NAG Library Routine Document

d06daf  (dim2_transform_affine)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{nv}$ – IntegerInput

On entry: the total number of vertices in the input mesh.

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

2:     $\mathbf{nedge}$ – IntegerInput

On entry: the number of the boundary or interface edges in the input mesh.

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

3:     $\mathbf{nelt}$ – IntegerInput

On entry: the number of triangles in the input mesh.

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

4:     $\mathbf{ntrans}$ – IntegerInput

On entry: the number of transformations of the input mesh.

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

5:     $\mathbf{itype}\left(\mathbf{ntrans}\right)$ – Integer arrayInput

On entry: $\mathbf{itype}\left(\mathit{i}\right)$, 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 $\mathit{i}=1,2,\dots ,\mathbf{ntrans}$.

6:     $\mathbf{trans}\left(6,\mathbf{ntrans}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: 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 $k,l=1,2$.

7:     $\mathbf{coori}\left(2,\mathbf{nv}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: $\mathbf{coori}\left(1,\mathit{i}\right)$ 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.

On exit: see Section 9.

8:     $\mathbf{edgei}\left(3,\mathbf{nedge}\right)$ – Integer arrayInput/Output

On entry: 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 $\mathit{j}=1,2,\dots ,\mathbf{nedge}$.

On exit: see Section 9.

9:     $\mathbf{conni}\left(3,\mathbf{nelt}\right)$ – Integer arrayInput/Output

On entry: 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 $\mathit{j}=1,2,\dots ,\mathbf{nelt}$.

On exit: see Section 9.

10:   $\mathbf{cooro}\left(2,\mathbf{nv}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{cooro}\left(1,\mathit{i}\right)$ 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.

11:   $\mathbf{edgeo}\left(3,\mathbf{nedge}\right)$ – Integer arrayOutput

On exit: 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 $\mathit{j}=1,2,\dots ,\mathbf{nedge}$.

12:   $\mathbf{conno}\left(3,\mathbf{nelt}\right)$ – Integer arrayOutput

On exit: 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 $\mathit{j}=1,2,\dots ,\mathbf{nelt}$.

13:   $\mathbf{itrace}$ – IntegerInput

On entry: the level of trace information required from d06daf.

$\mathbf{itrace}\le 0$

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 x04abf).

14:   $\mathbf{rwork}\left(\mathbf{lrwork}\right)$ – Real (Kind=nag_wp) arrayWorkspace

15:   $\mathbf{lrwork}$ – IntegerInput

On entry: the dimension of the array rwork as declared in the (sub)program from which d06daf is called.

Constraint: $\mathbf{lrwork}\ge 12\times \mathbf{ntrans}$.

16:   $\mathbf{ifail}$ – IntegerInput/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of ifail on exit.

On exit: $\mathbf{ifail}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6

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	$\mathbf{lrwork}<12\times \mathbf{ntrans}$.

$\mathbf{ifail}=2$

A serious error has occurred in an internal call to an auxiliary routine. 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.

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

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

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example