Filing the entire volume of a cube with small cubes in MATLAB

Question

I have built a hollow cube in MATLAB, I want to completely fill its volume with small cubes. Then I want to find a way to access these cubes and make paths through them, i.e

Answer 1

Same principle, we build a big cube, then a small one in the corner, then we repeat the small cube building with a small offset until we are full. The main difference with the old code is that this time the step change of each coordinate set is controlled (function of x,y,z coordinates of the small cube) instead of being random.

%%
clf; figure(1); format compact 
h(1) = axes('Position',[0.2 0.2 0.6 0.6]);

%These are the different 8 vertices of the cube, each is defined by its 3 x y z coordinates:
vert = [ 1  1 -1; -1  1 -1; -1  1  1; 1  1  1; -1 -1  1; 1 -1  1; 1 -1 -1; -1 -1 -1];

%These are the 6 faces of the cube, each is defined by connecting 4 of the available vertices:
fac = [1 2 3 4; 4 3 5 6; 6 7 8 5; 1 2 8 7; 6 7 1 4; 2 3 5 8];

%// How many small cube do we want
MainCubeSide = 2 ;              %// dimension of the side of the main cube
nCubeOnSide = 5 ;               %// number of small cube in one "row/column" of the main cube
nCubesTotal = nCubeOnSide^3  ;  %// total number of small cube

% define the Main container cube
MainCube.Vertices = vert *(2/MainCubeSide) ; %// because the cube as defined above has already a side=2
MainCube.Faces = fac ;
MainCube.FaceColor = 'w' ;

hMainCube = patch(MainCube);  %// patch function for the first big cube. 
axis([-1, 1, -1, 1, -1, 1]);
axis equal;
hold on;
material metal;
alpha('color');
alphamap('rampdown');
view(138,24)
%view(3);


%% // generate all the coordinates of each cube first
dstep = MainCubeSide / nCubeOnSide ;                                                 %// step size for small cube vertices
vElem = bsxfun(@plus, vert / nCubeOnSide , -( MainCubeSide/2 - dstep/2)*[1 1 1] )  ; %// elementary cube vertices

%%
hold on;
coords = zeros( size(vElem,1),size(vElem,2), nCubesTotal ) ;  %// To store the coordinates
colors = zeros( nCubesTotal , 3 ) ;                           %// To store the colours
hcube  = zeros( nCubesTotal , 1 ) ;                           %// To store the handles of the patch objects

iNeighbour = zeros( nCubesTotal , 6 ) ;   %// To save the index of the neighbours
idc = permute( reshape(1:nCubesTotal,nCubeOnSide,nCubeOnSide,nCubeOnSide) , [3 2 1] ) ;

%// For each cube ...
iCube = 0 ;
for iline=1:nCubeOnSide         %// Lines
    for icol=1:nCubeOnSide      %// Columns
        for ih=1:nCubeOnSide    %// Slice (height)
            iCube = iCube + 1 ;

            %// Take the base corner coordinates and add an offset to each coordinate
            coords(:,:,iCube) = bsxfun(@plus, vElem , dstep*[(iline-1) (icol-1) (ih-1)]);

            %// Save the colour
            colors(iCube,:) = rand(1,3) ; 

            %// Draw the cube
            hcube(iCube) = patch('Faces', fac, 'Vertices', coords(:,:,iCube), 'FaceColor', colors(iCube,:) ) ;

            drawnow     %// just for intermediate display, you can comment these 2 lines
            pause(0.05) %// just for intermediate display, you can comment these 2 lines

            %// save adjacent cubes indices
            ixAdj = [iline-1 iline+1 icol-1 icol+1 ih-1 ih+1] ;  %// indices of adjacent cubes
            idxFalse = (ixAdj<1) | (ixAdj>nCubeOnSide) ;  %// detect cube which would be "out" of the main cube
            ixAdj(idxFalse) = 1 ;                                %// just to not get an "indexing" error at this stage
            iNeighbour(iCube,:) = [idc(ixAdj(1),icol,ih)    idc(ixAdj(2),icol,ih) ...
                                   idc(iline,ixAdj(3),ih)   idc(iline,ixAdj(4),ih) ...
                                   idc(iline,icol,ixAdj(5)) idc(iline,icol,ixAdj(6)) ] ;
            iNeighbour(iCube,idxFalse) = NaN ;
        end
    end
end

This code saves the handle of each cube in the variable hcube so you can do assignment of property in bulk on all the cubes if you want. For example delete(hcube) will delete all the small cubes in one go, or set(hcube,'Facealpha',0.5) will make all cubes half transparent.

You can also set/change properties on only a subset of them hcube(idx_subset) = .... This is where knowing the adjacent cubes by their index could be useful, but your adjacency question is not completely defined yet.

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Edit: I have added the neighbour tracking in the main loop. It is probably not the most efficient way to do it but it does keep an index of all the neighbour fro each elementary cube. The iNeighbour variable (size: nCubesx6) hold the handle index of each neighbour (6 possible neighbours). When a neighbour didn't exist, I opted to place a NaN instead. To retrieve the index of the neighbour directly without the NaNs, I defined a helper anonymous function:

getNeighbourIndex = @(idx) iNeighbour(idx,~isnan(iNeighbour(idx,:))) ;

Which now help you track all the neighbour of a given cube. For example:

set(hcube,'Visible','off')  %// turn off all small cubes
CubeOfInterest = 111 ;      %// select one cube
%// display the main cube of interest, and it's neighbours in transparency
set(hcube(CubeOfInterest),'Visible','on','FaceColor','r','FaceAlpha',1) 
set(hcube(getNeighbourIndex(CubeOfInterest)),'Visible','on','FaceColor','g','FaceAlpha',.05)

As you can see, all the neighbours are there, whether we are close to a wall or not.