[Master Index] [Index for PublicToolbox/Sphere_Tessellation]

# sphere_tri

## (PublicToolbox/Sphere_Tessellation/sphere_tri.m in BrainStorm 2.0 (Alpha))

### Function Synopsis

    [FV] = sphere_tri(shape,maxlevel,r,winding)

### Help Text

     sphere_tri - generate a triangle mesh approximating a sphere

     Usage: FV = sphere_tri(shape,Nrecurse,r,winding)

       shape is a string, either of the following:
       'ico'   starts with icosahedron (most even, default)
       'oct'   starts with octahedron
       'tetra' starts with tetrahedron (least even)

       Nrecurse is int >= 0, setting the recursions (default 0)

       r is the radius of the sphere (default 1)

       winding is 0 for clockwise, 1 for counterclockwise (default 0).  The
       matlab patch command gives outward surface normals for clockwise
       order of vertices in the faces (viewed from outside the surface).

       FV has fields FV.vertices and FV.faces.  The vertices
       are listed in clockwise order in FV.faces, as viewed
       from the outside in a RHS coordinate system.

     The function uses recursive subdivision.  The first
     approximation is an platonic solid, either an  icosahedron,
     octahedron or a tetrahedron.  Each level of refinement
     subdivides each triangle face by a factor of 4 (see also
     mesh_refine).  At each refinement, the vertices are
     projected to the sphere surface (see sphere_project).

     A recursion level of 3 or 4 is a good sphere surface, if
     gouraud shading is used for rendering.

     The returned struct can be used in the patch command, eg:

     % create and plot, vertices: [2562x3] and faces: [5120x3]
     FV = sphere_tri('ico',4,1);
     lighting phong; shading interp; figure;
     patch('vertices',FV.vertices,'faces',FV.faces,...
           'facecolor',[1 0 0],'edgecolor',[.2 .2 .6]);
     axis off; camlight infinite; camproj('perspective');

     See also: mesh_refine, sphere_project

### Cross-Reference Information

This function calls

    mesh_refine_tri4                   C:\BrainStorm_2001\PublicToolbox\Sphere_Tessellation\mesh_refine_tri4.m
    sphere_project                     C:\BrainStorm_2001\PublicToolbox\Sphere_Tessellation\sphere_project.m

### Listing of function C:\BrainStorm_2001\PublicToolbox\Sphere_Tessellation\sphere_tri.m

    function [FV] = sphere_tri(shape,maxlevel,r,winding)

    % sphere_tri - generate a triangle mesh approximating a sphere
    %
    % Usage: FV = sphere_tri(shape,Nrecurse,r,winding)
    %
    %   shape is a string, either of the following:
    %   'ico'   starts with icosahedron (most even, default)
    %   'oct'   starts with octahedron
    %   'tetra' starts with tetrahedron (least even)
    %
    %   Nrecurse is int >= 0, setting the recursions (default 0)
    %
    %   r is the radius of the sphere (default 1)
    %
    %   winding is 0 for clockwise, 1 for counterclockwise (default 0).  The
    %   matlab patch command gives outward surface normals for clockwise
    %   order of vertices in the faces (viewed from outside the surface).
    %
    %   FV has fields FV.vertices and FV.faces.  The vertices
    %   are listed in clockwise order in FV.faces, as viewed
    %   from the outside in a RHS coordinate system.
    %
    % The function uses recursive subdivision.  The first
    % approximation is an platonic solid, either an  icosahedron,
    % octahedron or a tetrahedron.  Each level of refinement
    % subdivides each triangle face by a factor of 4 (see also
    % mesh_refine).  At each refinement, the vertices are
    % projected to the sphere surface (see sphere_project).
    %
    % A recursion level of 3 or 4 is a good sphere surface, if
    % gouraud shading is used for rendering.
    %
    % The returned struct can be used in the patch command, eg:
    %
    % % create and plot, vertices: [2562x3] and faces: [5120x3]
    % FV = sphere_tri('ico',4,1);
    % lighting phong; shading interp; figure;
    % patch('vertices',FV.vertices,'faces',FV.faces,...
    %       'facecolor',[1 0 0],'edgecolor',[.2 .2 .6]);
    % axis off; camlight infinite; camproj('perspective');
    %
    % See also: mesh_refine, sphere_project
    %

    % $Revision: 1 $ $Date: 5/20/04 3:58p $

    % Licence:  GNU GPL, no implied or express warranties
    % Jon Leech (leech @ cs.unc.edu) 3/24/89
    % icosahedral code added by Jim Buddenhagen (jb1556@daditz.sbc.com) 5/93
    % 06/2002, adapted from c to matlab by Darren.Weber_at_radiology.ucsf.edu
    % 05/2004, reorder of the faces for the 'ico' surface so they are indeed
    % clockwise!  Now the surface normals are directed outward.  Also reset the
    % default recursions to zero, so we can get out just the platonic solids.
    %
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    if ~exist('shape','var') || isempty(shape),
        shape = 'ico';
    end
    fprintf('...creating sphere tesselation based on %s\n',shape);

    % default maximum subdivision level
    if ~exist('maxlevel','var') || isempty(maxlevel) || maxlevel < 0,
        maxlevel = 0;
    end

    % default radius
    if ~exist('r','var') || isempty(r),
        r = 1;
    end

    if ~exist('winding','var') || isempty(winding),
        winding = 0;
    end

    % -----------------
    % define the starting shapes

    shape = lower(shape);

    switch shape,
    case 'tetra',

        % Vertices of a tetrahedron
        sqrt_3 = 0.5773502692;

        tetra.v = [  sqrt_3,  sqrt_3,  sqrt_3 ;   % +X, +Y, +Z  - PPP
                    -sqrt_3, -sqrt_3,  sqrt_3 ;   % -X, -Y, +Z  - MMP
                    -sqrt_3,  sqrt_3, -sqrt_3 ;   % -X, +Y, -Z  - MPM
                     sqrt_3, -sqrt_3, -sqrt_3 ];  % +X, -Y, -Z  - PMM

        % Structure describing a tetrahedron
        tetra.f = [ 1, 2, 3;
                    1, 4, 2;
                    3, 2, 4;
                    4, 1, 3 ];

        FV.vertices = tetra.v;
        FV.faces    = tetra.f;

    case 'oct',

        % Six equidistant points lying on the unit sphere
        oct.v = [  1,  0,  0 ;  %  X
                  -1,  0,  0 ;     % -X
                   0,  1,  0 ;  %  Y
                   0, -1,  0 ;     % -Y
                   0,  0,  1 ;     %  Z
                   0,  0, -1 ];    % -Z

        % Join vertices to create a unit octahedron
        oct.f = [ 1 5 3 ;    %  X  Z  Y  -  First the top half
                  3 5 2 ;    %  Y  Z -X
                  2 5 4 ;    % -X  Z -Y
                  4 5 1 ;    % -Y  Z  X
                  1 3 6 ;    %  X  Y -Z  -  Now the bottom half
                  3 2 6 ;    %  Y  Z -Z
                  2 4 6 ;    % -X  Z -Z
                  4 1 6 ];   % -Y  Z -Z

        FV.vertices = oct.v;
        FV.faces    = oct.f;

    case 'ico',

        % Twelve vertices of icosahedron on unit sphere
        tau = 0.8506508084; % t=(1+sqrt(5))/2, tau=t/sqrt(1+t^2)
        one = 0.5257311121; % one=1/sqrt(1+t^2) , unit sphere

        ico.v( 1,:) = [  tau,  one,    0 ]; % ZA
        ico.v( 2,:) = [ -tau,  one,    0 ]; % ZB
        ico.v( 3,:) = [ -tau, -one,    0 ]; % ZC
        ico.v( 4,:) = [  tau, -one,    0 ]; % ZD
        ico.v( 5,:) = [  one,   0 ,  tau ]; % YA
        ico.v( 6,:) = [  one,   0 , -tau ]; % YB
        ico.v( 7,:) = [ -one,   0 , -tau ]; % YC
        ico.v( 8,:) = [ -one,   0 ,  tau ]; % YD
        ico.v( 9,:) = [   0 ,  tau,  one ]; % XA
        ico.v(10,:) = [   0 , -tau,  one ]; % XB
        ico.v(11,:) = [   0 , -tau, -one ]; % XC
        ico.v(12,:) = [   0 ,  tau, -one ]; % XD

        % Structure for unit icosahedron
        ico.f = [  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 ];

        FV.vertices = ico.v;
        FV.faces    = ico.f;
    end

    % -----------------
    % refine the starting shapes with subdivisions
    if maxlevel,

        % Subdivide each starting triangle (maxlevel) times
        for level = 1:maxlevel,

            % Subdivide each triangle and normalize the new points thus
            % generated to lie on the surface of a sphere radius r.
            FV = mesh_refine_tri4(FV);
            FV.vertices = sphere_project(FV.vertices,r);

            % An alternative might be to define a min distance
            % between vertices and recurse or use fminsearch

        end
    end

    if winding,
        fprintf('...returning counterclockwise vertex order (viewed from outside)\n');
        FV.faces = FV.faces(:,[1 3 2]);
    else
        fprintf('...returning clockwise vertex order (viewed from outside)\n');
    end

    return

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