Here is the formula in polar coordinates:
Parameters influence: . m: rotational symmetry . n1, n2, n3: form . a, b: expansion coefficientsFor further details see the Math Trek article [1]Johan Gielis has founded Genicap Corp. [2] 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:
(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 440Ro: This is a brillant addition GS! Well DONE!
