# csdms-contrib/slepian_echo

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 function varargout=cube2sphere(lfin,alfa,bita,gama,eo,sc) % [x,y,z,J,N,dxi,deta,legs]=CUBE2SPHERE(lfin,alfa,bita,gama,eo,sc) % % Constructs a cubed sphere on 6 faces with 2^lfin[+1] nodes in each % direction. Reference is Ronchi et al., doi:10.1006/jcph.1996.0047 % % INPUT: % % lfin Number of subdivisions [default: 6] % alfa First Euler angle of wholesale tilt of all tiles [defaulted] % beta Second Euler angle of wholesale tilt of all tiles [defaulted] % gama Third Euler angle of wholesale tilt of all tiles [defaulted] % eo 0 even number of points [default] % 1 odd number of points % sc 0 regular cubed sphere [default] % 1 superchunk cubed sphere % % OUTPUT: % % x,y,z The NxNx6 matrices with the Cartesian coordinates for the six faces % J The NxN Jacobian of the transformation for a single face % N The number of points in one grid dimension, i.e. 2^lfin+1 % dxi,deta The spacing in xi and eta % legs A legend for the different chunk faces % % SEE ALSO: PLOTONCUBE, IMAGELETTER, SPHERE2CUBE, PLOTONCHUNK, CUBEMATS, CUBEJAC % % EXAMPLE: % % cube2sphere('demo1') % cube2sphere('demo2') % cube2sphere('demo3') % % VOLUME CALCULATION: % sum(6*dxi*deta*sum(sum(J(1:end-1,1:end-1))).*(rads/1000).^2*dr) % % Last modified by fjsimons-at-alum.mit.edu, 1/20/2011 % First define the defaults of all input variables defval('lfin',6) defval('alfa',[]); defval('bita',[]); defval('gama',[]); defval('eo',0); defval('sc',0); if ~isstr(lfin) % Be in the know what the final dimension will be N=2^lfin+eo; % Get the rotation matrices [rottot,mats,legs]=cubemats(alfa,bita,gama); % Produce the Jacobian, the single-face coordinates and the angular spacing [J,coordd,dxi,deta]=cubejac(N,N,sc); % Initialize the coordinate matrices [x,y,z]=deal(zeros(N,N,6)); % Loop over all faces for f=1:6 % Do the coordinate transform all at once, see SPHERE2CUBE for the inverse stuff=rottot*mats{f}*coordd; % Now distribute over the three three-dimensional vectors x(:,:,f)=reshape(stuff(1,:),N,N); y(:,:,f)=reshape(stuff(2,:),N,N); z(:,:,f)=reshape(stuff(3,:),N,N); end % Check that this is indeed still the unit sphere difer(x.^2+y.^2+z.^2-1,9,[],NaN) % The first and last rows/columns of J are exact repeats difer(J(1,:)-J(end,:),[],[],NaN) difer(J(:,1)-J(:,end),[],[],NaN) % Check that the integral over the surface is OK % The quality of this approximation depends on the number of % subdivisions carried out - see Ronchi eq. (20); deta*dxi is the % infinitesimal area right at the center of the block, where distortion % is minimal. But of course for finite subdivisions, this "rectangular" % area is not exactly to what it would be on the sphere % disp(sprintf('Nominal area of sphere is %6.4f ; compare at %6.4f',... % 6*dxi*deta*sum(sum(J(1:end-1,1:end-1))),4*pi)) % Provide output varns={x,y,z,J,N,dxi,deta,legs}; varargout=varns(1:nargout); elseif strcmp(lfin,'demo1') % Run the program [x,y,z,J]=cube2sphere; % Define some colors cols={'c','m','y','k','r','b'}; % Plot in succession for in=1:6 p{in}=plot3(x(:,:,in),y(:,:,in),z(:,:,in),'o',... 'MarkerF',cols{in},'MarkerE',cols{in}); hold on; end axis equal; hold off; axis off; set([p{:}],'MarkerS',3) elseif strcmp(lfin,'demo2') % Run the program with no initial tilt [x,y,z,J,N]=cube2sphere(3,0,0,0); % Plot in succession as a mesh for in=1:6 p{in}=mesh(x(:,:,in),y(:,:,in),z(:,:,in),in*ones(N,N)); hold on; end colormap(gray(6)) axis equal; hold off; axis off; elseif strcmp(lfin,'demo3') clf % Run the program with no initial tilt [x,y,z,J,N,dx,de,legs]=cube2sphere(3,0,0,0); % Plot if you like pols={'+','o','v','^','s','d'}; for ind=1:6 plot3(x(:,:,ind),y(:,:,ind),z(:,:,ind),... 'marker',pols{ind},'color','k') axis equal; axis([-1 1 -1 1 -1 1]); view(122,40); title(legs{ind}) xlabel('x'); ylabel('y'); zlabel('z') pause hold on end hold off end