naginterfaces.library.mesh.dim2_​transform_​affine

naginterfaces.library.mesh.dim2_transform_affine(mode, itype, trans, coori, edgei, conni, itrace, io_manager=None)

dim2_transform_affine is a utility which performs an affine transformation of a given mesh.

For full information please refer to the NAG Library document for d06da

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/d06/d06daf.html

Parameters

modeint

If $mode=1$, the arguments $coori$, $edgei$ and $conni$ are overwritten on exit by the output values described in $cooro$, $edgeo$ and $conno$ respectively. In this case $cooro$, $edgeo$ and $conno$ are not referenced, and you can save storage space.

If $mode\ne 1$, no such aliasing is assumed.

itypeint, array-like, shape $\left(\text{ntrans}\right)$

$itype[\text{i}-1]$, for $\text{i}=1,2,\dots ,\text{ntrans}$, indicates the type of each transformation as follows:

$itype[i-1]=0$

Identity transformation.

$itype[i-1]=1$

Translation.

$itype[i-1]=2$

Symmetric transformation with respect to a user-supplied line.

$itype[i-1]=3$

Rotation.

$itype[i-1]=4$

Scaling.

$itype[i-1]=10$

User-supplied analytic transformation.

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

transfloat, array-like, shape $(6,\text{ntrans})$

The arguments for each transformation. For $i=1,2,\dots ,\text{ntrans}$, $trans[0,i-1]$ to $trans[5,i-1]$ contain the arguments of the $i$th transformation.

If $itype[i-1]=0$, elements $trans[0,i-1]$ to $trans[5,i-1]$ are not referenced.

If $itype[i-1]=1$, the translation vector is $\overrightarrow{u}=\left(\begin{array}{c}a\\ b\end{array}\right)$, where $a=trans[0,i-1]$ and $b=trans[1,i-1]$, while elements $trans[2,i-1]$ to $trans[5,i-1]$ are not referenced.

If $itype[i-1]=2$, the user-supplied line is the curve {$(x,y)\in R^2$; such that $ax+by+c=0$}, where $a=trans[0,i-1]$, $b=trans[1,i-1]$ and $c=trans[2,i-1]$, while elements $trans[3,i-1]$ to $trans[5,i-1]$ are not referenced.

If $itype[i-1]=3$, the centre of the rotation is $(x_0,y_0)$ where $x_0=trans[0,i-1]$ and $y_0=trans[1,i-1]$, $\theta =trans[2,i-1]$ is its angle in degrees, while elements $trans[3,i-1]$ to $trans[5,i-1]$ are not referenced.

If $itype[i-1]=4$, $a=trans[0,i-1]$ is the scaling coefficient in the $x$-direction, $b=trans[1,i-1]$ is the scaling coefficient in the $y$-direction, and $(x_0,y_0)$ are the scaling centre coordinates, with $x_0=trans[2,i-1]$ and $y_0=trans[3,i-1]$; while elements $trans[4,i-1]$ to $trans[5,i-1]$ are not referenced.

If $itype[i-1]=10$, 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$a_{kl}=trans[2\times (k-1)+l-1,i-1]$, and $b_k=trans[4+k-1,i-1]$ with $k,l=1,2$.

coorifloat, array-like, shape $(2,\text{nv})$

$coori[0,\text{i}-1]$ contains the $x$ coordinate of the $\text{i}$th vertex of the input mesh, for $\text{i}=1,2,\dots ,\text{nv}$; while $coori[1,i-1]$ contains the corresponding $y$ coordinate.

edgeiint, array-like, shape $(3,\text{nedge})$

The specification of the boundary or interface edges. $edgei[0,j-1]$ and $edgei[1,j-1]$ contain the vertex numbers of the two end points of the $j$th boundary edge. $edgei[2,j-1]$ is a user-supplied tag for the $j$th boundary edge.

conniint, array-like, shape $(3,\text{nelt})$

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

itraceint

The level of trace information required from dim2_transform_affine.

$itrace\le 0$

No output is generated.

$itrace\ge 1$

Details of each transformation, the matrix $A$ and the vector $B$ of the final transformation, which is the composition of all the $\text{ntrans}$ transformations, are printed.

io_managerFileObjManager, optional

Manager for I/O in this routine.

