],'FontSize',fontSize);
xlabel('X','FontSize',fontSize); ylabel('Y','FontSize',fontSize); zlabel('Z','FontSize',fontSize);
hp=patch('Faces',F,'Vertices',V);
set(hp,'FaceColor','r','FaceAlpha',faceAlpha,'lineWidth',edgeWidth,'edgeColor',edgeColor);
set(gca,'FontSize',fontSize);
view(3); axis tight;  axis equal;  grid on;

subplot(2,3,4); hold on;
title(['Cube 3 ref. it.'],'FontSize',fontSize);
xlabel('X','FontSize',fontSize); ylabel('Y','FontSize',fontSize); zlabel('Z','FontSize',fontSize);
hp=patch('Faces',F,'Vertices',V);
set(hp,'FaceColor','r','FaceAlpha',faceAlpha,'lineWidth',edgeWidth,'edgeColor',edgeColor);
set(gca,'FontSize',fontSize);
view(3); axis tight;  axis equal;  grid on;

subplot(2,3,2); hold on;
title(['Rhombic dodecahedron 0 ref. it.'],'FontSize',fontSize);
xlabel('X','FontSize',fontSize); ylabel('Y','FontSize',fontSize); zlabel('Z','FontSize',fontSize);
hp=patch('Faces',F,'Vertices',V);
set(hp,'FaceColor','b','FaceAlpha',faceAlpha,'lineWidth',edgeWidth,'edgeColor',edgeColor);
set(gca,'FontSize',fontSize);
view(3); axis tight;  axis equal;  grid on;

subplot(2,3,5); hold on;
title(['Rhombic dodecahedron 3 ref. it.'],'FontSize',fontSize);
xlabel('X','FontSize',fontSize); ylabel('Y','FontSize',fontSize); zlabel('Z','FontSize',fontSize);
hp=patch('Faces',F,'Vertices',V);
set(hp,'FaceColor','b','FaceAlpha',faceAlpha,'lineWidth',edgeWidth,'edgeColor',edgeColor);
set(gca,'FontSize',fontSize);
view(3); axis tight;  axis equal;  grid on;

subplot(2,3,3); hold on;
title(['geoSphere 0 ref. it.'],'FontSize',fontSize);
xlabel('X','FontSize',fontSize); ylabel('Y','FontSize',fontSize); zlabel('Z','FontSize',fontSize);
hp=patch('Faces',F,'Vertices',V);
set(hp,'FaceColor','g','FaceAlpha',faceAlpha,'lineWidth',edgeWidth,'edgeColor',edgeColor);
set(gca,'FontSize',fontSize);
view(3); axis tight;  axis equal;  grid on;

subplot(2,3,6); hold on;
title(['geoSphere 2 ref. it.'],'FontSize',fontSize);
xlabel('X','FontSize',fontSize); ylabel('Y','FontSize',fontSize); zlabel('Z','FontSize',fontSize);
hp=patch('Faces',F,'Vertices',V);
set(hp,'FaceColor','g','FaceAlpha',faceAlpha,'lineWidth',edgeWidth,'edgeColor',edgeColor);
set(gca,'FontSize',fontSize);
view(3); axis tight;  axis equal;  grid on;
```

GIBBON www.gibboncode.org

Kevin Mattheus Moerman, gibbon.toolbox@gmail.com

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GIBBON: The Geometry and Image-based Bioengineering add-On. A toolbox for image segmentation, image-based modeling, meshing, and finite element analysis.

Copyright (C) 2019 Kevin Mattheus Moerman

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