New subject: merry x-mas

24 Dec 2002

      Dear reader 

Intro 

About three or four years ago I downloaded fractint and winfract. Since then, every year, during wintertime I'm playing with those programs. It's the formula parser that has my main interest. I can subdivide this in three things: 1. What are the possibilities of the parser? 2. Do I understand what it is doing? 3. Can I produce pictures, which are somehow interesting and/or beautiful? 

Now I feel the need to show some of my "investigations" to people who are interested. I'm not sure this list is the right place, because emails I read are mainly about implementation-problems. Well, we will see. Quit normal, during my investigations, a lot of questions pop up. Questions I couldn't answer. Maybe someone can. 

Short formulas and graphs of real valued functions 

This first time I like to show is that you can make really short formulas. Here's one: 

formula#1 

jhline   { :
           |Imag(Pixel)|<p1
         }
I like it: there is no initiation, nothing in the looping part, there is only a criteria to stop. The formula draws the x-as. The thickness of the line depends on the parameter p1. 

So I ask myself: "What is the shortest, meaningful formula possible". I came with this: 

formula#2 

jhshortest { :
    Pixel<0
  }
This formula divides the plane in two half-planes. Apart from the name there are 10 symbols.

Naturally this gives my first question: 

question#1 

Is there a shorter formula possible? Does anybody know a shorter one? 

Because Pixel=( , ) is a complex number or otherwise interpreted a point of the plain, the statement Pixel<0 
has mathematical spoken no meaning: there is no ordering in Z or R2. Nevertheless the generator interprets the statement. So 
question#2 

How interprets the formulagenerator things like z<1? 

For sure these formulas have nothing to do with fractals. But they have to do with fractint. And the first one gave me a marvellous idea (at least that's what I think): you can use the formulagenerator for drawing the graph of real valued functions in a rather simple way. Here are two formulas doing that. The first draws a parabola, the second draws the graphs of functions of the form f(x)=(ax+b)/(cx^2+dx+e) 

formula#3 

jhparabola      {;Jos Hendriks,2002
                :
                |real(Pixel)^2-imag(Pixel)-1.25|<p1
                }
formula#4 

jhgraphs1   {;Jos Hendriks,2002
            ;graph of y= (a1*x+a2*x)/(a3*x^2+a4*x+a5)\
            ;imag(p3) gives the thickness
            :
            |(real(p1)*real(Pixel)+imag(p1))/\
            (real(p2)*real(Pixel)^2+imag(p2)*\
            real(Pixel)+real(p3))-imag(Pixel)|<imag(p3)||\
            |real(Pixel)|<.001|||imag(Pixel)|<.001
            }
In these formulas is the amount of typework not greater then in a "math-program". The rendering time is not too long: a few seconds or less. Moreover, and I think the most interesting part: The unusual way fractint draws the graph (it's scanning if a point belongs to the graph or not) makes a graph a quite elegant drawing because the thickness of the line is not everywhere the same. So you can create elegant curves. About this maybe another time. 

For convenience I have included some parameter files, because some picture parameters are important. For instance maxiter=2. I have the frms also repeated. In this way you can transport all at once into a par file.

p.s:don't blame me for mistakes I make in writing English: it's not my native language

frm:jhline {   ;Jos Hendriks,2002
                    ;draws the line y=0, p1 gives the thickness of the line
                    ;watch out: only two colors are used
                    ;take for instance maxiter=2, inside=0, outside=iter 
                    :
                    |Imag(Pixel)|<p1
           }

x-as {;Jos Hendriks,2002
reset=2000 type=formula formulaname=jhline
center-mag=0/0/0.6666667 params=0.001/0 float=y maxiter=2 inside=0
colors=00000e
}

frm:jhparabola{ ;Jos Hendriks,2002, 
                ;draws a parabola. Thickness depends on p1.
                ;use maxiter=2,inside color=0, outside=iter
                :
                |real(Pixel)^2-imag(Pixel)-1.25|<p1
              }

jhparabola1 {;Jos Hendriks,2002
reset=2000 type=formula formulafile=sier.frm formulaname=jhparabola
center-mag=0.0301786/0.369842/0.4959278 params=0.1/0 float=y
maxiter=2 inside=0 colors=00000e
}

frm:jhgraphs1 { ;Jos Hendriks,2002
                ;graph of y= (a1*x+a2*x)/(a3*x^2+a4*x+a5)\
                ;imag(p3) gives the thickness
                ;also the x- and y-axes are drawn
                :
                |(real(p1)*real(Pixel)+imag(p1))/\
                (real(p2)*real(Pixel)^2+imag(p2)*\
                real(Pixel)+real(p3))-imag(Pixel)|<imag(p3)||\
                |real(Pixel)|<.001|||imag(Pixel)|<.001
              {

graph1 { ;Jos Hendriks,2002, broken polynomial function
; also x- en y-axes are visible
reset=2000 type=formula formulafile=sier.frm formulaname=jhgraphs1
passes=1 center-mag=-0.791138/1.96774/0.2142747
params=0/5/-1/2/1/0.02 float=y maxiter=2 inside=0 cyclerange=1/1
colors=00000e
}

graph2 { ; Jos Hendriks,2002, broken polynomial function
; also x- en y-as are visible
reset=2000 type=formula formulafile=sier.frm formulaname=jhgraphs1
passes=1 center-mag=-0.791138/1.96774/0.2142747
params=0/5/1/-2/-3/0.05 float=y maxiter=2 inside=0 cyclerange=1/1
colors=00000e
}
graph3 { ; Jos Hendriks,2002, broken polynomial function
; also x- en y-axes are visible
reset=2000 type=formula formulafile=sier.frm formulaname=jhgraphs1
passes=1 center-mag=-0.791138/1.96774/0.2142747
params=0/5/1/0/1/0.05 float=y maxiter=2 inside=0 cyclerange=1/1
colors=00000e 
}

graph4 { ; Jos Hendriks,2002, broken polynomial function
; also x- en y-as are visible
reset=2000 type=formula formulafile=sier.frm formulaname=jhgraphs1
passes=1 center-mag=-0.791138/1.96774/0.2142747
params=-5/5/1/0/1/0.05 float=y maxiter=2 inside=0 cyclerange=1/1
colors=00000e
}

graph5 { ; Jos Hendriks,2002, 
;with correct choosen parameters graph of linear f: a line
; also x- en y-axes are visible
reset=2000 type=formula formulafile=sier.frm formulaname=jhgraphs1
passes=1 center-mag=-0.791138/1.96774/0.2142747
params=1/3/0/0/1/0.05 float=y maxiter=2 inside=0 cyclerange=1/1
colors=00000e
}

investigations#1

jos hendriks

Guy Marson

jos hendriks

Guy Marson

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