How can I create a Triangulated sphere with faces of triangles with the same area each one.
I want something like this,
http://imageshack.us/a/img198/5041/71183923.png
and I searched and saw I could use the function isosurfaceof MATLAB, but the triangles don't have equal area and form and they are essentially squares divided in two with the square diagonal.
here's my code:
>> [X,Y,Z] = meshgrid(-10:10,-10:10,-10:10);
>> V = sqrt(X.^2+Y.^2+Z.^2);
>> fv = isosurface(X,Y,Z,V)
>> p = patch(fv);
>> set(p,'EdgeColor','g')
and the result is below:
Use this function from file exchange
function [p, t] = TriSphere(N, R)
% TRISPHERE: Returns the triangulated model of a sphere using the
% icosaedron subdivision method.
%
% INPUT:
% N (integer number) indicates the number of subdivisions,
% it can assumes values between 0-inf. The greater N the better will look
% the surface but the more time will be spent in surface costruction and
% more triangles will be put in the output model.
%
% OUTPUT:
% In p (nx3) and t(mx3) we can find points and triangles indexes
% of the model. The sphere is supposed to be of unit radius and centered in
% (0,0,0). To obtain spheres centered in different location, or with
% different radius, is just necessary a traslation and a scaling
% trasformation. These operation are not perfomed by this code beacuse it is
% extrimely convinient, in order of time perfomances, to do this operation
% out of the function avoiding to call the costruction step each time.
%
% NOTE:
% This function is more efficient than the matlab command sphere in
% terms of dimension fo the model/ accuracy of recostruction. This due to
% well traingulated model that requires a minor number of patches for the
% same geometrical recostruction accuracy. Possible improvement are possible
% in time execution and model subdivision flexibilty.
%
% EXAMPLE:
%
% N=5;
%
% [p,t] = TriSphere(N);
%
% figure(1) axis equal hold on trisurf(t,p(:,1),p(:,2),p(:,3)); axis vis3d
% view(3)
% Author: Giaccari Luigi Created:25/04/2009%
% For info/bugs/questions/suggestions : [email protected]
% ORIGINAL NAME: BUILDSPHERE
%
% Adjusted by Rody Oldenhuis (speed/readability)
% error traps
error(nargchk(1,1,nargin));
error(nargoutchk(1,2,nargout));
if ~isscalar(N)
error('Buildsphere:N_mustbe_scalar',...
'Input N must be a scalar.');
end
if round(N) ~= N
error('Buildsphere:N_mustbe_scalar',...
'Input N must be an integer value.');
end
% Coordinates have been taken from Jon Leech' files
% Twelve vertices of icosahedron on unit sphere
tau = 0.8506508083520400; % t = (1+sqrt(5))/2, tau = t/sqrt(1+t^2)
one = 0.5257311121191336; % one = 1/sqrt(1+t^2) (unit sphere)
p = [
+tau +one +0 % ZA
-tau +one +0 % ZB
-tau -one +0 % ZC
+tau -one +0 % ZD
+one +0 +tau % YA
+one +0 -tau % YB
-one +0 -tau % YC
-one +0 +tau % YD
+0 +tau +one % XA
+0 -tau +one % XB
+0 -tau -one % XC
+0 +tau -one]; % XD
% Structure for unit icosahedron
t = [
5 8 9
5 10 8
6 12 7
6 7 11
1 4 5
1 6 4
3 2 8
3 7 2
9 12 1
9 2 12
10 4 11
10 11 3
9 1 5
12 6 1
5 4 10
6 11 4
8 2 9
7 12 2
8 10 3
7 3 11 ];
% possible quick exit
if N == 0, return, end
% load pre-generated trispheres (up to 8 now...)
if N <= 8
S = load(['TriSphere', num2str(N), '.mat'],'pts','idx');
p = S.pts; t = S.idx;
if nargin == 2, p = p*R; end
return
else
% if even more is requested (why on Earth would you?!), make sure to START
% from the maximum pre-loadable trisphere
S = load('TriSphere8.mat','pts','idx');
p = S.pts; t = S.idx; clear S; N0 = 10;
end
% how many triangles/vertices do we have?
nt = size(t,1); np = size(p,1); totp = np;
% calculate the final number of points
for ii=N0:N, totp = 4*totp - 6; end
% initialize points array
p = [p; zeros(totp-12, 3)];
% determine the appropriate class for the triangulation indices
numbits = 2^ceil(log(log(totp+1)/log(2))/log(2));
castToInt = ['uint',num2str(numbits)];
% issue warning when required
if numbits > 64
warning('TriSphere:too_many_notes',...
['Given number of iterations would require a %s class to accurately ',...
'represent the triangulation indices. Using double instead; Expect ',...
'strange results!']);
castToInt = @double;
else
castToInt = str2func(castToInt);
end
% refine icosahedron N times
for ii = N0:N
% initialize inner loop
told = t;
t = zeros(nt*4, 3);
% Use sparse. Yes, its slower in a loop, but for N = 6 the size is
% already ~10,000x10,000, growing by a factor of 4 with every
% increasing N; its simply too memory intensive to use zeros().
peMap = sparse(np,np);
ct = 1;
% loop trough all old triangles
for j = 1:nt
% some helper variables
p1 = told(j,1);
p2 = told(j,2);
p3 = told(j,3);
x1 = p(p1,1); x2 = p(p2,1); x3 = p(p3,1);
y1 = p(p1,2); y2 = p(p2,2); y3 = p(p3,2);
z1 = p(p1,3); z2 = p(p2,3); z3 = p(p3,3);
% first edge
% -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
% preserve triangle orientation
if p1 < p2, p1m = p1; p2m = p2; else p2m = p1; p1m = p2; end
% If the point does not exist yet, calculate the new point
p4 = peMap(p1m,p2m);
if p4 == 0
np = np+1;
p4 = np;
peMap(p1m,p2m) = np;%#ok
p(np,1) = (x1+x2)/2;
p(np,2) = (y1+y2)/2;
p(np,3) = (z1+z2)/2;
end
% second edge
% -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
% preserve triangle orientation
if p2 < p3; p2m = p2; p3m = p3; else p2m = p3; p3m = p2; end
% If the point does not exist yet, calculate the new point
p5 = peMap(p2m,p3m);
if p5 == 0
np = np+1;
p5 = np;
peMap(p2m,p3m) = np;%#ok
p(np,1) = (x2+x3)/2;
p(np,2) = (y2+y3)/2;
p(np,3) = (z2+z3)/2;
end
% third edge
% -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
% preserve triangle orientation
if p1 < p3; p1m = p1; p3m = p3; else p3m = p1; p1m = p3; end
% If the point does not exist yet, calculate the new point
p6 = peMap(p1m,p3m);
if p6 == 0
np = np+1;
p6 = np;
peMap(p1m,p3m) = np;%#ok
p(np,1) = (x1+x3)/2;
p(np,2) = (y1+y3)/2;
p(np,3) = (z1+z3)/2;
end
% allocate new triangles
% refine indexing
% p1
% /\
% /t1\
% p6/____\p4
% /\ /\
% /t4\t2/t3\
% /____\/____\
% p3 p5 p2
t(ct,1) = p1; t(ct,2) = p4; t(ct,3) = p6; ct = ct+1;
t(ct,1) = p4; t(ct,2) = p5; t(ct,3) = p6; ct = ct+1;
t(ct,1) = p4; t(ct,2) = p2; t(ct,3) = p5; ct = ct+1;
t(ct,1) = p6; t(ct,2) = p5; t(ct,3) = p3; ct = ct+1;
end % end subloop
% update number of triangles
nt = ct-1;
end % end main loop
% normalize all points to 1 (or R)
p = bsxfun(@rdivide, p, sqrt(sum(p.^2,2)));
if (nargin == 2), p = p*R; end
% convert t to proper integer class
t = castToInt(t);
end % funciton TriSphere