[GS] - ''Superformula'' was proposed by Johan Gielis (a Belgian botanist) in 1997. He has modified 
and generalized the superellipse formula discovered by Gabriel Lam��© in 1818 (a French mathematician).
Superformula consists in a simple 2D analytical expression allowing to draw a wide variety of 
geometric and natural shapes (starfish, petals, snowflakes) by choosing suitable values relevant 
to few parameters.

[http://gersoo.free.fr/wiki/w12977/sformula.jpg]

Here is the formula in polar coordinates:

[http://gersoo.free.fr/wiki/w12977/superformula.jpg]

Parameters influence:
. m: rotational symmetry
. n1, n2, n3: form
. a, b: expansion coefficients

For further details see the Math Trek article [http://www.sciencenews.org/articles/20030503/mathtrek.asp]

Johan Gielis has founded Genicap Corp. [http://www.genicap.com/] to develop and to market graphics 
software based on this formula. The ''Superformula'' may reduce the size of complex 2D and 3D 
graphic files by more than 90 percent in CAD and graphic design applications.

A few examples:

[http://gersoo.free.fr/wiki/w12977/examples.gif]

      (a,b,m,n1,n2,n3)
 (1)  (1, 1.8, 6, 1, 2, 1.4)
 (2)  (1.5, 2, 8, 0.9, 1.8, 0.8)
 (3)  (1, 2, 10, 0.9, 1.7, 1.1)
 (4)  (1.6, 1, 12, 1.5, 2, 7.5)
 (5)  (2.7, 2.6, 6, 12, 8.3, 5.3)
 (6)  (1.5, 2, 2, 1.7, 0.7, 2)
 (7)  (1.19, 2.8, 32, 3, 0.9, 6)
 (8)  (0.9, 10, 4, 4.2, 17, 1.5)
-

 # superformula.tcl 
 # Author:      Gerard Sookahet
 # Date:        16 Nov 2004 
 # Description: Plot Superformula based upon Johan Gielis equations 
 #              See http://www.sciencenews.org/articles/20030503/mathtrek.asp
 
 package require Tk
 
 bind all <Escape> {exit}
 
 proc About {} {
  set w .about
  catch {destroy $w}
  toplevel $w
  wm title $w "About Superformula"
  message $w.msg -justify center -aspect 250 -relief sunken -bg blue -fg white \
         -text "Superformula\n\nGerard Sookahet\n\nNovember 2004"
  button $w.bquit -text " OK " -command {destroy .about}
  eval pack [winfo children $w]
 }
 # Init or reset data
 proc Init { w mid } {
     global a b m k
     global n1 n2 n3
 
  $w.c delete all
 
  set a 1.0
  set b 1.0
  set m 6.0
  set k 2
  set n1 3.0
  set n2 1.0
  set n3 1.0
 
  PlotSuperformula $w $mid 0
 }
 # Filled polygons or not
 proc FilledState { w mid } {
     global fs
     global poly
  if $fs {$w.c itemconfigure $poly -outline lightblue -width 1 -fill blue} else {
          $w.c itemconfigure $poly -outline lightblue -width 1 -fill black 
  }
 }
 
 proc DisplayState { w mid } {
     global rf
  if $rf {Init $w $mid} 
 }
 
 proc Main { N } {
     global a b m k
     global n1 n2 n3
     global rf fs
 
  set w .sp
  catch {destroy $w}
  toplevel $w
  wm withdraw .
  wm title $w "Superformula"
  wm geometry $w +100+10
  $w config -bg orange
 
  set f0 [frame $w.f0]
  set mid [expr {$N/2}]
  pack [canvas $w.c -width $N -height $N -bg black] $f0 -side left
  Init $w $mid
 
  set f1 [frame $f0.f1 -relief sunken -bd 1]
  pack $f1 -fill x
  scale $f1.sca -from 0.1 -to 10 -length 340 -resolution 0.01 -label "a" \
                -orient vert -bd 1 -showvalue true -variable a \
                -command "PlotSuperformula $w $mid"
  scale $f1.scb -from 0.1 -to 10 -length 340 -resolution 0.01 -label "b" \
                -orient vert -bd 1 -showvalue true -variable b \
                -command "PlotSuperformula $w $mid"
  scale $f1.scm -from 0 -to 34 -length 340 -resolution 0.2 -label "m" \
                -orient vert -bd 1 -showvalue true -variable m \
                -command "PlotSuperformula $w $mid"
  scale $f1.scn1 -from 0.1 -to 17 -length 340 -resolution 0.1 -label "n1" \
                 -orient vert -bd 1 -showvalue true -variable n1 \
                 -command "PlotSuperformula $w $mid"
  scale $f1.scn2 -from 0 -to 17 -length 340 -resolution 0.1 -label "n2" \
                 -orient vert -bd 1 -showvalue true -variable n2 \
                 -command "PlotSuperformula $w $mid"
  scale $f1.scn3 -from 0 -to 17 -length 340 -resolution 0.1 -label "n3" \
                 -orient vert -bd 1 -showvalue true -variable n3 \
                 -command "PlotSuperformula $w $mid"
  scale $f1.sck -from 1 -to 34 -length 340 -label "pi" \
                -orient vert -bd 1 -showvalue true -variable k \
                -command "PlotSuperformula $w $mid"
  eval pack [winfo children $f1] -side left
 
  set f2 [frame $f0.f2 -relief sunken -bd 1]
  pack $f2 -fill x
  checkbutton $f2.cbf -text "Filled shape" -variable fs \
             -command "FilledState $w $mid"
  checkbutton $f2.cbr -text "Refresh display" -variable rf \
             -command "DisplayState $w $mid"
  eval pack [winfo children $f2] -side left
 
  set f3 [frame $f0.f3 -relief sunken -bd 1]
  pack $f3 -fill x
  button $f3.bc -text Clear -width 6 -bg blue -fg white \
        -command "$w.c delete all"
  button $f3.ba -text About -width 6 -bg blue -fg white \
        -command About
  button $f3.bq -text Quit -width 5 -bg blue -fg white -command exit
  eval pack [winfo children $f3] -side left
 }
 
 proc PlotSuperformula { w mid v } {
     global a b m k
     global n1 n2 n3
     global rf fs
     global poly
 
  if $rf {$w.c delete all}
 
  set kpi [expr {$k*3.14159265358}]
  set N 400
  set s 0.4
 
  set l {}
  for {set i 0} {$i<=$N} {incr i} {
     set phi [expr {$i*$kpi/$N}]
 
     set t  [expr {$m*$phi/4}]
     set t1 [expr {abs(cos($t)/$a)}]
     set t1 [expr {pow($t1,$n2)}]
     set t2 [expr {abs(sin($t)/$b)}]
     set t2 [expr {pow($t2,$n3)}]
 
     set r [expr {pow($t1+$t2,1/$n1)}]
     if {[expr {abs($r)}] <= 0.000001} {
       lappend l 0 0
     } else {
       set r [expr {1/$r}]
       lappend l [expr {round($mid + $s*$mid*$r*cos($phi))}] \
                 [expr {round($mid + $s*$mid*$r*sin($phi))}]
     }
  }
  set poly [eval $w.c create polygon $l] 
  if $fs {$w.c itemconfigure $poly -outline lightblue -width 1 -fill blue} else {
          $w.c itemconfigure $poly -outline lightblue -width 1 -fill black 
  }
 }
 
 set rf 1
 set fs 1
 Main 440

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[Ro]: This is a brillant addition [GS]!  Well DONE!


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[[
[Category Graphics] | 
[Category Mathematics] |
[Category Plotting]
]]