任意の精度まで球の三角形分割を生成できる関数がここにあります。buildsphereこれは、残念ながらインターネットから完全に姿を消してしまった Giaccari Luigiの機能に基づいています (この機能と共に)。
そこで、ここと私の File Exchange アカウントに投稿します。承認されたら、このコードをリンクに置き換えます。
私のコンピューターには、この関数が依存する事前生成されたモデルがいくつかあることに注意してください。実行する前に、そのセクションを削除する必要があります。
生成された球体のプロットについては、Gunther Struyf の回答を参照してください。
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 : giaccariluigi@msn.com
% 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
