How can I create a Delanay point in a cube?

I want to generate Mesh for a cube,but the points inside the cube can only be regularly distributed so far, and I want the surface and volume meshes to be regular triangles and regular tetrahedra.Matlab only. What do I need to do?

I would appreciate some help :)

Answer 1

[Bruno Luong]

A cube can be devided in 6 tetrahedra

nx=4;
ny=3;
nz=2;

grid = arrayfun(@(n)linspace(0,n,n+1),[nx,ny,nz], 'unif', 0);
[grid{:}] = ndgrid(grid{:});
grid = cat(4,grid{:});
grid = reshape(grid,[],3);

Tcube = [...
     5, 1, 3, 2;
     7, 5, 3, 2;
     7, 6, 5, 2;
     7, 4, 6, 2;
     7, 3, 4, 2;
     7, 8, 6, 4];
[Tx, Ty, Tz]=ind2sub([2 2 2],Tcube);
Tx = Tx-1;
Ty = Ty-1;
Tz = Tz-1;
C = zeros([nx,ny,nz,size(Tcube)]);
for k=1:nz
    for j=1:ny
        for i=1:nx
            Cijk = sub2ind([nx,ny,nz]+1,i+Tx,j+Ty,k+Tz);
            C(i,j,k,:,:) = reshape(Cijk,[1,1,1,size(Tcube)]);
        end
    end
end
C = reshape(C, [], 4);
TR = triangulation(C,grid);

% Graphical plot
patch('Faces', TR.ConnectivityList, 'Vertices', TR.Points,'FaceColor', 'g', 'FaceAlpha', 0.5);
view(3);
axis equal

[Chenglin Li]

Thank you, but this is similar to the situation I generated. They are all right triangles. What I want is an equilateral triangle

[Bruno Luong]

Equilateral triangles has 60 degree at vertex. How do you make non overlapping covering the 90 degree at the corner of the cubes by such angle?

Same impossibility when reasoning with solid-angle of the regular tetrahedron and corner.

[Chenglin Li]

You don't need all triangles to be equilateral, something like this works best.

[Bruno Luong]

Then any meshing package would give more or less equilateral elements. Have you tried if pde toolbox have such mesh built function? (I don't own the pakage license)

[Chenglin Li]

I am currently looking at the PDE toolbox, but the generated mesh is generated according to the.STL file, I want to set the geometric boundary to generate the equilateral triangle mesh.

[Bruno Luong]

You always speak about "triangles" as if you want to mesh the surface and not the volume.

In 2D this FileExchange gives very doof result.

[Chenglin Li]

I have seen this 2D program and it works very well，but I still want to generate regular tetrahedral volume mesh in 3D, but the grid points in the volume are not clear how to generate.

Answer 2

[Shivam]

To triangulate a 3D point cloud you need the BallPivoting algorithm:

https://vgc.poly.edu/~csilva/papers/tvcg99.pdf

[Chenglin Li]

Thanks,is there a similar MATLAB program, I want to see how it is generated from the program.