manual/trafo.tex

   1 \chapter{Module trafo: linear transformations}
   2
   3 \label{trafo}
   4
   5 With the  \verb|trafo| module \PyX\ supports linear transformations, which can
   6 then be applied to canvases,  B\'ezier paths and other objects. It consists
   7 of the main class \verb|trafo| representing a general linear
   8 transformation and subclasses thereof, which provide special operations
   9 like translation, rotation, scaling, and mirroring.
  10
  11 \section{Class trafo}
  12
  13 The \verb|trafo| class represents a general linear
  14 transformation, which is defined for a vector $\vec{x}$ as
  15 \[
  16   \vec{x}' = \mathsf{A}\, \vec{x} + \vec{b}\ ,
  17 \]
  18 where $\mathsf{A}$ is the transformation matrix and $\vec{b}$ the
  19 translation vector. The transformation matrix must not be singular,
  20 \textit{i.e.} we require $\det \mathsf{A} \ne 0$.
  21
  22
  23
  24 Multiple \verb|trafo| instances can be multiplied, corresponding to a
  25 consecutive application of the respective transformation. Note that
  26 \verb|trafo1*trafo2| means that \verb|trafo1| is applied after
  27 \verb|trafo2|, \textit{i.e.} the new transformation is given
  28 by $\mathsf{A} = \mathsf{A}_1 \mathsf{A}_2$ and
  29 $\vec{b} = \mathsf{A}_1 \vec{b}_2 + \vec{b}_1$.  Use the \verb|trafo|
  30 methods described below, if you prefer thinking the other way round.
  31 The inverse of a transformation can be obtained via the \verb|trafo|
  32 method \verb|inverse()|, defined by the inverse $\mathsf{A}^{-1}$ of
  33 the transformation matrix and the translation vector
  34 $-\mathsf{A}^{-1}\vec{b}$.
  35
  36 The methods of the \verb|trafo| class are summarized in the following
  37 table.
  38
  39 \medskip
  40 \begin{tabularx}{\linewidth}{>{\hsize=.8\hsize}X>{\raggedright\arraybackslash\hsize=1.2\hsize}X}
  41 \texttt{trafo} method & function \\
  42 \hline
  43 \texttt{\_\_init\_\_(matrix=((1,0),(0,1)),\newline
  44 \phantom{\_\_init\_\_(}vector=(0,0)):} & create new \texttt{trafo}
  45 instance with transformation \texttt{matrix} and \texttt{vector}.
  46 \\
  47 \texttt{apply(x, y)} & apply \texttt{trafo} to point vector
  48 $(\mathtt{x}, \mathtt{y})$.\\
  49 \texttt{inverse()} & returns inverse transformation of
  50 \texttt{trafo}.\\
  51 \texttt{mirrored(angle)} & returns \texttt{trafo} followed by mirroring
  52 at line through $(0,0)$ with  direction \texttt{angle} in degrees.\\
  53 \texttt{rotated(angle, \newline\phantom{rotate(}x=None, y=None)} &
  54 returns \texttt{trafo} followed by rotation by \texttt{angle} degrees
  55 around point $(\mathtt{x},
  56 \mathtt{y})$, or $(0,0)$, if not given.\\
  57 \texttt{scaled(sx, sy=None,\newline\phantom{scale(}x=None, y=None)} &
  58 returns \texttt{trafo} followed by
  59 scaling with scaling factor \texttt{sx} in $x$-direction, \texttt{sy} in
  60 $y$-direction ($\mathtt{sy}=\mathtt{sx}$, if not given) with scaling
  61 center $(\mathtt{x}, \mathtt{y})$, or $(0,0)$, if not given.\\
  62 \texttt{translated(x, y)} & returns \texttt{trafo} followed by
  63 translation by vector $(\mathtt{x}, \mathtt{y})$.\\
  64 \texttt{slanted(a, angle=0, x=None, y=None)} & returns \texttt{trafo}
  65 followed by XXX\\
  66 \end{tabularx}
  67 \medskip
  68
  69
  70
  71 \section{Subclasses of trafo}
  72
  73 The \verb|trafo| module provides a number of subclasses of
  74 the \verb|trafo| class, each of which corresponds to one \verb|trafo|
  75 method. They are listed in the following table:
  76
  77 \medskip
  78 \begin{tabularx}{\linewidth}{>{\hsize=.7\hsize}X>{\raggedright\arraybackslash\hsize=1.3\hsize}X}
  79   \texttt{trafo} subclass & function \\
  80   \hline
  81   \texttt{mirror(angle)} & mirroring at line through $(0,0)$
  82   with direction  \texttt{angle} in degrees.\\
  83   \texttt{rotate(angle, \newline\phantom{rotation(}x=None, y=None)} &
  84   rotation by \texttt{angle} degrees around point $(\mathtt{x},
  85   \mathtt{y})$, or $(0,0)$, if not given.\\
  86   \texttt{scale(sx, sy=None,\newline\phantom{scaling(}x=None, y=None)} &
  87   scaling with scaling factor \texttt{sx} in $x$-direction,
  88   \texttt{sy} in $y$-direction ($\mathtt{sy}=\mathtt{sx}$, if not
  89   given) with scaling
  90   center $(\mathtt{x}, \mathtt{y})$, or $(0,0)$, if not given.\\
  91   \texttt{translate(x, y)} &
  92   translation by vector $(\mathtt{x}, \mathtt{y})$.\\
  93   \texttt{slant(a, angle=0, x=None, y=None)} & XXX \\
  94 \end{tabularx}
  95 \medskip
  96
  97
  98 % \section{Examples}
  99
 100
 101
 102 %%% Local Variables:
 103 %%% mode: latex
 104 %%% TeX-master: "manual.tex"
 105 %%% End: