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\input macro.tex
\def\today{\ifcase\month\or
  January\or February\or March\or April\or May\or June\or
  July\or August\or September\or October\or November\or December\fi
  \space\number\day, \number\year}
\baselineskip 14pt
\rm
\line{\sevenrm a97.tex[106,phy] \today\hfill}

\font\rmn=cmr9

\bigskip
\line{\bf Printing Geometrical Images.\hfil}


To print characters along a line in the output page within a nested
iteration of the form

\smallskip
{\obeylines\obeyspaces\let =\ \tt
        FOR R:= 1 TO RMAX DO
           BEGIN
           FOR C:= 1 TO CMAX DO
           (* write correct character for (R,C)*);
           WRITELN
           END.
}

\smallskip\noindent
find the equation $f(R,C)=0$ for the line, and use the condition

\smallskip
{\obeylines\obeyspaces\let =\ \tt
        IF f(R,C)=0
        THEN WRITE('*')
        ELSE WRITE(' ')

{\baselineskip10pt
          1 2 3 4 5
         ----------
        1|        *
        2|      *
        3|    *
        4|  *
        5|*
}

        IF R+C=6 (* F = R+C - 6 *)
        THEN WRITE('*')
        ELSE WRITE(' ')
}

\smallskip
To print characters on one side of a line, use the appropriate case of

\smallskip
{\obeylines\obeyspaces\let =\ \tt
        IF f(R,C)>0
        IF f(R,C)<0
        IF f(R,C)>=0
        IF f(R,C)<=0
}

\smallskip\noindent
where the latter two are appropriate if the dividing line is itself to be dark.

\smallskip
{\obeylines\obeyspaces\let =\ \tt
{\baselineskip10pt
          1 2 3 4 5
         ----------
        1|* * * * *
        2|* * * *
        3|* * *
        4|* *
        5|*
}

        IF R+C <= 6 THEN ...

{\baselineskip10pt
          1 2 3 4 5 6 7
         --------------
        1|
        2|
        3|
        4|
        5|
}

        IF SQR(R-7)+SQR(C-8)<=18 THEN ...
}

\smallskip\noindent
Shaded regions defined by several lines can be put together using {\tt AND} 
and {\tt OR} on the conditions for the separate lines:

\smallskip
{\obeylines\obeyspaces\let =\ \tt
{\baselineskip10pt
          -8                            8
          -------------------------------
        -8|                             |
          |                             |
          |                             |
          |                             |
          |                             |
          |                             |
          |                             |
          |                             |
          |                             |
          |                             |
          |                             |
          |                             |
          |                             |
         8|                             |
	  -------------------------------
}

	IF (R+C>=0) OR (SQR(R)+SQR(C)<=25) THEN ...

{\baselineskip10pt
          -8                            8
          -------------------------------
        -8|                             |
          |                             |
          |                             |
          |                             |
          |                             |
          |                             |
          |                             |
          |                             |
          |                             |
          |                             |
          |                             |
          |                             |
          |                             |
         8|                             |
	  -------------------------------
}

	IF (R+C>=0) AND (SQR(R)+SQR(C)<=25) THEN ...
}

\smallskip
In the first image, where the line and circle cross, three of the adjacent
regions are shaded, so {\tt OR} is correct. In the second, only one is shaded, so
{\tt AND} is correct.

A common situation has two of the four regions, diagonally opposite, shaded
at a crossing. If the equations of the lines are $f(R,C)=0$ and $g(R,C)=0$,
test the product $f(R,C)*g(R,C)$:

\figbox 2truein:

{\obeylines\obeyspaces\let =\ \tt
        R-C=0              R+C-8=0              (R-C)*(R+C-8)>0
}

\smallskip\noindent
If you want the dividing lines to be dark, use {\tt >=} or {\tt <= }:

\smallskip
{\obeylines\obeyspaces\let =\ \tt
        program PinWheel(output);
           const Size = 10;
           var Col, Row : integer;
        begin  (* PinWheel *)
           for Row:= - Size to Size do begin
              for Col:= -Size to Size do
                 if (Col * Row * (Col-Row)*(Col+Row)) >= 0 then write('*')
                 else write(' ');
              writeln;
           end;
        end.  (* PinWheel *)
}

