Simplex Wave + Explanations

This block experiments SimplexNoise (see also 3, 2, 1). More particularly, how to produce a continous radial noise.

Work initiated by this block from elenstar, Drawing Vector Field, and the outstanding Neonflames.

Acknowledgments to:

<html>
<head>
<meta charset="utf-8">
<title>Simplex Wave + Explanations</title>
<meta content="How to produce a continuous radial noise with Simplex" name="description">
<style>
  
  #under-construction {
    display: none;
    position: absolute;
    top: 200px;
    left: 300px;
    font-size: 40px;
  }
  
  div.step {
    position: relative;
  }
  
  div.require {
    position: absolute;
    left: 150px;
    bottom: -10px;
  }
  
  .desc {
    position: absolute;
    left: 300px;
    padding: 10px;
  }
  
  .desc .highlighted {
    color: red;
  }
  
  .title {
   	font-weight: bold;
  }
  
  ul {
    margin: 0px;
    padding-left: 25px
  }
  
  canvas {
    margin: 10px;
    border: solid lightgrey 1px;
    border-radius: 5px;
    box-shadow: 2px 2px 6px grey;
  }

</style>
<body>
  <div id="under-construction">
    UNDER CONSTRUCTION
  </div>
  
  <div id="radial-wave" class="step">
    <div class="desc">
      <div class="title">Radial wave</div>
      <ul>
        <li>radial path with a continuous and balanced noise</li>
        <li>emerge from the summation of 2 continuous-but-unbalanced radial noises</li>
      </ul>
    </div>
    <canvas></canvas>
  	<div id="traces" class="require">&downarrow; mixes</div>
  </div>
  
  <div id="continuous-radial-noises" class="step">
    <div class="desc">
      <div class="title">Continuous-but-unbalanced radial noises</div>
      <ul>
        <li>discontinuities of the 2 underlying radial noises are eliminated by weighting each radial noise with an appropriate function (I use a cos-based function)</li>
        <li>red segments highlight where underlying radial noises are discontinuous</li>
      </ul>
    </div>
    <canvas></canvas>
    <div id="uses" class="require">&downarrow; weights</div>
  </div>
  
  <div id="radial-noises" class="step">
    <div class="desc">
      <div class="title">Non-continuous radial noises</div>
      <ul>
        <li>applies the underlying noises in a radial way</li>
        <li>red segments highlight where corresponding underlying noises are discontinuous</li>
        <li>the characteristics of the underlying noises makes discontinuities diametrically opposed</li>
      </ul>
    </div>
    <canvas></canvas>
  	<div id="transforms" class="require">&downarrow; applies</div>
  </div>
  
  <div id="noises" class="step">
    <div class="desc">
      <div class="title">SimplexNoises</div>
      <ul>
        <li>noise values are computed thanks to SimplexNoise</li>
        <li>noise values depend on angles (each 4°)</li>
        <li>the second noise is shifted by &pi;, so that discontinuities arises at distinct angles</li>
      </ul>
    </div>
    <canvas></canvas>
  </div>
  <script src="https://d3js.org/d3.v4.min.js"></script>
  <script src="simplex-noise.min.js"></script>
  <script>
    var _2PI = 2*Math.PI,
        _3PI = 3*Math.PI;
    
    var cbw = 1, //canvas border width
        cm = 10, //canvas margin
    		totalWidth = 300,
        totalHeight = 500,
        width = totalWidth-(cm+cbw)*2,
        height = (totalHeight-((cm+cbw)*2)*4)/4,
        midWidth = width/2,
        midHeight = height/2,
        firstWaveXAlign = midWidth/2,
        secondWaveXAlign = 3*midWidth/2;
    
    var simplex = new SimplexNoise(),
    		angleBasedLength = 1, //lower values make more heratic waves' shapes
    		timeBasedLength = 5000,	//lower values make radiations' shapes evolving faster
        noiseStrength = midHeight/4;
    
    var vertexCount = 360/4,		// segment per wave
        waveRadius = midHeight - noiseStrength,
        wave = [];
    
    var noiseScale= d3.scaleLinear().domain([0, 3*Math.PI]).range([-midWidth/2+10, midWidth/2-10]);
    
    initLayout();
    initWave();
    
    var radialWaveContext = document.querySelector("#radial-wave canvas").getContext("2d"),
        continuousRadialNoisesContext = document.querySelector("#continuous-radial-noises canvas").getContext("2d"),
        radialNoisesContext = document.querySelector("#radial-noises canvas").getContext("2d"),
        noisesContext = document.querySelector("#noises canvas").getContext("2d");
    
    d3.interval(function(elapsed) {
      updateWave(elapsed);
      redrawRadialWave();
      redrawContinuousRadialNoises();
      redrawRadialNoises();
      redrawNoises();
    });
    
    function initLayout() {
      d3.selectAll("canvas")
        .attr("width", width)
        .attr("height", height);
    }
    
    function initWave() {
      var angle, oppositeAngle, noise;
      
      for (var v=0; v<vertexCount; v++) {
        angle = _2PI*v/vertexCount;
        wave[v] = {
          angle: angle,												// [0, vertexCount] -> [0, 2PI[; discontinuous around 0 rad; will produce noise values discontinued around 0 rad;
          oppositeAngle : (angle+Math.PI)%_2PI,	// [0, vertexCount] -> [PI, ..., 2PI, 0, ..., PI[; discontinuous around v = 0; will produce noise values discontinued around PI;
          continuityCoef: (Math.cos(angle+Math.PI)+1)/2,	// allows to produce a continous noise from 'angle'-based noise and 'oppositeAngle'-based noise; allows to eliminate discontinuities around 0 rad and PI;
          cos: Math.cos(angle),
          sin: Math.sin(angle),
          angleNoise: 0,											// 'angle'-based noise; updated at each frame
          oppositeAngleNoise: 0								// 'opppositeAngle'-based noise; updated at each frame
        };
      }
    }
    
    function updateWave(elapsed) {
      var timeBasedChange = elapsed/timeBasedLength;
      var vertex;
      //begin: update each waves's shape using SimplexNoise
      for (var v=0; v<vertexCount; v++) {
        vertex = wave[v];
        vertex.angleNoise = noiseStrength*simplex.noise2D(vertex.angle/angleBasedLength, timeBasedChange);
        vertex.oppositeAngleNoise = noiseStrength*simplex.noise2D(vertex.oppositeAngle/angleBasedLength, timeBasedChange+10);
      }
      //end: update each waves's shape using SimplexNoise
    }
    
    function redrawRadialWave() {
      var noisedRadius;
      
      radialWaveContext.setTransform(1, 0, 0, 1, 0, 0);
      radialWaveContext.translate(midWidth, midHeight);
      
      //begin: delete existing image
      radialWaveContext.clearRect(-width/2, -height/2, width, height)
      //begin: delete existing image
      
      //begin: draw reference circles
      radialWaveContext.strokeStyle = 'lightgrey';
      radialWaveContext.beginPath();
      radialWaveContext.arc(0, 0, waveRadius, 0, _2PI);
      radialWaveContext.stroke();
      //end: draw reference circles
      
      //begin: insert new wave's shape
      radialWaveContext.strokeStyle = 'grey';
      radialWaveContext.beginPath();
      for (var v=0; v<vertexCount; v++) {
        noisedRadius = (waveRadius+wave[v].continuityCoef*wave[v].angleNoise+(1-wave[v].continuityCoef)*wave[v].oppositeAngleNoise);
        radialWaveContext.lineTo(
          noisedRadius*wave[v].cos,
          noisedRadius*wave[v].sin
        );
      }
      radialWaveContext.closePath();
      radialWaveContext.stroke();
      //end: insert new wave's shape
    }
    
    function redrawContinuousRadialNoises() {
      var noisedRadius;
      
      //begin: delete existing image
      continuousRadialNoisesContext.setTransform(1, 0, 0, 1, 0, 0);
      continuousRadialNoisesContext.clearRect(0, 0, width, height)
      //begin: delete existing image

//begin: draw reference circles
      continuousRadialNoisesContext.strokeStyle = 'lightgrey';
      continuousRadialNoisesContext.beginPath();
      continuousRadialNoisesContext.arc(firstWaveXAlign, midHeight, waveRadius, 0, _2PI);
      continuousRadialNoisesContext.moveTo(secondWaveXAlign+waveRadius, midHeight);
      continuousRadialNoisesContext.arc(secondWaveXAlign, midHeight, waveRadius, 0, _2PI);
      continuousRadialNoisesContext.stroke();
      //end: draw reference circles

//begin: draw first radial noise
      continuousRadialNoisesContext.strokeStyle = 'grey';
      continuousRadialNoisesContext.setTransform(1, 0, 0, 1, 0, 0);
      continuousRadialNoisesContext.translate(firstWaveXAlign, midHeight);
      continuousRadialNoisesContext.beginPath();
      for (var v=0; v<vertexCount; v++) {
        noisedRadius = waveRadius+wave[v].continuityCoef*wave[v].angleNoise;
        continuousRadialNoisesContext.lineTo(
          noisedRadius*wave[v].cos,
          noisedRadius*wave[v].sin
        );
      }
      continuousRadialNoisesContext.stroke();
      
      //begin: draw junction of first radial noise
      continuousRadialNoisesContext.strokeStyle = 'red';
      continuousRadialNoisesContext.beginPath();
      for (var v=vertexCount-1; v<vertexCount+1; v++) {
        noisedRadius = waveRadius+wave[v%vertexCount].continuityCoef*wave[v%vertexCount].angleNoise;
        continuousRadialNoisesContext.lineTo(
          noisedRadius*wave[v%vertexCount].cos,
          noisedRadius*wave[v%vertexCount].sin
        );
      }
      continuousRadialNoisesContext.closePath();
      continuousRadialNoisesContext.stroke();
      //end: draw junction of first radial noise
      //end: draw first radial noise
      
      //begin: draw second radial noise
      continuousRadialNoisesContext.strokeStyle = 'grey';
      continuousRadialNoisesContext.setTransform(1, 0, 0, 1, 0, 0);
      continuousRadialNoisesContext.translate(secondWaveXAlign, midHeight);
      continuousRadialNoisesContext.beginPath();
      for (var v=0; v<vertexCount; v++) {
        noisedRadius = waveRadius+(1-wave[v].continuityCoef)*wave[v].oppositeAngleNoise;
        continuousRadialNoisesContext.lineTo(
          noisedRadius*wave[v].cos,
          noisedRadius*wave[v].sin
        );
      }
      continuousRadialNoisesContext.closePath();
      continuousRadialNoisesContext.stroke();
      
      //begin: draw junction of second radial noise
      continuousRadialNoisesContext.strokeStyle = 'red';
      continuousRadialNoisesContext.beginPath();
      for (var v=vertexCount/2-1; v<vertexCount/2+1; v++) {
        noisedRadius = waveRadius+(1-wave[v].continuityCoef)*wave[v].oppositeAngleNoise;
        continuousRadialNoisesContext.lineTo(
          noisedRadius*wave[v].cos,
          noisedRadius*wave[v].sin
        );
      }
      continuousRadialNoisesContext.stroke();
      //end: draw junction of second radial noise
      //end: draw second radial noise
    }
    
    function redrawRadialNoises() {
      var noisedRadius;
      
      //begin: delete existing image
      radialNoisesContext.setTransform(1, 0, 0, 1, 0, 0);
      radialNoisesContext.clearRect(0, 0, width, height)
      //begin: delete existing image

//begin: draw reference circles
      radialNoisesContext.strokeStyle = 'lightgrey';
      radialNoisesContext.beginPath();
      radialNoisesContext.arc(firstWaveXAlign, midHeight, waveRadius, 0, _2PI);
      radialNoisesContext.moveTo(secondWaveXAlign+waveRadius, midHeight);
      radialNoisesContext.arc(secondWaveXAlign, midHeight, waveRadius, 0, _2PI);
      radialNoisesContext.stroke();
      //end: draw reference circles
      
      //begin: draw first radial noise
      radialNoisesContext.strokeStyle = 'grey';
      radialNoisesContext.setTransform(1, 0, 0, 1, 0, 0);
      radialNoisesContext.translate(firstWaveXAlign, midHeight);
      radialNoisesContext.beginPath();
      for (var v=0; v<vertexCount; v++) {
        noisedRadius = waveRadius+wave[v].angleNoise;
        radialNoisesContext.lineTo(
          noisedRadius*wave[v].cos,
          noisedRadius*wave[v].sin
        );
      }
      radialNoisesContext.stroke();
      
      //begin: draw junction of first radial noise
      radialNoisesContext.strokeStyle = 'red';
      radialNoisesContext.beginPath();
      for (var v=vertexCount-1; v<vertexCount+1; v++) {
        noisedRadius = waveRadius+wave[v%vertexCount].angleNoise;
        radialNoisesContext.lineTo(
          noisedRadius*wave[v%vertexCount].cos,
          noisedRadius*wave[v%vertexCount].sin
        );
      }
      radialNoisesContext.stroke();
      //end: draw junction of first radial noise
      //end: draw first radial noise
      
      //begin: draw second radial noise
      radialNoisesContext.strokeStyle = 'grey';
      radialNoisesContext.setTransform(1, 0, 0, 1, 0, 0);
      radialNoisesContext.translate(secondWaveXAlign, midHeight);
      radialNoisesContext.beginPath();
      for (var v=0; v<vertexCount; v++) {
        noisedRadius = waveRadius+wave[v].oppositeAngleNoise;
        radialNoisesContext.lineTo(
          noisedRadius*wave[v].cos,
          noisedRadius*wave[v].sin
        );
      }
      radialNoisesContext.closePath();
      radialNoisesContext.stroke();
      
      //begin: draw junction of second radial noise
      radialNoisesContext.strokeStyle = 'red';
      radialNoisesContext.beginPath();
      for (var v=vertexCount/2-1; v<vertexCount/2+1; v++) {
        noisedRadius = waveRadius+wave[v].oppositeAngleNoise;
        radialNoisesContext.lineTo(
          noisedRadius*wave[v].cos,
          noisedRadius*wave[v].sin
        );
      }
      radialNoisesContext.stroke();
      //end: draw junction of second radial noise
      //end: draw second radial noise
    }
    
    function redrawNoises() {
      //begin: delete existing image
      noisesContext.setTransform(1, 0, 0, 1, 0, 0);
      noisesContext.clearRect(0, 0, width, height);
      //begin: delete existing image
      
      //begin: draw reference axes
      noisesContext.strokeStyle = 'lightgrey';
      noisesContext.beginPath();
      noisesContext.moveTo(firstWaveXAlign+noiseScale(0), midHeight);
      noisesContext.lineTo(firstWaveXAlign+noiseScale(3*Math.PI), midHeight);
      noisesContext.moveTo(secondWaveXAlign+noiseScale(0), midHeight);
      noisesContext.lineTo(secondWaveXAlign+noiseScale(3*Math.PI), midHeight);
      noisesContext.stroke();
      
      var textY = 3*midHeight/2;
      noisesContext.fillStyle = 'lightgrey';
      noisesContext.textAlign = "start";
			noisesContext.fillText("0", firstWaveXAlign+noiseScale(0), textY);
			noisesContext.fillText("0", secondWaveXAlign+noiseScale(0), textY);
      noisesContext.textAlign = "center";
      noisesContext.fillText("π", firstWaveXAlign+noiseScale(Math.PI), textY);
      noisesContext.fillText("π", secondWaveXAlign+noiseScale(Math.PI), textY);
      noisesContext.fillText("2π", firstWaveXAlign+noiseScale(_2PI), textY);
      noisesContext.fillText("2π", secondWaveXAlign+noiseScale(_2PI), textY);
      noisesContext.textAlign = "end";
      noisesContext.fillText("3π", firstWaveXAlign+noiseScale(_3PI), textY);
      noisesContext.fillText("3π", secondWaveXAlign+noiseScale(_3PI), textY);
      //end: draw reference axes
			
      //begin: draw first noise
      noisesContext.setTransform(1, 0, 0, 1, 0, 0);
      noisesContext.translate(firstWaveXAlign, midHeight);
      
      //begin: draw junction of first noise
      noisesContext.strokeStyle = 'pink';
      noisesContext.beginPath();
      for (var v=vertexCount-1; v<vertexCount+1; v++) {
        noisesContext.lineTo(
          noiseScale(wave[v%vertexCount].angle),
          wave[v%vertexCount].angleNoise
        );
      }
      noisesContext.stroke();
      //end: draw junction of first noise
      
      noisesContext.strokeStyle = 'grey';
      noisesContext.beginPath();
      for (var v=0; v<vertexCount; v++) {
        noisesContext.lineTo(
          noiseScale(wave[v].angle),
          wave[v].angleNoise
        );
      }
      noisesContext.stroke();
      //end: draw first noise
      
      //begin: draw second noise
      noisesContext.setTransform(1, 0, 0, 1, 0, 0);
      noisesContext.translate(secondWaveXAlign, midHeight);
      
      //begin: draw junction of second noise
      noisesContext.strokeStyle = 'pink';
      noisesContext.beginPath();
      for (var v=vertexCount/2-1; v<vertexCount/2+1; v++) {
        noisesContext.lineTo(
          noiseScale(Math.PI+wave[v%vertexCount].oppositeAngle),
          wave[v%vertexCount].oppositeAngleNoise
        );
      }
      noisesContext.stroke();
      //end: draw junction of second noise
      
      noisesContext.strokeStyle = 'grey';
      noisesContext.beginPath();
      for (var v=vertexCount/2; v<vertexCount; v++) {
        noisesContext.lineTo(
          noiseScale(Math.PI+wave[v].oppositeAngle),
          wave[v].oppositeAngleNoise
        );
      }
      for (var v=0; v<vertexCount/2; v++) {
        noisesContext.lineTo(
          noiseScale(Math.PI+wave[v].oppositeAngle),
          wave[v].oppositeAngleNoise
        );
      }
      noisesContext.stroke();
      //end: draw second noise
    }
  </script>
</body>
</html>

https://d3js.org/d3.v4.min.js