Returns

coorifloat, ndarray, shape $(2,\text{nv})$

See Further Comments.

edgeiint, ndarray, shape $(3,\text{nedge})$

If $mode=1$, $edgei$ holds the output values described in $edgeo$.

conniint, ndarray, shape $(3,\text{nelt})$

If $mode=1$, $conni$ holds the output values described in $conno$.

cooroNone or float, ndarray, shape $(:,:)$

$cooro[0,\text{i}-1]$ will contain the $x$ coordinate of the $\text{i}$th vertex of the transformed mesh, for $\text{i}=1,2,\dots ,\text{nv}$; while $cooro[1,i-1]$ will contain the corresponding $y$ coordinate.

edgeoNone or int, ndarray, shape $(:,:)$

The specification of the boundary or interface edges of the transformed mesh. If the number of symmetric transformations is even or zero then $edgeo[\text{i}-1,\text{j}-1]=edgei[\text{i}-1,\text{j}-1]$, for $\text{j}=1,2,\dots ,\text{nedge}$, for $\text{i}=1,2,\dots ,3$; otherwise $edgeo[0,\text{j}-1]=edgei[1,\text{j}-1]$,$edgeo[1,\text{j}-1]=edgei[0,\text{j}-1]$ and $edgeo[2,\text{j}-1]=edgei[2,\text{j}-1]$, for $\text{j}=1,2,\dots ,\text{nedge}$.

connoNone or int, ndarray, shape $(:,:)$

The connectivity of the transformed mesh between triangles and vertices. If the number of symmetric transformations is even or zero then $conno[\text{i}-1,\text{j}-1]=conni[\text{i}-1,\text{j}-1]$, for $\text{j}=1,2,\dots ,\text{nelt}$, for $\text{i}=1,2,\dots ,3$; otherwise $conno[0,\text{j}-1]=conni[0,\text{j}-1]$, $conno[1,\text{j}-1]=conni[2,\text{j}-1]$ and $conno[2,\text{j}-1]=conni[1,\text{j}-1]$, for $\text{j}=1,2,\dots ,\text{nelt}$.

Raises

NagValueError

(errno $1$)

On entry, $\text{nv}=⟨value⟩$.

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

(errno $1$)

On entry, $itype[\text{I}-1]=⟨value⟩$ and $\text{I}=⟨value⟩$.

Constraint: $itype[\text{I}-1]=0$, $1$, $2$, $3$, $4$ or $10$.

(errno $1$)

On entry, $\text{nedge}=⟨value⟩$.

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

(errno $1$)

On entry, $\text{nelt}=⟨value⟩$ and $\text{nv}=⟨value⟩$.

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

(errno $1$)

On entry, $edgei(\text{I},\text{J})=⟨value⟩$, $\text{I}=⟨value⟩$, $\text{J}=⟨value⟩$ and $\text{nv}=⟨value⟩$.

Constraint: $edgei(\text{I},\text{J})\ge 1$ and $edgei(\text{I},\text{J})\le \text{nv}$.

(errno $1$)

On entry, $conni(\text{I},\text{J})=⟨value⟩$, $\text{I}=⟨value⟩$, $\text{J}=⟨value⟩$ and $\text{nv}=⟨value⟩$.

Constraint: $conni(\text{I},\text{J})\ge 1$ and $conni(\text{I},\text{J})\le \text{nv}$.

(errno $1$)

On entry, $\text{ntrans}=⟨value⟩$.

Constraint: $\text{ntrans}>0$.

(errno $1$)

On entry, the end points of the edge $\text{J}$ have the same index $\text{I}$: $\text{J}=⟨value⟩$ and $\text{I}=⟨value⟩$.

(errno $1$)

On entry, vertices $2$ and $3$ of the triangle $\text{K}$ have the same index $\text{I}$: $\text{K}=⟨value⟩$ and $\text{I}=⟨value⟩$.

(errno $1$)

On entry, vertices $1$ and $3$ of the triangle $\text{K}$ have the same index $\text{I}$: $\text{K}=⟨value⟩$ and $\text{I}=⟨value⟩$.

(errno $1$)

On entry, vertices $1$ and $2$ of the triangle $\text{K}$ have the same index $\text{I}$: $\text{K}=⟨value⟩$ and $\text{I}=⟨value⟩$.

(errno $2$)

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

Notes

dim2_transform_affine 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 $2\times 2$ 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 function is partly derived from material in the MODULEF package from INRIA (Institut National de Recherche en Informatique et Automatique).