How to express PSTricks point in terms of a shifted variable and function?

Question

For a trivial example, suppose that \f(x) is a quadratic curve defined as follows

\def\f(#1){-#1*(#1-8)/4}

For each element \ix in {0,1,2,3,4,5,6,7,8} there is a point (\ix,\f(\ix)) that will be the center of a square of 1 cm square.

How to express the bottom-left & top-right points of \psframe in terms of \ix and \f(\ix)?

NOTE: Please don't suggest other simpler solutions such as using \rput to put the squares because I am interested in PSTricks point expression. RPN notation should be avoided if it is possible.

I give you the skeleton (not working) as below to save your time.

\documentclass[pstricks,border=12pt]{standalone}
\usepackage{pst-plot}%loads multido automatically

\begin{document}

\def\f(#1){-#1*(#1-8)/4}

\begin{pspicture}
\multido{\ix=0+1}{9}{
% the following expression is wrong!
\psframe[dimen=middle](*\ix-0.5 {\f(x)-0.5})(*\ix+0.5 {\f(x)+0.5})
}
\psplot[algebraic]{0}{8}{\f(x)}
\end{pspicture}

\end{document}

The output I want to get is as follows:

Answer 1

\listfiles
\documentclass{article}
\usepackage{pst-plot}
\SpecialCoor
\begin{document}

\def\f(#1){-#1*(#1-8)/4}

\begin{pspicture}(-1,-2)(10,5)
\psforeach{\ix}{0,1,..,9}{%
  \rput(*\ix\space {\f(x)}){\psframe[dimen=middle](-0.5,-0.5)(0.5,0.5)}}
\psplot[algebraic]{0}{8}{\f(x)}
\end{pspicture}

\begin{pspicture}(-1,-2)(10,5)
\psforeach{\ix}{0,1,..,9}{%
  \psframe[dimen=middle](*{\ix\space 0.5 sub}  {\f(\ix)-0.5})
                        (*{\ix\space 0.5 add} {\f(\ix)+0.5})}
\psplot[algebraic]{0}{8}{ \f(x) }
\end{pspicture}

\end{document}

Answer 2

Use a postscript formulation of your algebraic function, and then reference the coordinates using the "raw PS" notation:

\documentclass{article}
\usepackage{pst-plot}% http://ctan.org/pkg/pst-plot
% The following package (multido) is loaded by pst-plot
%\usepackage{multido}% http://ctan.org/pkg/multido
\begin{document}

\def\f(#1){-#1*(#1-8)/4}% function f(x)=-x*(x-8)/4
% PS definition of f(x): x neg x 8 sub 4 div mul
\def\halfsize{0.5\space}% Modify this for smaller squares

\begin{pspicture}
  \multido{\ix=0+1}{9}{%
    \psframe[linewidth=0.5pt,linecolor=red!50]% Red squares
      (!\ix\space \halfsize sub \ix\space neg \ix\space 8 sub 4 div mul \halfsize sub)% bottom left
      (!\ix\space \halfsize add \ix\space neg \ix\space 8 sub 4 div mul \halfsize add)% upper right
  }
  \psplot[algebraic]{0}{8}{\f(x)}% Plot function
\end{pspicture}

\end{document}

The postscript formulation of \f(x)=-x*(x-8)/4 is

x neg x 8 sub 4 div mul

Now you can specify the node positions using postscript notation/coordinates (!<x> <y>) (excuse the ASCII art):

         (     x - 0.5     ,      - x    *      (x - 8) / 4          - 0.5 )

         |     x - 0.5     |      - x    *      (x - 8) / 4          - 0.5 |
         |     x - 0.5     |      - x    *      (x - 8) / 4        |0.5| - |
         |     x - 0.5     |      - x    |      (x - 8) / 4        |0.5| - |
         |     x - 0.5     |      - x    |      x - 8    |         |0.5| - |
         |    x    |0.5| - |    x    | - |    x    |8| - |4| / | * |0.5| - |
(x,y) ~ (!\ix\space 0.5 sub \ix\space neg \ix\space 8 sub 4 div mul 0.5 sub)

It is required that you use \space after control sequences to introduce a space, otherwise elements are concatenated, leading to incorrect output (or sometimes input that doesn't compile). Also, note that there's no comma , separating the coordinates <x> and <y>; they are merely placed on a stack - first <x> and then <y>.