# geoSphere

Below is a demonstration of the features of the geoSphere function

## Syntax

[F,V,Vs]=geoSphere(n,r,solidType);

## Description

Use geoSphere to generate triangulated spheres with nearly geodesic triangle distributions. The density of the triangulation can be controlled through a particular choice of n (number of mesh refinement steps).

## Examples

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

Plot Settings

```fontSize=15;
faceAlpha=1;
edgeColor=0.2*ones(1,3);
edgeWidth=1.5;
```

## Building a geodesic dome based on the icosahedron

The function inputs are n and r which define the mesh refinement and radius respectively. The mesh refinement number n defines the number of subtriangulation (see function subTri) iterations performed on an icosahedron.

```r=1; %sphere radius
n=2; %Refinements
[F,V,~]=geoSphere(n,r);
[Fi,Vi,~]=geoSphere(0,r);
```

Visualize sphere

```cFigure; hold on;
subplot(1,2,1); hold on;
gpatch(Fi,Vi,'rw','r',0.8,2);
plotV(Vi,'r.','MarkerSize',50);
camlight headlight;
axisGeom(gca,fontSize);

subplot(1,2,2); hold on;
gpatch(F,V,'bw','k',1,2);
plotV(Vi,'r.','MarkerSize',50);
camlight headlight;
axisGeom(gca,fontSize);
drawnow;
``` Below is a visualisation for n=0:1:3. The function outputs the geodesic dome faces (F) and vertices (V) and also the spherical coordinates of the vertices (Vs) (this output is suppressed in the example below).

```cFigure; % Open figure for plotting

%Defining triangulated geodesic domes with different densities
r=1; %sphere radius
n=0:1:3; %Refinements
pColors=gjet(numel(n));
for q=1:1:numel(n)
[F,V,~]=geoSphere(n(q),r);
subplot(2,2,q); hold on;
title([num2str(n(q)),' refinement iterations'],'FontSize',fontSize);
gpatch(F,V,pColors(q,:));
% patchNormPlot(F,V);
camlight headlight;
axisGeom(gca,fontSize);
end
drawnow;
``` ## Using other solid types

Other platonic solids can also be used as a starting tesselation. However these may not be as geodesic as the result for the icosahedron and dodecahedron.

```%e.g. using a cube
solidTypes=1:5;

cFigure; % Open figure for plotting
titleCell={'tetrahedron','cube','octahedron','icosahedron','dodecahedron'};
pColors=gjet(numel(solidTypes));
for q=solidTypes
[F,V,~]=geoSphere(0,r,q);
subplot(2,3,q); hold on;
title(['Based on: ',titleCell{q}],'FontSize',fontSize);
gpatch(F,V,pColors(q,:));
% patchNormPlot(F,V);
camlight headlight;
axisGeom(gca,fontSize);
end
drawnow;
```  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

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see http://www.gnu.org/licenses/.