\bigskip
{\baselineskip5pt
\halign{\qquad\qquad\lft{\tt #}\cr
*\phantom{*********}***********\cr
**\phantom{********}********** \cr
***\phantom{*******}*********  \cr
****\phantom{******}********   \cr
*****\phantom{*****}*******    \cr
******\phantom{****}******     \cr
*******\phantom{***}*****      \cr
********\phantom{**}****       \cr
*********\phantom{*}***        \cr
************\cr
*********************\cr
\phantom{*********}************\cr
\phantom{********}***\phantom{*}*********\cr
\phantom{*******}****\phantom{**}********\cr
\phantom{******}*****\phantom{***}*******\cr
\phantom{*****}******\phantom{****}******\cr
\phantom{****}*******\phantom{*****}*****\cr
\phantom{***}********\phantom{******}****\cr
\phantom{**}*********\phantom{*******}***\cr
\phantom{*}**********\phantom{********}**\cr
***********\phantom{*********}*\cr}
}

\figbox 3truein:

{\obeylines\obeyspaces\let =\ \tt
        program PinWheel(output)
	   const Size = 10;
	   var Col, Row : integer;
	begin (* PinWheel *)
	   for Row:= - Size to Size do begin
	      for Col:= -Size to Size do
		 if (Col * Row * (Col-Row)*(Col+Row))) >= 0 then write ('*')
		 else write(' ');
	      writeln;
	   end;
	end. (* PinWheel *)
}

\smallskip
In the above image, the oval line f=0 separates the image into an outer part
where the test $g<0$ must be made, and an inner part where h<0 must be tested.
We therefore make the test on~$f$ first.

\smallskip
{\obeylines\obeyspaces\let =\ \tt
	IF f<0 THEN
	   IF h<0 THEN WRITE(' ') ELSE WRITE('*')
	ELSE
	   IF g<0 THEN WRITE(' ') ELSE WRITE('*')
}

\smallskip
Rule of good programming practice: Wherever you can, split a problem into
pieces such that some of the difficult aspects of the problem are restricted
to each piece. In short: {\tt DIVIDA UT IMPERE (DIVIDE AND CONQUER)}.

After having designed a program that prints an image, we decide to overlay
it with a one-inch grid of asterisks. Using the condition

\smallskip
{\obeylines\obeyspaces\let =\ \tt
        IF (R MOD 6=0) OR (C MOD 10=0)
        THEN WRITE('*')
        ELSE whatever we had before
}

\smallskip\noindent
gives priority to the grid, and thereby lays it over the other image.

\figbox 1truein:

To print alternate dark and white diagonal
stripes, each five characters wide, notice
that the sum R+C indicates which diagonal we are on, and diagonals separated
by multiples of ten are the same shade, so we may test {\tt (R+C) MOD 10}.

The correct condition is

\smallskip
{\obeylines\obeyspaces\let =\ \tt
	IF (R+C) MOD 10<5
	THEN WRITE(' ')
	ELSE WRITE('*')
}

\figbox 3truein:

\centerline{The repeated pattern}

\smallskip
{\obeylines\obeyspaces\let =\ \tt
	IF (R MOD 6=0) OR (C MOD 10=0)
	THEN WRITE('*')
	ELSE
	   IF (R MOD 12-6) * (C MOD 20-10)>0
	   (*1*)
	   THEN WRITE('/')
	   ELSE WRITE(' ')
}

{\rmn
{\narrower\smallskip\noindent
{\bf Drill:} what happens if we insert

{\obeylines\obeyspaces\let =\ \tt
	AND (R+C)MOD 3=0,
}
at (*1*)?
\smallskip}
}


\bigskip
\line{\copyright 1984 Robert W. Floyd; 
First draft (not published) March 29, 1984\hfil}
%revised: Date; subsequently revised.\hfill}

\bye