Matrix Functions

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

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:

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.

$FMX(…)

Matrix creation Returns: Sequence

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

where <rotation> is defined as:

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

$FMXINV(m)

Matrix inversion Returns: Sequence

This function inverts the matrix sequence m, then returns the result as a matrix sequence.

$FMXMULT(m1,m2)

Matrix multiplication Returns: Sequence

This 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.

$FMXTRFP(m,p)

Point transformation by matrix Returns: Sequence

This 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.

$FMXTRFV(m,v)

Vector transformation by matrix Returns: Sequence

This 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.

$FMXTRSP(m)

Matrix transposition Returns: Sequence

This function transposes the matrix sequence m, then returns the result as a matrix sequence.

$FMXTYPE(m[,type])

Matrix type Returns: Numeric

This 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