Matrix Functions

Matrix functions work with sequences of length 12 that define the APT standard rotation and translation components of a matrix. For example:

$\begin{array}{llll} a_1, & a_2, & a_3, & a_4, \\ a_5, & a_6, & a_7, & a_8, \\ a_9, & a_{10}, & a_{11}, & a_{12} \end{array}$

Matrix functions can also work with square matrices of any dimension, defined as a series of numbers that represent a matrix organized in row-column order. For example:

$\begin{array}{lll} a_1, & a_2, & a_3 \\ a_4, & a_5, & a_6 \\ a_7, & a_8, & a_9 \end{array}$

Matrix functions that return a matrix, vector or point sequence do so by returning a sequence with the same length as that of the input matrix, vector or point sequence.

Matrices can be constructed using the $FSEQ function or the { } sequence operators. For example, the following are equivalent:

M1=$FSEQ(1,0,0,0,0,1,0,0,0,0,1,0)
M1={1,0,0,0,0,1,0,0,0,0,1,0}

It is important to note that matrices are sequences and not arrays. Matrix functions do not work with arrays.

The $FMX Function

$\textbf{{\char36}FMX}\textbf{(}\icamhsp{0.707mm}\mathtt{<\!matrix\!>}\icamhsp{0.707mm}\textbf{)}$

Matrix creation

Returns: Sequence

The $FMX function returns a 12 element matrix sequence as defined by the standard <matrix> post-processor command syntax listed below:

$\bigl[\,\textbf{INVERS},\bigr]\begin{pmatrix}\!\begin{array}{l}\Bigl[\,\mathit{x_{1}},\mathit{y_{1}}\,\bigl[,\mathit{z_{1}}\,\bigr]\,\Bigr]\,\bigl[\,\mathtt{<\!rotation\!>}\,\bigr]\\\;\\\textbf{MSYS}\,\bigl[,\textbf{LAST}\,\bigr]\\\;\\\mathit{a_{1}},\mathit{a_{2}},\mathit{a_{3}},\mathit{a_{4}},\mathit{a_{5}},\mathit{a_{6}},\mathit{a_{7}},\mathit{a_{8}},\mathit{a_{9}},\mathit{a_{10}},\mathit{a_{11}},\mathit{a_{12}}\end{array}\!\end{pmatrix}$

where <rotation> is defined as:

$\raise0.10ex\hbox{$\icamrange{2.66}{0}\begin{bmatrix},\!\begin{pmatrix}\!\begin{array}{l}\textbf{XYROT}\\\textbf{YZROT}\\\textbf{ZXROT}\end{array}\!\end{pmatrix}\!,\mathit{angle}\end{bmatrix}$}\icamhsp{0.707mm}$ $\raise0.03ex\hbox{$\Bigl[,\textbf{AT},\mathit{x_{c}},\mathit{y_{c}}\,\bigl[,\mathit{z_{c}}\,\bigr]\,\Bigr]$}\icamhsp{0.707mm}$ $\raise0.03ex\hbox{$\Bigl[,\textbf{TRANS},\mathit{x_{2}},\mathit{y_{2}}\,\bigl[,\mathit{z_{2}}\,\bigr]\,\Bigr]$}$

A full description of the <matrix> syntax can be found in “The ORIGIN Command [MLEPC]”.

The $FMXINV Function

$\textbf{{\char36}FMXINV}\textbf{(}\icamhsp{0.707mm}\mathit{m}\icamhsp{0.707mm}\textbf{)}$

Matrix inversion

Returns: Sequence

The $FMXINV function inverts the matrix sequence m, then returns the result as a matrix sequence.

The $FMXMULT Function

$\textbf{{\char36}FMXMULT}\textbf{(}\icamhsp{0.707mm}\mathit{m_{1}},\mathit{m_{2}}\icamhsp{0.707mm}\textbf{)}$

Matrix multiplication

Returns: Sequence

The $FMXMULT function multiplies matrix m1 by m2, then returns the result as a matrix sequence. A diagnostic will be output and macro processing will fail if both matrices do not have the same dimensions.

The $FMXTRFP Function

$\textbf{{\char36}FMXTRFP}\textbf{(}\icamhsp{0.707mm}\mathit{m},\mathit{p}\icamhsp{0.707mm}\textbf{)}$

Point transformation by matrix

Returns: Sequence

The $FMXTRFP function transforms a point p, defined as a sequence of arbitrary length, by the matrix m, then returns the translated and rotated result as a point sequence. This function only solves for the minimum number of common dimensions. Unsolved excess point dimensions are not transformed in the output point sequence.

The $FMXTRFV Function

$\textbf{{\char36}FMXTRFV}\textbf{(}\icamhsp{0.707mm}\mathit{m},\mathit{v}\icamhsp{0.707mm}\textbf{)}$

Vector transformation by matrix

Returns: Sequence

The $FMXTRFV function transforms a vector v, defined as a sequence of arbitrary length, by the matrix m, then returns the rotated result as a vector sequence. This function only solves for the minimum number of common dimensions. Unsolved excess vector dimensions are not transformed in the output vector sequence.

The $FMXTRSP Function

$\textbf{{\char36}FMXTRSP}\textbf{(}\icamhsp{0.707mm}\mathit{m}\icamhsp{0.707mm}\textbf{)}$

Matrix transposition

Returns: Sequence

The $FMXTRSP function transposes the matrix sequence m, then returns the result as a matrix sequence.

The $FMXTYPE Function

$\textbf{{\char36}FMXTYPE}\textbf{(}\icamhsp{0.707mm}\mathit{m}\,\bigl[,\mathit{type}\,\bigr]\icamhsp{0.707mm}\textbf{)}$

Matrix type

Returns: Sequence

The $FMXTYPE function returns a numeric value that identifies the transformation type of a matrix m. Return values are as follows:

0:

The matrix consists entirely of zeroes

1:

Identity

10:

Translation

100:

Rotation (9 decimal digit or better accuracy)

200:

Rotation and/or scaling and/or mirroring

1000:

Scale

10000:

Mirror

Translation is only checked for 3x4 and homogeneous 4x4 matrixes. A matrix with combined translation and rotation will return the value 110. Similarly, a matrix with combined scaling and translation will return 1010 and a matrix with combined mirroring and translation will return 10010. Scaling and mirroring cannot be detected when combined with rotation. A matrix containing such a combined transformation, or a matrix containing rotation defined using vectors with 8 or fewer digits of precision, will return a type 200 (or 210 if combined with translation).

An optional type keyword can be given to test for a specific transformation type. A value of zero is returned if the transformation is not present, otherwise the return value is as listed above. The type keyword must be one of the following:

TRANS:

Translation

ROTATE:

Rotation

SCALE:

Scaling

MIRROR:

Mirroring

For example, the following IF block tests a matrix M for various transformations:
IF/$FMXTYPE(M).EQ.1
  identity
ELSEIF/$FMXTYPE(M,TRANS).GT.0
  translation
ELSEIF/…
  …
ENDOF/IF