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Enabling the back and forward buttons

Building on information presented in Adding Mandelbrot image grayscale, we describe here how to enable the browser's back and forward buttons.

The technique used to enable the browser's back and forward buttons is to get and set the page's URL hash string, as in:


http://samples.msdn.microsoft.com/Workshop/samples/mandelbrot/mandelbrotExplorer.html#0.722,-0.178,0.846,0.246,1

In general, whenever the URL changes, the browser's back and forward buttons become enabled. Thus, each unique Mandelbrot image (page) will have a unique hash string.

The hash string, #0.722,-0.178,0.846,0.246,1, can be interpreted as follows:

  • globals.reMax = 0.722
  • globals.reMin = -0.178
  • globals.ImMax = 0.846
  • globals.ImMin = 0.246
  • globals.grayscaleFactor = 1

We'll discuss this in more detail later, but for now, globals.grayscaleFactor indicates how much to lighten a given image (a value of 1 has no effect).

Mandelbrot 4 is similar to Mandelbrot 5, as shown next:

Mandelbrot 5


<!DOCTYPE html>
<html>

<head>
  <meta http-equiv="X-UA-Compatible" content="IE=10" />
  <meta content="text/html; charset=utf-8" http-equiv="Content-Type">
  <title>Mandelbrot 5</title>
  <style>
    html, body {
      margin: 0;
      padding: 0;
      text-align: center;
    }
    
    canvas {
      border: 2px solid black;
    }    
    
    table {
      margin: 0 auto; /* Center the table. */
    }
    
    #messageBox {
      text-align: left;
    }
        
    #elapsedTime {
      width: 23em;
      text-align: right;
    }    

    button {
      width: 5em;
    }
    
    #filenameForm {
      visibility: hidden; /* As opposed to "display: none", keep room for this hidden element in the layout. */
    }
}
  </style>      
</head>

<body>
  <h1>Mandelbrot 5</h1>
  <p>This example demonstrates how to enable the browser's back and forward buttons.</p>      
  <table>
    <tr>
      <td id="messageBox"></td>
      <td id="elapsedTime"></td>
    </tr>
  </table>
  <canvas width="600" height="400" oncontextmenu="return false;"> <!-- Because the hold gesture event can fire more than once, the 'oncontextmenu="return false;"' attribute is used to stop the right-click context menu from appearing inappropriately. -->
    Canvas not supported - upgrade your browser (after checking that your browser is in the correct mode).
  </canvas><br>
  <button type="button" id="resetButton">Reset</button>  
  <button type="button" id="lightenButton">Lighten</button>    
  <button type="button" id="saveButton">Save</button>
  <form id="filenameForm"> 
    Extensionless filename: <input id="filename" type="text"> <input type="submit" value="Submit">
  </form>
  <script>
    var RE_MAX = 1.1; // This value will be adjusted as necessary to ensure that the rendered Mandelbrot set is never skewed (that is, true to it's actual shape).
    var RE_MIN = -2.5;
    var IM_MAX = 1.2;
    var IM_MIN = -1.2;
    var MAX_ITERATIONS = 1200; // Increase this value to improve detection of complex c values that belong to the Mandelbrot set.
    var STATIC_ZOOM_BOX_FACTOR = 0.25; // Increase to make the double-click and hold gesture zoom box bigger.
    var DEFAULT_MESSAGE = "Click or click-and-drag to zoom."
      
      
      var globals = {}; // See the handleLoad function.
      
      window.addEventListener('load', handleLoad, false);
                
    /************************************************************************************************************************************************************/
    
    Number.prototype.format = function() {
    /* 
      Formats this integer so that it has commas in the expected places.
    */
    	var numberString = Math.round(this).toString(); // An integer value is assumed, so we ensure that it is indeed an integer.
    	var precompiledRegularExpression = /(\d+)(\d{3})/;
    	
    	while ( precompiledRegularExpression.test(numberString) ) {
    		numberString = numberString.replace(precompiledRegularExpression, '$1' + ',' + '$2'); // For this integer, inject ","'s at the appropriate locations.
    	} // while
    	
    	return numberString;
    } // Number.prototype.format

    /*----------------------------------------------------------------------------------------------------------------------------------------------------------*/    

    function handleLoad() {          
      var canvas = document.getElementsByTagName('canvas')[0];
      var canvasWidth = canvas.width;
      var canvasHeight = canvas.height;      
      var ctx = canvas.getContext('2d');
      
      document.getElementsByTagName('table')[0].width = canvasWidth; // Make the table's width the same as the canvas's width. 
      document.getElementById('messageBox').innerHTML = DEFAULT_MESSAGE;            

      globals.canvas = canvas;
      globals.canvas.context = ctx;
      globals.canvas.context.imageDataObject = ctx.createImageData(canvasWidth, canvasHeight); // Create an appropriately sized but empty canvas image data object.
      
      globals.staticZoomBoxWidth = STATIC_ZOOM_BOX_FACTOR * canvasWidth; // Maintains the original canvas width/height ratio.
      globals.staticZoomBoxHeight = STATIC_ZOOM_BOX_FACTOR * canvasHeight; // Maintains the original canvas width/height ratio.      
      
      globals.pointer = {};
      globals.pointer.down = false;  
                 
      window.addEventListener('hashchange', handleHashChange, false); // This event handler executes whenever the URL hash string changes.
      
      canvas.addEventListener('mousedown', handlePointer, false);
      canvas.addEventListener('mousemove', handlePointer, false);
      canvas.addEventListener('mouseup', handlePointer, false);    
            
      document.getElementById('resetButton').addEventListener('click', handleResetButton, false);
      document.getElementById('lightenButton').addEventListener('click', handleLightenButton, false);    
      document.getElementById('saveButton').addEventListener('click', handleSaveButton, false);        
      document.getElementById('filenameForm').addEventListener('submit', handleFormSubmit, false);    
      
      ctx.fillStyle = "rgba(255, 0, 0, 0.3)"; // The color and opacity of the zoom box. This is what gets saved when calling ctx.save().          
 
      handleHashChange();
    } // handleLoad
    
    /*----------------------------------------------------------------------------------------------------------------------------------------------------------*/    
    
    function handleHashChange() {
      var hashValues = getHashValues(); // This function examines window.location.hash but doesn't change it.
      
      if (hashValues) {
        globals.ReMax = hashValues.ReMax;
        globals.ReMin = hashValues.ReMin;
        globals.ImMax = hashValues.ImMax;
        globals.ImMin = hashValues.ImMin;
        globals.grayscaleFactor = hashValues.grayscaleFactor;
      }
      else {
        globals.ReMax = adjusted_RE_MAX();
        globals.ReMin = RE_MIN;
        globals.ImMax = IM_MAX;
        globals.ImMin = IM_MIN;     
        globals.grayscaleFactor = 1; // Multiplying any value by 1 has no effect.
      } // if-else
      
      drawMandelbrot(globals.ReMax, globals.ReMin, globals.ImMax, globals.ImMin, globals.grayscaleFactor);
    } // handelHashChange    

    /*----------------------------------------------------------------------------------------------------------------------------------------------------------*/        

    function getHashValues() {
      var dirtyComplexPlaneExtremaString = (window.location.hash).replace('#', ''); // Remove the leading "#" character from the string.
      var complexPlaneExtremaString = dirtyComplexPlaneExtremaString.split(','); // Returns an array. Assumes the following string form: "ReMax,ReMin,ImMax,ImMin,grayscaleFactor" (note that if grayscaleFactor is 1, the image's grayscale is not affected).
      
      var ReMax = parseFloat( complexPlaneExtremaString[0] ); 
      var ReMin = parseFloat( complexPlaneExtremaString[1] ); 
      var ImMax = parseFloat( complexPlaneExtremaString[2] ); 
      var ImMin = parseFloat( complexPlaneExtremaString[3] );
      var grayscaleFactor = parseFloat( complexPlaneExtremaString[4] );
      
      if ( isNaN(ReMax) || isNaN(ReMin) || isNaN(ImMax) || isNaN(ImMin) || isNaN(grayscaleFactor) ) { 
        return null;
      } // if 
      
      return {ReMax: ReMax, ReMin: ReMin, ImMax: ImMax, ImMin: ImMin, grayscaleFactor: grayscaleFactor};
    } // getHashValues
        
