ndgrid build matrices or N-D arrays by replicating some template vectors

## Syntax

                   [X, Y] = ndgrid(x)
                   [X, Y] = ndgrid(x,y)
                [X, Y, Z] = ndgrid(x,y,z)
             [X, Y, Z, T] = ndgrid(x,y,z,t)
        [X1, X2, ..., Xm] = ndgrid(x1,x2,...,xm)

## Arguments

x, y, z, ...

  vectors of any data types. They may have distinct data types. By default, `y=x`.

X, Y, Z, ...

  matrices in case of 2 input arguments, or hypermatrices otherwise. They all have the same sizes: `size(x,"*")` rows, `size(y,"*")` columns, `size(z,"*")` layers, etc. They have the datatypes of respective input vectors: `typeof(X)==typeof(x)`, `typeof(Y)==typeof(y)`, etc.

## Description

The first application of `ndgrid` is to build a grid of nodes meshing the 2D or 3D or N-D space according to 2, 3, or more sets `x`, `y`, etc.. of "template" coordinates sampled along each direction/dimension of the space that you want to mesh.

Hence, the matrix or hypermatrix `X` is made by replicating the vector `x` as all its columns ; the matrix or hypermatrix `Y` is made by replicating the vector `y` as all its rows ; `Z` is made of replicating the vector `z` along all its local thicknesses (3rd dimension); etc.

    --> [X, Y] = ndgrid([1 3 4], [0 2 4 6])
     X  =
       1.   1.   1.   1.
       3.   3.   3.   3.
       4.   4.   4.   4.

     Y  =
       0.   2.   4.   6.
       0.   2.   4.   6.
       0.   2.   4.   6.

Then, the coordinates of the node(i,j) in the 2D space will be simply `[x(i), y(j)]` now given by `[X(i,j), Y(i,j)]`. As well, the coordinates of a `node(i,j,k)` of a 3D grid will be `[x(i), y(j), z(k)]` now given by `[X(i,j,k), Y(i,j,k), Z(i,j,k)]`.

This replication scheme can be generalized to any number of dimensions, as well to any type of uniform data. Let's for instance consider 2 attributes:

1. The first is a number, to be chosen from the vector say `n= [ 3 7 ]`
2. The second is a letter, to be chosen from the vector say `c= ["a" "e" "i" "o" "u" "y"]`

Then we want to build the set of all {n,c} possible pairs. It will just be the 2D grid:

    --> [N, C] = ndgrid([3 7],["a" "e" "i" "o" "u" "y"])
     C  =
    !a  e  i  o  u  y  !
    !a  e  i  o  u  y  !

     N  =
       3.   3.   3.   3.   3.   3.
       7.   7.   7.   7.   7.   7.

Then, the object(i,j) will have the properties `{n(i) c(j)}` that now can be addressed with `{N(i,j) C(i,j)}`. This kind of grid may be useful to initialize an array of structures.

Following examples show how to use `X, Y, Z` in most frequent applications.

## Examples

Example #1:

    // Create a simple 2d grid
    x = linspace(-10,2,40);
    y = linspace(-5,5,40);
    [X,Y] = ndgrid(x,y);

    // Compute ordinates Z(X,Y) on the {X, Y} grid and plot Z(X,Y)
    Z = X - 3*X.*sin(X).*cos(Y-4) ;

    clf()
    plot3d(x,y,Z, flag=[color("green") 2 4], alpha=7, theta=60); show_window()

x = linspace(-10,2,40); y = linspace(-5,5,40); [X,Y] = ndgrid(x,y); Z = X - 3*X.*sin(X).*cos(Y-4) ; clf() plot3d(x,y,Z, flag=[color("green") 2 4], alpha=7, theta=60); show_window()

Example #2:

    // Create a simple 3d grid
    nx = 10; ny = 6; nz = 4;
    x = linspace(0,2,nx);
    y = linspace(0,1,ny);
    z = linspace(0,0.5,nz);
    [X,Y,Z] = ndgrid(x,y,z);

    // Try to display this 3d grid
    XF=[]; YF=[]; ZF=[];

    for k=1:nz
       [xf,yf,zf] = nf3d(X(:,:,k),Y(:,:,k),Z(:,:,k));
       XF = [XF xf]; YF = [YF yf]; ZF = [ZF zf];
    end

    for j=1:ny
       [xf,yf,zf] = nf3d(matrix(X(:,j,:),[nx,nz]),...
                         matrix(Y(:,j,:),[nx,nz]),...
                         matrix(Z(:,j,:),[nx,nz]));
       XF = [XF xf]; YF = [YF yf]; ZF = [ZF zf];
    end

    clf()
    plot3d(XF,YF,ZF, flag=[0 6 3], 66, 61,leg="X@Y@Z")
    xtitle("A 3d grid !"); show_window()

nx = 10; ny = 6; nz = 4; x = linspace(0,2,nx); y = linspace(0,1,ny); z = linspace(0,0.5,nz); [X,Y,Z] = ndgrid(x,y,z); XF=[]; YF=[]; ZF=[]; for k=1:nz [xf,yf,zf] = nf3d(X(:,:,k),Y(:,:,k),Z(:,:,k)); XF = [XF xf]; YF = [YF yf]; ZF = [ZF zf]; end for j=1:ny [xf,yf,zf] = nf3d(matrix(X(:,j,:),[nx,nz]),... matrix(Y(:,j,:),[nx,nz]),... matrix(Z(:,j,:),[nx,nz])); XF = [XF xf]; YF = [YF yf]; ZF = [ZF zf]; end plot3d(XF,YF,ZF, flag=[0 6 3], 66, 61, leg="X@Y@Z") xtitle("A 3d grid !");

Example #3: Creates a table of digrams:

    [c1, c2] = ndgrid(["a" "b" "c"], ["a" "b" "c" "d" "e" "f" "g" "h"])
    c1+c2

    --> [c1, c2] = ndgrid(["a" "b" "c"], ["a" "b" "c" "d" "e" "f" "g" "h"])
     c2  =
    !a  b  c  d  e  f  g  h  !
    !a  b  c  d  e  f  g  h  !
    !a  b  c  d  e  f  g  h  !

     c1  =
    !a  a  a  a  a  a  a  a  !
    !b  b  b  b  b  b  b  b  !
    !c  c  c  c  c  c  c  c  !

    --> c1+c2
     ans  =
    !aa  ab  ac  ad  ae  af  ag  ah  !
    !ba  bb  bc  bd  be  bf  bg  bh  !
    !ca  cb  cc  cd  ce  cf  cg  ch  !

## See also

- meshgrid
- kron
- feval
- nf3d

## History

6.0.0 Extension to all homogeneous datatypes ([], booleans, encoded integers, polynomials, rationals, strings). Revision of the help page.
6.0.1 `ndgrid(x)` is now accepted: Default `y=x`.