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# ndgrid

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

### Syntax

```[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.

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()```

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()```

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  !
```