    /*----------------------------------------------------------------------------------------------------------------------------------------------------------*/        

    function adjusted_RE_MAX() {    
      var ReMax = globals.canvas.width * ( (IM_MAX - IM_MIN) / globals.canvas.height ) + RE_MIN;
      
      if (RE_MAX != ReMax) {
        alert("RE_MAX has been adjusted to: " + ReMax); // The user should never see this if RE_MAX is set correctly above.
      } // if

      return ReMax;
    } // adjusted_RE_MAX    
    
    /*----------------------------------------------------------------------------------------------------------------------------------------------------------*/    
    
    function drawMandelbrot(ReMax, ReMin, ImMax, ImMin, grayscaleFactor) {            
      document.getElementById('messageBox').innerHTML = "Calculating..."; // This isn't displayed until the drawMandelbrot function block exits. 
      document.getElementById('elapsedTime').innerHTML = ""; // Erase the prior run's statistics.           
      
      if (window.setImmediate) {
        window.setImmediate(calculateMandelbrot); // Allow the drawMandelbrot function to immediately terminate, thus printing "Calculating..." to the screen.       
      }
      else {
        window.setTimeout(calculateMandelbrot, 0); // Allow the drawMandelbrot function to immediately terminate, thus printing "Calculating..." to the screen.      
      }           
      
      function calculateMandelbrot() {
        var startTime = new Date(); // Report how long it takes to render this particular region of the Mandelbrot set.             
        
        var canvas = globals.canvas;
        var canvasWidth = canvas.width;
        var canvasHeight = canvas.height;
        var ctx = canvas.context;
        var imageDataObjectData = ctx.imageDataObject.data; // imageDataObjectData is a reference to, not a copy of, ctx.imageDataObject.data
        
        var maxPixelGrayscaleValue = 0; // This will contain the lightest shade of gray in the drawn Mandelbrot image.
        
        var x_coefficient = (ReMax - ReMin) / canvasWidth; // Keep the below loops as computation-free as possible.
        var y_coefficient = (ImMin - ImMax) / canvasHeight; // Keep the below loops as computation-free as possible.
  
        var iterationSum = 0;
        var currentPixel = 0;          
        for (var y = 0; y < canvasHeight; y++) {
          var c_Im = (y * y_coefficient) + ImMax; // Note that c = c_Re + c_Im*i
          
          for (var x = 0; x < canvasWidth; x++) {
            var c_Re = (x * x_coefficient) + ReMin // Convert the canvas x-coordinate to a complex plane Re-coordinate. c_Re represents the real part of a c value.
            
            var z_Re = 0; // The first z value (Zo) must be 0.
            var z_Im = 0; // The first z value (Zo) must be 0. Note that z = z_Re + z_Im*i
            
            var c_belongsToMandelbrotSet = true; // Assume that the c associated with Zn belongs to the Mandelbrot set (i.e., Zn remains bounded under iteration of Zn+1 = (Zn)^2 + c).
            var exponentialSmoothingSum = 0;
            for (var iterationCount = 1; iterationCount <= MAX_ITERATIONS; iterationCount++) {
              iterationSum++; // Keep track of how many iterations were performed in total so we can report this to the user.
            
              var z_Re_squared = z_Re * z_Re; // A small speed optimization.
              var z_Im_squared = z_Im * z_Im; // A small speed optimization.
              
              exponentialSmoothingSum += Math.exp( -(z_Re_squared + z_Im_squared) ); // Technically, this should be e^(-|z|). However, avoiding the expensive square root operation doesn't really affect the resulting image.              
              if (exponentialSmoothingSum >= 255) { // Don't cycle through the gray colors.
                exponentialSmoothingSum = 255;
              }
      
              if (z_Re_squared + z_Im_squared > 4) { // Checks if |z^2| is greater than 2. This approach avoids the expensive square root operation.
                c_belongsToMandelbrotSet = false; // This complex c value is not part of the Mandelbrot set (because it will always tend towards infinity under iteration).
                break; // Immediately check the next c value to see if it belongs to the Mandelbrot set or not.
              } // if
              
              // The next two lines perform Zn+1 = (Zn)^2 + c (recall that (x + yi)^2 = x^2 - y^2 + 2xyi, thus the real part is x^2 - y^2 and the imaginary part is 2xyi).
              z_Im = (2 * z_Re * z_Im) + c_Im; // We must calculate the next value of z_Im first because it depends on the current value of z_Re (not the next value of z_Re).
              z_Re = z_Re_squared - z_Im_squared + c_Re; // Calculate the next value of z_Re.
            } // for   
            
            if (c_belongsToMandelbrotSet) { // This complex c value is probably part of the Mandelbrot set because Zn did not tend toward infinity within MAX_ITERATIONS iterations.
              imageDataObjectData[currentPixel++] = 0; // Red. Note that there are 255 possible shades of red, green, blue, and alpha (i.e., opacity).
              imageDataObjectData[currentPixel++] = 0; // Green.
              imageDataObjectData[currentPixel++] = 0; // Blue.
              imageDataObjectData[currentPixel++] = 255; // Alpha (i.e., 0% transparency).
            } 
            else { // This complex c value is definitely not part of the Mandelbrot set because Zn would tend toward infinity under iteration (i.e., |Zn| > 2).
              var pixelGrayscaleValue = 255 - exponentialSmoothingSum % 256; // Force the value of exponentialSmoothingSum to be between 0 and 255 inclusively. Note that all values for red, green, and blue are identical when using a grayscale.
              var adjustedPixelGrayscaleValue = pixelGrayscaleValue * grayscaleFactor; // Avoids doing this more than once.
              
              imageDataObjectData[currentPixel++] = adjustedPixelGrayscaleValue; // Because we mod by 256, the value of exponentialSmoothingSum will always be between 0 and 255.
              imageDataObjectData[currentPixel++] = adjustedPixelGrayscaleValue; // If exponentialSmoothingSum is 255 (it's maximum possible value), then 255 % 256 = 255.
              imageDataObjectData[currentPixel++] = adjustedPixelGrayscaleValue; // When exponentialSmoothingSum is 255, we have 255 - 255 = 0, so the shade values for RGB are all set to 0 (that is, the c-value pixel is rendered black - indicating that this particular c-value very slowly tends towards infinity).
              imageDataObjectData[currentPixel++] = 255; // Always draw the c-value pixels with no transparency.
              
              if (pixelGrayscaleValue > maxPixelGrayscaleValue) {
                maxPixelGrayscaleValue = pixelGrayscaleValue; // Determine the lightest shade of gray in case the user clicks the Lighten button.
              } // if
            } // if-else
          } // for
        } // for        
        
        globals.maxPixelGrayscaleValue = maxPixelGrayscaleValue; // Store the lightest shade of gray in case the user clicks the Lighten button.      
        ctx.putImageData(ctx.imageDataObject, 0, 0); // Render our carefully constructed canvas image data array to the canvas.
            
        var elapsedMilliseconds = (new Date()) - startTime;
        document.getElementById('elapsedTime').innerHTML = iterationSum.format() + " iterations in " + (elapsedMilliseconds / 1000).toFixed(2) + " seconds"; // Note that the UI element is not updated until after this block terminates (which is the desired behavior).            
        document.getElementById('messageBox').innerHTML = DEFAULT_MESSAGE; // Erase the "Calculating..." message and replace it with the default message.        
      } // calculateMandelbrot
    } // drawMandelbrot
    
    /*----------------------------------------------------------------------------------------------------------------------------------------------------------*/
    
    function xToRe(x) {
      var x_coefficient = (globals.ReMax - globals.ReMin) / globals.canvas.width; 
      
      return (x * x_coefficient) + globals.ReMin; // Converts a canvas x-coordinate value to the associated complex plane Re-coordinate.
    } // xToRe
    
    /*----------------------------------------------------------------------------------------------------------------------------------------------------------*/    

    function yToIm(y) {
      var y_coefficient = (globals.ImMin - globals.ImMax) / globals.canvas.height; 
      
      return (y * y_coefficient) + globals.ImMax; // Converts a canvas y-coordinate value to the associated complex plane Im-coordinate.
    } // yToIm
    
