quadSphere

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;

Building a quadrangulated sphere

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
r=1; %sphere radius
n=0:1:3; %Refinements
pColors=autumn(numel(n));
for q=1:1:numel(n);
    [F,V]=quadSphere(n(q),r,2);

    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);
    camlight headlight;
    set(gca,'FontSize',fontSize);
    view(3); axis tight;  axis equal;  grid on;
end

Demo showing initial shapes and subquadrangulated results

hf=cFigure;

[F,V]=quadSphere(0,r,2);
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);
camlight headlight;
set(gca,'FontSize',fontSize);
view(3); axis tight;  axis equal;  grid on;

[F,V]=quadSphere(3,r,2);
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);
camlight headlight;
set(gca,'FontSize',fontSize);
view(3); axis tight;  axis equal;  grid on;

[F,V]=quadSphere(0,r,5);
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);
camlight headlight;
set(gca,'FontSize',fontSize);
view(3); axis tight;  axis equal;  grid on;

[F,V]=quadSphere(3,r,5);
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);
camlight headlight;
set(gca,'FontSize',fontSize);
view(3); axis tight;  axis equal;  grid on;

[F,V]=quadSphere(0,r,6);
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);
camlight headlight;
set(gca,'FontSize',fontSize);
view(3); axis tight;  axis equal;  grid on;

[F,V]=quadSphere(2,r,6);
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);
camlight headlight;
set(gca,'FontSize',fontSize);
view(3); axis tight;  axis equal;  grid on;

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Kevin Mattheus Moerman, [email protected]