Enveloppe

function setup() {

  createCanvas(windowWidth, windowHeight);

}

​

function draw() {

  background(255);

  translate(windowWidth/2, windowHeight/2)

  scale(windowHeight/2, -windowHeight/2)

  // Drawing the line network defined by 

  // F(x,y,t) = 0

  // where 

  // F(x,y,t) = t*y + (1-t)*x - t*(1-t)

  let N = 10

  stroke(0)

  strokeWeight(2/windowHeight)

  for(let n=0; n<=N; ++n) {

    let t = n/N

    let P01 = createVector(0,1-t)

    let P12 = createVector(t,0)

    line(P01.x, P01.y, P12.x, P12.y)

  }

  // Let's compute the partial derivative w.r.t. t

  // dF/dt(x,y,t) = t*y + (1-t)*x -t*(1-t)

  // The envelope conditions are :

  //  | F(x,y,t) = 0           (1)

  //  | dF/dt(x,y,t) = 0       (2)

  // Rearranging (2) => t = (x-y+1)/2

  // Reinjecting into (1) => y² -2*(x+1)*y + (x-1)² = 0

  // Solving for y, we obtain :

  //   y = x+1 +/- sqrt(x)      (x>0)

  // By symetry of the envelope conditions (using t'=1-t),

  // one could use the alternate formula : x = y+1 +/- sqrt(y)

  N = 100

  stroke(255,0,0)

  noFill()

  strokeWeight(4/windowHeight)

  beginShape()

  for(let n=0; n<=N; ++n) {

    //*

    let x = n/N

    let y = x+1-2*sqrt(x)

    //let y = x+1+2*sqrt(x)

    /*/

    let y = n/N

    //let x = y+1-2*sqrt(y)

    let x = y+1+2*sqrt(y)

    //*/

    vertex(x,y)

  }

  endShape()

}