Implicit Line and Circle

This example uses an implicit representations of

a line and a circle to draw a picture. 5000 circles 

are placed at random locations. Points inside a specific 

circle are colored green. Points above the line are 

colored red. Points below the line are colored blue.

​

Colors include transparency to give a blending effect.

*/

​

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/*

This function uses the imlicit definition of a line

(y-y1)*(x2-x1)-(x-x1)*(y2-y1)=0

connecting (x1,y1) and (x2,y2). A positive return 

value puts (x,y) on one side of the line. A negative

value puts it on the other.

*/

function impLine(x,y,x1,y1,x2,y2){

  return((y-y1)*(x2-x1)-(x-x1)*(y2-y1));

}

​

/*

This function uses the implicit definition of a circle

(x-x1)*(x-x1)+(y-y1)*(y-y1)-r*r=0

centered at (x1,y1) with radius r. A positive return

value means that (x,y) is outside of the circle. A 

negative value puts (x,y) inside the circle.

*/

function impCircle(x,y,x1,y1,r){

  return((x-x1)*(x-x1)+(y-y1)*(y-y1)-r*r);

}

​

function setup(){

  color(0,0,0,0);   //this makes the outline of the

                    //circles invisible

}

function draw(){

  var x,y;      //coordinates of circle

  var d;        //value from LINE function

  var e;        //value for CIRCLE function

  var i;        //index for for loop

  //repeat 5000 times

  for (i=0;i<5000;i=i+1){

    //random point

    x=width*random();

    y=height*random();

    //we call impLine with the line 

    //connecting (0,300) and (400,100)

    d=impLine(x,y,0,300,400,100);

    //call impCircle with circle 

    //at (200,200) with radius 100

    e=impCircle(x,y,200,200,100);

    if (e<0)                    //inside circle

      fillcolor(0,255,0,0.1)

    else if (d<0)               //above the line

      fillcolor(255,0,0,0.1);

    else                        //below or on the line

      fillcolor(0,0,255,0.1);

    circle(x,y,10);

    fill();

   }

}

​