NAG Toolbox: nag_mesh_2d_transform_affine (d06da)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{itype}\left(\mathbf{ntrans}\right)$ – int64int32nag_int array

$\mathbf{itype}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{ntrans}$, indicates the type of each transformation as follows:

$\mathbf{itype}\left(i\right)=0$

Identity transformation.

$\mathbf{itype}\left(i\right)=1$

Translation.

$\mathbf{itype}\left(i\right)=2$

Symmetric transformation with respect to a user-supplied line.

$\mathbf{itype}\left(i\right)=3$

Rotation.

$\mathbf{itype}\left(i\right)=4$

Scaling.

$\mathbf{itype}\left(i\right)=10$

User-supplied analytic transformation.

Note that the transformations are applied in the order described in itype.

Constraint: $\mathbf{itype}\left(\mathit{i}\right)=0$, $1$, $2$, $3$, $4$ or $10$, for $\mathit{i}=1,2,\dots ,\mathbf{ntrans}$.

2:     $\mathrm{trans}\left(6,\mathbf{ntrans}\right)$ – double array

The arguments for each transformation. For $i=1,2,\dots ,\mathbf{ntrans}$, $\mathbf{trans}\left(1,i\right)$ to $\mathbf{trans}\left(6,i\right)$ contain the arguments of the $i$th transformation.

If $\mathbf{itype}\left(i\right)=0$, elements $\mathbf{trans}\left(1,i\right)$ to $\mathbf{trans}\left(6,i\right)$ are not referenced.

If $\mathbf{itype}\left(i\right)=1$, the translation vector is $\overrightarrow{u}=\left(\begin{array}{c}a\\ b\end{array}\right)$, where $a=\mathbf{trans}\left(1,i\right)$ and $b=\mathbf{trans}\left(2,i\right)$, while elements $\mathbf{trans}\left(3,i\right)$ to $\mathbf{trans}\left(6,i\right)$ are not referenced.

If $\mathbf{itype}\left(i\right)=2$, the user-supplied line is the curve {$\left(x,y\right)\in {ℝ}^2$; such that $ax+by+c=0$}, where $a=\mathbf{trans}\left(1,i\right)$, $b=\mathbf{trans}\left(2,i\right)$ and $c=\mathbf{trans}\left(3,i\right)$, while elements $\mathbf{trans}\left(4,i\right)$ to $\mathbf{trans}\left(6,i\right)$ are not referenced.

If $\mathbf{itype}\left(i\right)=3$, the centre of the rotation is $\left(x_0,y_0\right)$ where $x_0=\mathbf{trans}\left(1,i\right)$ and $y_0=\mathbf{trans}\left(2,i\right)$, $\theta =\mathbf{trans}\left(3,i\right)$ is its angle in degrees, while elements $\mathbf{trans}\left(4,i\right)$ to $\mathbf{trans}\left(6,i\right)$ are not referenced.

If $\mathbf{itype}\left(i\right)=4$, $a=\mathbf{trans}\left(1,i\right)$ is the scaling coefficient in the $x$-direction, $b=\mathbf{trans}\left(2,i\right)$ is the scaling coefficient in the $y$-direction, and $\left(x_0,y_0\right)$ are the scaling centre coordinates, with $x_0=\mathbf{trans}\left(3,i\right)$ and $y_0=\mathbf{trans}\left(4,i\right)$; while elements $\mathbf{trans}\left(5,i\right)$ to $\mathbf{trans}\left(6,i\right)$ are not referenced.

If $\mathbf{itype}\left(i\right)=10$, the user-supplied analytic affine transformation $Y=A\times X+B$ is such that $A={\left(a_{kl}\right)}_{1\le k,l\le 2}$ and $B={\left(b_k\right)}_{1\le k\le 2}$ where$a_{kl}=\mathbf{trans}\left(2\times \left(k-1\right)+l,i\right)$, and $b_k=\mathbf{trans}\left(4+k,i\right)$ with $k,l=1,2$.

3:     $\mathrm{coori}\left(2,\mathbf{nv}\right)$ – double array

$\mathbf{coori}\left(1,\mathit{i}\right)$ contains the $x$ coordinate of the $\mathit{i}$th vertex of the input mesh, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$; while $\mathbf{coori}\left(2,i\right)$ contains the corresponding $y$ coordinate.

4:     $\mathrm{edgei}\left(3,\mathbf{nedge}\right)$ – int64int32nag_int array

The specification of the boundary or interface edges. $\mathbf{edgei}\left(1,j\right)$ and $\mathbf{edgei}\left(2,j\right)$ contain the vertex numbers of the two end points of the $j$th boundary edge. $\mathbf{edgei}\left(3,j\right)$ is a user-supplied tag for the $j$th boundary edge.

Constraint: $1\le \mathbf{edgei}\left(\mathit{i},\mathit{j}\right)\le \mathbf{nv}$ and $\mathbf{edgei}\left(1,\mathit{j}\right)\ne \mathbf{edgei}\left(2,\mathit{j}\right)$, for $\mathit{i}=1,2$ and $\mathit{j}=1,2,\dots ,\mathbf{nedge}$.

5:     $\mathrm{conni}\left(3,\mathbf{nelt}\right)$ – int64int32nag_int array

The connectivity of the input mesh between triangles and vertices. For each triangle $\mathit{j}$, $\mathbf{conni}\left(\mathit{i},\mathit{j}\right)$ gives the indices of its three vertices (in anticlockwise order), for $\mathit{i}=1,2,3$ and $\mathit{j}=1,2,\dots ,\mathbf{nelt}$.

Constraints:

• $1\le \mathbf{conni}\left(i,j\right)\le \mathbf{nv}$;
• $\mathbf{conni}\left(1,j\right)\ne \mathbf{conni}\left(2,j\right)$;
• $\mathbf{conni}\left(1,\mathit{j}\right)\ne \mathbf{conni}\left(3,\mathit{j}\right)$ and $\mathbf{conni}\left(2,\mathit{j}\right)\ne \mathbf{conni}\left(3,\mathit{j}\right)$, for $\mathit{i}=1,2,3$ and $\mathit{j}=1,2,\dots ,\mathbf{nelt}$.

6:     $\mathrm{itrace}$ – int64int32nag_int scalar

The level of trace information required from nag_mesh_2d_transform_affine (d06da).

$\mathbf{itrace}\le 0$

No output is generated.

$\mathbf{itrace}\ge 1$

Details of each transformation, the matrix $A$ and the vector $B$ of the final transformation, which is the composition of all the ntrans transformations, are printed on the current advisory message unit (see nag_file_set_unit_advisory (x04ab)).

Optional Input Parameters

1:     $\mathrm{nv}$ – int64int32nag_int scalar

Default: the dimension of the array coori.

The total number of vertices in the input mesh.

Constraint: $\mathbf{nv}\ge 3$.

2:     $\mathrm{nedge}$ – int64int32nag_int scalar

Default: the dimension of the array edgei.

The number of the boundary or interface edges in the input mesh.

Constraint: $\mathbf{nedge}\ge 1$.

3:     $\mathrm{nelt}$ – int64int32nag_int scalar

Default: the dimension of the array conni.

The number of triangles in the input mesh.

Constraint: $\mathbf{nelt}\le 2\times \mathbf{nv}-1$.

4:     $\mathrm{ntrans}$ – int64int32nag_int scalar

Default: the dimension of the arrays itype, trans. (An error is raised if these dimensions are not equal.)

The number of transformations of the input mesh.

Constraint: $\mathbf{ntrans}\ge 1$.

Output Parameters

1:     $\mathrm{coori}\left(2,\mathbf{nv}\right)$ – double array

see Further Comments.

2:     $\mathrm{edgei}\left(3,\mathbf{nedge}\right)$ – int64int32nag_int array

See Further Comments.

3:     $\mathrm{conni}\left(3,\mathbf{nelt}\right)$ – int64int32nag_int array

See Further Comments.

4:     $\mathrm{cooro}\left(2,\mathbf{nv}\right)$ – double array

$\mathbf{cooro}\left(1,\mathit{i}\right)$ will contain the $x$ coordinate of the $\mathit{i}$th vertex of the transformed mesh, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$; while $\mathbf{cooro}\left(2,i\right)$ will contain the corresponding $y$ coordinate.

5:     $\mathrm{edgeo}\left(3,\mathbf{nedge}\right)$ – int64int32nag_int array

The specification of the boundary or interface edges of the transformed mesh. If the number of symmetric transformations is even or zero then$\mathbf{edgeo}\left(\mathit{i},\mathit{j}\right)=\mathbf{edgei}\left(\mathit{i},\mathit{j}\right)$, for $\mathit{i}=1,2,3$ and $\mathit{j}=1,2,\dots ,\mathbf{nedge}$; otherwise $\mathbf{edgeo}\left(1,\mathit{j}\right)=\mathbf{edgei}\left(2,\mathit{j}\right)$,$\mathbf{edgeo}\left(2,\mathit{j}\right)=\mathbf{edgei}\left(1,\mathit{j}\right)$ and $\mathbf{edgeo}\left(3,\mathit{j}\right)=\mathbf{edgei}\left(3,\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,\mathbf{nedge}$.

6:     $\mathrm{conno}\left(3,\mathbf{nelt}\right)$ – int64int32nag_int array

The connectivity of the transformed mesh between triangles and vertices. If the number of symmetric transformations is even or zero then$\mathbf{conno}\left(\mathit{i},\mathit{j}\right)=\mathbf{conni}\left(\mathit{i},\mathit{j}\right)$, for $\mathit{i}=1,2,3$ and $\mathit{j}=1,2,\dots ,\mathbf{nelt}$; otherwise $\mathbf{conno}\left(1,\mathit{j}\right)=\mathbf{conni}\left(1,\mathit{j}\right)$, $\mathbf{conno}\left(2,\mathit{j}\right)=\mathbf{conni}\left(3,\mathit{j}\right)$ and $\mathbf{conno}\left(3,\mathit{j}\right)=\mathbf{conni}\left(2,\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,\mathbf{nelt}$.

7:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   $\mathbf{ifail}=1$

On entry,	$\mathbf{nv}<3$;
or	$\mathbf{nelt}>2\times \mathbf{nv}-1$;
or	$\mathbf{nedge}<1$;
or	$\mathbf{edgei}\left(i,j\right)<1$ or $\mathbf{edgei}\left(i,j\right)>\mathbf{nv}$ for some $i=1,2$ and $j=1,2,\dots ,\mathbf{nedge}$;
or	$\mathbf{edgei}\left(1,j\right)=\mathbf{edgei}\left(2,j\right)$ for some $j=1,2,\dots ,\mathbf{nedge}$;
or	$\mathbf{conni}\left(i,j\right)<1$ or $\mathbf{conni}\left(i,j\right)>\mathbf{nv}$ for some $i=1,2,3$ and $j=1,2,\dots ,\mathbf{nelt}$;
or	$\mathbf{conni}\left(1,j\right)=\mathbf{conni}\left(2,j\right)$ or $\mathbf{conni}\left(1,j\right)=\mathbf{conni}\left(3,j\right)$ or $\mathbf{conni}\left(2,j\right)=\mathbf{conni}\left(3,j\right)$ for some $j=1,2,\dots ,\mathbf{nelt}$;
or	$\mathbf{ntrans}<1$;
or	$\mathbf{itype}\left(i\right)\ne 0$, $1$, $2$, $3$, $4$ or $10$ for some $i=1,2,\dots ,\mathbf{ntrans}$;
or	$\mathit{lrwork}<12\times \mathbf{ntrans}$.

   $\mathbf{ifail}=2$

A serious error has occurred in an internal call to an auxiliary function. Check the input mesh especially the triangles/vertices and the edges/vertices connectivities as well as the details of each transformations.

   $\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

   $\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: d06da_example

function d06da_example


fprintf('d06da example results\n\n');

coor1 = zeros(2,400);
for j = 1:20
  for i = 1:20
    coor1(1,(j-1)*20+i) = (i-1)/19;
    coor1(2,(j-1)*20+i) = (j-1)/19;
  end
end
edge1 = ones(3, 76, 'int64');
edge1(1, 1:76) = int64([1:19,20*(1:19),401-(1:19),401-20*(1:19)]);
edge1(2, 1:76) = int64([2:19,20*(1:19),401-(1:19),401-20*(1:19),1]);
conn1 = zeros(3, 722, 'int64');
ind = -1;
for i=1:379
  if (rem(i, 20) ~= 0)
    ind = ind+2;
    conn1(1, ind)   = int64(i);
    conn1(1, ind+1) = int64(i);
    conn1(2, ind)   = int64(i+1);
    conn1(2, ind+1) = int64(i+21);
    conn1(3, ind)   = int64(i+21);
    conn1(3, ind+1) = int64(i+20);
  end
end
reft1 = ones(722, 1, 'int64');
reft2 = reft1;
reft2(:) = int64(2);
itype = [int64(1)];
itrace = int64(1);

% Transform the first domain to obtain an overlapping second domain
trans = [15/19; 17/19; 0; 0; 0; 0];
[coor1, edge1, conn1, coor2, edge2, conn2, ifail] = ...
d06da( ...
       itype, trans, coor1, edge1, conn1, itrace);

% Restitch the meshes
[nv3, nelt3, nedge3, coor3, edge3, conn3, reft3, ifail] = ...
d06db( ...
       coor1, edge1, conn1, reft1, coor2, edge2, conn2, reft2, itrace);

% Plot the result
fig1 = figure;
triplot(transpose(double(conn3(:,1:nelt3))), coor3(1,:), coor3(2,:));
title ('Interior mesh of the two overlapping squares geometry');

% Now consider a partitioned second domain
trans = [1; 0; 0; 0; 0; 0];
[coor1, edge1, conn1, coor2, edge2, conn2, ifail] = ...
d06da( ...
       itype, trans, coor1, edge1, conn1, itrace);

% Restitch the meshes
[nv3, nelt3, nedge3, coor3, edge3, conn3, reft3, ifail] = ...
d06db( ...
       coor1, edge1, conn1, reft1, coor2, edge2, conn2, reft2, itrace);


% Plot the result
fig2 = figure;
triplot(transpose(double(conn3(:,1:nelt3))), coor3(1,:), coor3(2,:));
title ('Interior mesh of the two partitioned squares geometry');

d06da example results


 Transformation   1: translation
  translation vector:  0.7895      0.8947

 Final transformation matrix y = A*x + b:
    1.000       0.000         0.7895
    0.000       1.000         0.8947

 Transformation   1: translation
  translation vector:   1.000       0.000

 Final transformation matrix y = A*x + b:
    1.000       0.000          1.000
    0.000       1.000          0.000