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 \begin{displaymath}
  16   \vec{x}' = \mathsf{A}\, \vec{x} + \vec{b}\ ,
  17 \end{displaymath}
  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{tableii}{l|l}{textrm}{\texttt{trafo} method}{function}
  41 \lineii{\texttt{\_\_init\_\_(matrix=((1,0),(0,1)), vector=(0,0)):}}{create new \texttt{trafo} instance with transformation \texttt{matrix} and \texttt{vector}.}
  42 \lineii{\texttt{apply(x, y)}}{apply \texttt{trafo} to point vector $(\mathtt{x}, \mathtt{y})$.}
  43 \lineii{\texttt{inverse()}}{returns inverse transformation of \texttt{trafo}.}
  44 \lineii{\texttt{mirrored(angle)}}{returns \texttt{trafo} followed by mirroring at line through $(0,0)$ with  direction \texttt{angle} in degrees.}
  45 \lineii{\texttt{rotated(angle, x=None, y=None)}}{returns \texttt{trafo} followed by rotation by \texttt{angle} degrees around point $(\mathtt{x}, \mathtt{y})$, or $(0,0)$, if not given.}
  46 \lineii{\texttt{scaled(sx, sy=None, x=None, y=None)}}{returns \texttt{trafo} followed by scaling with scaling factor \texttt{sx} in $x$-direction, \texttt{sy} in $y$-direction ($\mathtt{sy}=\mathtt{sx}$, if not given) with scaling center $(\mathtt{x}, \mathtt{y})$, or $(0,0)$, if not given.}
  47 \lineii{\texttt{translated(x, y)}}{returns \texttt{trafo} followed by translation by vector $(\mathtt{x}, \mathtt{y})$.}
  48 \lineii{\texttt{slanted(a, angle=0, x=None, y=None)}}{returns \texttt{trafo} followed by XXX}
  49 \end{tableii}
  50 \medskip
  51
  52
  53
  54 \section{Subclasses of trafo}
  55
  56 The \verb|trafo| module provides a number of subclasses of
  57 the \verb|trafo| class, each of which corresponds to one \verb|trafo|
  58 method. They are listed in the following table:
  59
  60 \medskip
  61 \begin{tableii}{l|l}{textrm}{\texttt{trafo} subclass}{function}
  62 \lineii{\texttt{mirror(angle)}}{mirroring at line through $(0,0)$ with direction  \texttt{angle} in degrees.}
  63 \lineii{\texttt{rotate(angle, x=None, y=None)}}{rotation by \texttt{angle} degrees around point $(\mathtt{x}, \mathtt{y})$, or $(0,0)$, if not given.}
  64 \lineii{\texttt{scale(sx, sy=None, x=None, y=None)}}{scaling with scaling factor \texttt{sx} in $x$-direction, \texttt{sy} in $y$-direction ($\mathtt{sy}=\mathtt{sx}$, if not given) with scaling center $(\mathtt{x}, \mathtt{y})$, or $(0,0)$, if not given.}
  65 \lineii{\texttt{translate(x, y)}}{translation by vector $(\mathtt{x}, \mathtt{y})$.}
  66 \lineii{\texttt{slant(a, angle=0, x=None, y=None)}}{XXX}
  67 \end{tableii}
  68 \medskip
  69
  70
  71 % \section{Examples}
  72
  73
  74
  75 %%% Local Variables:
  76 %%% mode: latex
  77 %%% TeX-master: "manual.tex"
  78 %%% End: