NAG Library Routine Document

D06DAF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

1 Purpose

D06DAF is a utility which performs an affine transformation of a given mesh.

2 Specification

SUBROUTINE D06DAF (

NV, NEDGE, NELT, NTRANS, ITYPE, TRANS, COORI, EDGEI, CONNI, COORO, EDGEO, CONNO, ITRACE, RWORK, LRWORK, IFAIL)

INTEGER	NV, NEDGE, NELT, NTRANS, ITYPE(NTRANS), EDGEI(3,NEDGE), CONNI(3,NELT), EDGEO(3,NEDGE), CONNO(3,NELT), ITRACE, LRWORK, IFAIL
REAL (KIND=nag_wp)	TRANS(6,NTRANS), COORI(2,NV), COORO(2,NV), RWORK(LRWORK)

3 Description

D06DAF generates a mesh (coordinates, triangle/vertex connectivities and edge/vertex connectivities) resulting from an affine transformation of a given mesh. This transformation is of the form

Y = A \times X + B

, where

$Y$ , $X$ and $B$ are in $ℝ^{2}$ , and
$A$ is a real $2$ by $2$ matrix.

Such a transformation includes a translation, a rotation, a scale reduction or increase, a symmetric transformation with respect to a user-supplied line, a user-supplied analytic transformation, or a composition of several transformations.

This routine is partly derived from material in the MODULEF package from INRIA (Institut National de Recherche en Informatique et Automatique).

4 References

None.

5 Parameters

1: $NV$ – INTEGERInput

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

Constraint:

NV \geq 3

2: $NEDGE$ – INTEGERInput

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

Constraint:

NEDGE \geq 1

3: $NELT$ – INTEGERInput

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

Constraint:

NELT \leq 2 \times NV - 1

4: $NTRANS$ – INTEGERInput

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

Constraint:

NTRANS \geq 1

5: $ITYPE (NTRANS)$ – INTEGER arrayInput

On entry:

ITYPE (i)

, for

i = 1, 2, \dots, NTRANS

, indicates the type of each transformation as follows:

$ITYPE (i) = 0$: Identity transformation.
$ITYPE (i) = 1$: Translation.
$ITYPE (i) = 2$: Symmetric transformation with respect to a user-supplied line.
$ITYPE (i) = 3$: Rotation.
$ITYPE (i) = 4$: Scaling.
$ITYPE (i) = 10$: User-supplied analytic transformation.

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

Constraint:

ITYPE (i) = 0

1

2

3

4

10

, for

i = 1, 2, \dots, NTRANS

6: $TRANS (6, NTRANS)$ – REAL (KIND=nag_wp) arrayInput

On entry: the parameters for each transformation. For

i = 1, 2, \dots, NTRANS

TRANS (1, i)

TRANS (6, i)

contain the parameters of the

i

th transformation.

ITYPE (i) = 0

, elements

TRANS (1, i)

TRANS (6, i)

are not referenced.

ITYPE (i) = 1

, the translation vector is

\vec{u} = (\begin{matrix} a \\ b \end{matrix})

, where

a = TRANS (1, i)

and

b = TRANS (2, i)

, while elements

TRANS (3, i)

TRANS (6, i)

are not referenced.

ITYPE (i) = 2

, the user-supplied line is the curve {

(x, y) \in ℝ^{2}

; such that

a x + b y + c = 0

}, where

a = TRANS (1, i)

b = TRANS (2, i)

and

c = TRANS (3, i)

, while elements

TRANS (4, i)

TRANS (6, i)

are not referenced.

ITYPE (i) = 3

, the centre of the rotation is

(x_{0}, y_{0})

where

x_{0} = TRANS (1, i)

and

y_{0} = TRANS (2, i)

θ = TRANS (3, i)

is its angle in degrees, while elements

TRANS (4, i)

TRANS (6, i)

are not referenced.

ITYPE (i) = 4

a = TRANS (1, i)

is the scaling coefficient in the

x

-direction,

b = TRANS (2, i)

is the scaling coefficient in the

y

-direction, and

(x_{0}, y_{0})

are the scaling centre coordinates, with

x_{0} = TRANS (3, i)

and

y_{0} = TRANS (4, i)

; while elements

TRANS (5, i)

TRANS (6, i)

are not referenced.

ITYPE (i) = 10

, the user-supplied analytic affine transformation

Y = A \times X + B

is such that

A = {(a_{k l})}_{1 \leq k, l \leq 2}

and

B = {(b_{k})}_{1 \leq k \leq 2}

where

a_{k l} = TRANS (2 \times (k - 1) + l, i)

, and

b_{k} = TRANS (4 + k, i)

with

k, l = 1, 2

7: $COORI (2, NV)$ – REAL (KIND=nag_wp) arrayInput/Output

On entry:

COORI (1, i)

contains the

x

coordinate of the

i

th vertex of the input mesh, for

i = 1, 2, \dots, NV

; while

COORI (2, i)

contains the corresponding

y

coordinate.

On exit: see Section 9.

8: $EDGEI (3, NEDGE)$ – INTEGER arrayInput/Output

On entry: the specification of the boundary or interface edges.

EDGEI (1, j)

and

EDGEI (2, j)

contain the vertex numbers of the two end points of the

j

th boundary edge.

EDGEI (3, j)

is a user-supplied tag for the

j

th boundary edge.

Constraint:

1 \leq EDGEI (i, j) \leq NV

and

EDGEI (1, j) \neq EDGEI (2, j)

, for

i = 1, 2

and

j = 1, 2, \dots, NEDGE

On exit: see Section 9.

9: $CONNI (3, NELT)$ – INTEGER arrayInput/Output

On entry: the connectivity of the input mesh between triangles and vertices. For each triangle

j

CONNI (i, j)

gives the indices of its three vertices (in anticlockwise order), for

i = 1, 2, 3

and

j = 1, 2, \dots, NELT

Constraints:

$1 \leq CONNI (i, j) \leq NV$ ;
$CONNI (1, j) \neq CONNI (2, j)$ ;
$CONNI (1, j) \neq CONNI (3, j)$ and $CONNI (2, j) \neq CONNI (3, j)$ , for $i = 1, 2, 3$ and $j = 1, 2, \dots, NELT$ .

On exit: see Section 9.

10: $COORO (2, NV)$ – REAL (KIND=nag_wp) arrayOutput

On exit:

COORO (1, i)

will contain the

x

coordinate of the

i

th vertex of the transformed mesh, for

i = 1, 2, \dots, NV

; while

COORO (2, i)

will contain the corresponding

y

coordinate.

11: $EDGEO (3, NEDGE)$ – 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

EDGEO (i, j) = EDGEI (i, j)

, for

i = 1, 2, 3

and

j = 1, 2, \dots, NEDGE

; otherwise

EDGEO (1, j) = EDGEI (2, j)

EDGEO (2, j) = EDGEI (1, j)

and

EDGEO (3, j) = EDGEI (3, j)

, for

j = 1, 2, \dots, NEDGE

12: $CONNO (3, NELT)$ – 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

CONNO (i, j) = CONNI (i, j)

, for

i = 1, 2, 3

and

j = 1, 2, \dots, NELT

; otherwise

CONNO (1, j) = CONNI (1, j)

CONNO (2, j) = CONNI (3, j)

and

CONNO (3, j) = CONNI (2, j)

, for

j = 1, 2, \dots, NELT

13: $ITRACE$ – INTEGERInput

On entry: the level of trace information required from D06DAF.

$ITRACE \leq 0$: No output is generated.
$ITRACE \geq 1$: 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: $RWORK (LRWORK)$ – REAL (KIND=nag_wp) arrayWorkspace

15: $LRWORK$ – INTEGERInput

On entry: the dimension of the array RWORK as declared in the (sub)program from which D06DAF is called.

Constraint:

LRWORK \geq 12 \times NTRANS

16: $IFAIL$ – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ 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 parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,	$NV < 3$ ;
or	$NELT > 2 \times NV - 1$ ;
or	$NEDGE < 1$ ;
or	$EDGEI (i, j) < 1$ or $EDGEI (i, j) > NV$ for some $i = 1, 2$ and $j = 1, 2, \dots, NEDGE$ ;
or	$EDGEI (1, j) = EDGEI (2, j)$ for some $j = 1, 2, \dots, NEDGE$ ;
or	$CONNI (i, j) < 1$ or $CONNI (i, j) > NV$ for some $i = 1, 2, 3$ and $j = 1, 2, \dots, NELT$ ;
or	$CONNI (1, j) = CONNI (2, j)$ or $CONNI (1, j) = CONNI (3, j)$ or $CONNI (2, j) = CONNI (3, j)$ for some $j = 1, 2, \dots, NELT$ ;
or	$NTRANS < 1$ ;
or	$ITYPE (i) \neq 0$ , $1$ , $2$ , $3$ , $4$ or $10$ for some $i = 1, 2, \dots, NTRANS$ ;
or	$LRWORK < 12 \times NTRANS$ .

$IFAIL = 2$: 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.

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

D06DAF is not threaded by NAG in any implementation.

D06DAF makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

You may not wish to save the input mesh (COORI, EDGEI and CONNI) and could call D06DAF using the same parameters for the input and the output (transformed) mesh.