TITLE: Julia Sunrise
NAME: Paul Beach
COUNTRY: Canada
EMAIL: sniffyraven@fastmail.fm
WEBPAGE: http://www.amazon.com/gp/product/0595137318/ref=sr_11_1/102-1966828-1646500?%5Fe
ncoding=UTF8

TOPIC: Light and Fog
COPYRIGHT: I SUBMIT TO THE STANDARD RAYTRACING COMPETITION COPYRIGHT.
JPGFILE: j_sun.jpg
IMAGE DESCRIPTION: 

Math Fog

It is generally assumed that a signal will live forever if it is only sampled at
twice its highest frequency. If a lissajous figure is sampled and 'filtered'
through a linear ; and then a cubic sphere sweep, the results look somewhat
different. 

References:
Vibrations and Waves, A.P. French
Mathematical Foundations of Information Theory, A.I. Khinchin

The Scene

The sun is a Julia fractal. Presumably, the term fractal means that it is not a
differential equation. I propose yet another for the string of pearls and the
water serpent.

x = ( arctan y/x)**k
y = ( arctan x/y)**k

Three branches will spiral out to a limit cycle. A branch or projection can be
selected using the mod function. Code for the string of pearls is given below.


The water weeds are generated with this group, (Cyclic group of order Six).

x = x/y
y = y*x 

Continue the algebra six times to get back to the starting point. This is done
again in the complex plane, and etched on the moon. 

// Code for string of pearls
  #include "colors.inc" 
  background{ Gray40}
  camera {  
    location <.6, .7, -2>
    look_at <.6, .7, 1> }
  light_source { <3, 8, -20> color White }


  #macro arc_tan_1( )    
      
       #local k1 = 1 / ( 2 / pi  - .05 )   ;  
      #local ay = 2/ pi ;  #local ax = 2/ pi ;
     #local rad = 0;       
  
         #local Count1=0;
          #while (Count1 < 131)  
          #local mod3 = mod ( Count1 , 3) ;
          #local mod2 = mod ( Count1, 2 );    
          #local ax =  exp ( k1 * (ln (atan (ay / ax )))); 
          #local ay = exp (  k1 * (ln (atan(ax / ay )))); 
          
          #local nax =  exp ( k1 * (ln (atan (ay / ax )))); 
          #local nay = exp (  k1 * (ln (atan( nax / ay ))));  
          
          #local nax =  exp ( k1 * (ln (atan (nay / nax )))); 
          #local nay = exp (  k1 * (ln (atan( nax / nay ))));  
          
          #local nax =  exp ( k1 * (ln (atan (nay / nax )))); 
          #local nay = exp (  k1 * (ln (atan( nax / nay ))));
           
                          
                                
                  #if ( mod3 = 2)  
                                   cylinder {  <ax, ay> <nax , nay > , .005   
pigment { Gray60 filter .3 }finish { specular .5 ambient .26 reflection .18
diffuse .28} 
                 } 
                  #if (Count1 > 32) 
                  
                  sphere { < ax, ay  > .041 + rad  pigment 
                  { rgb < .7, .7 , ax> } finish { ambient .4 specular .3
reflection .4}  
                  }  
                  
                                     sphere { < ax, ay  > .045 + rad  pigment 
                  { White transmit .5} finish { ambient .8 specular .3 }  
                  }    
                   #local rad = rad + .0003;
                  #end 
                  #end  
           
           #local nax = 0 ;
           #local nay = 0;                              
                                              
        #declare Count1=Count1+1;
        #end 
#end
arc_tan_1( )