    /*----------------------------------------------------------------------------------------------------------------------------------------------------------*/

    function handlePointer(evt) {
      var canvasWidthHeightRatio = globals.canvas.width / globals.canvas.height;
      var ctx = globals.canvas.context;
      
      var canvasX;
      var canvasY;      
      
      if (evt.offsetX && evt.offsetY) {
        canvasX = evt.offsetX; // Not supported in Firefox.
        canvasY = evt.offsetY; // Does not assume that the canvas element is a direct descendent of the body element.
      } else {
        canvasX = evt.clientX - evt.target.offsetLeft; // Supported in Firefox.
        canvasY = evt.clientY - evt.target.offsetTop; // Assumes that the canvas element is a direct descendent of the body element.
      } // if-else
      
      var zoomBoxWidth;
      var zoomBoxHeight;
      
      var ReMax;
      var ReMin;
      var ImMax;
      var ImMin;
      
      var staticZoomBoxWidth = globals.staticZoomBoxWidth;
      var staticZoomBoxHeight = globals.staticZoomBoxHeight;
      var halfStaticZoomBoxWidth = staticZoomBoxWidth / 2;xx
      var halfStaticZoomBoxHeight = staticZoomBoxHeight / 2;
      
      switch (evt.type) {
        case 'mousedown':
          globals.pointer.down = true;      
          globals.pointer.x1 = canvasX;
          globals.pointer.y1 = canvasY;
          break;
        case 'mousemove':
          if (globals.pointer.down) {
            zoomBoxHeight = Math.abs(canvasY - globals.pointer.y1);  
            zoomBoxWidth = zoomBoxHeight * canvasWidthHeightRatio; // We must keep the zoom box dimensions proportional to the canvas dimensions in order to ensure that the resulting zoomed Mandelbrot image does not become skewed.
            ctx.putImageData(ctx.imageDataObject, 0, 0); // Assumes that an initial image of the Mandelbrot set is drawn before we get to this point in the code. The purpose of this line is to erase the prior zoom box rectangle before drawing the next zoom box rectangle.
            ctx.fillRect(globals.pointer.x1, globals.pointer.y1, zoomBoxWidth, zoomBoxHeight); // With a freshly painted image of the current Mandelbrot set in place (see prior line), draw a new zoom box rectangle.
          }
          break;
        case 'mouseup':
          globals.pointer.down = false;          
          
          zoomBoxHeight = Math.abs(canvasY - globals.pointer.y1); // Only allow the zoom box to be drawn from an upper-left corner position down to a lower-right corner position.
          zoomBoxWidth = zoomBoxHeight * canvasWidthHeightRatio; // Again, ensure that the width/height ratio of the zoom box is proportional to the canvas's (this simplifies the algorithm).          
          
          if (zoomBoxHeight == 0) { // No zoom box has been drawn, so honor the fixed sized zoom box.          
            ctx.fillRect(canvasX - halfStaticZoomBoxWidth, canvasY - halfStaticZoomBoxHeight, staticZoomBoxWidth, staticZoomBoxHeight); // Just leave this on the screen.
                         
            ReMin = xToRe(canvasX - halfStaticZoomBoxWidth); // Center the static zoom box about the point (evt.offsetX, evt.offsetY).
            ImMax = yToIm(canvasY - halfStaticZoomBoxHeight); 
            
            ReMax = xToRe(canvasX + halfStaticZoomBoxWidth);
            ImMin = yToIm(canvasY + halfStaticZoomBoxHeight);
          } 
          else { // A (possibly tiny) zoom box has been drawn, so honor it.
            ReMin = xToRe(globals.pointer.x1); // Convert the mouse's x-coordinate value (on the canvas) to the associated Re-coordinate value in the complex plane.
            ImMax = yToIm(globals.pointer.y1); // Convert the mouse's y-coordinate value (on the canvas) to the associated Im-coordinate value in the complex plane.
                                      
            ReMax = xToRe(zoomBoxWidth + globals.pointer.x1); // Convert the zoom box's final x-coordinate value to the associated Re-coordinate value in the complex plane.  
            ImMin = yToIm(zoomBoxHeight + globals.pointer.y1);  // Convert the zoom box's final y-coordinate value to the associated Re-coordinate value in the complex plane.            
          } // if-else        
        
          window.location.hash = ReMax + "," + ReMin + "," + ImMax + "," + ImMin + "," + globals.grayscaleFactor; // This triggers the handleHashChange event handler which, among other things, is responsible for drawing the Mandelbrot set.
          break; 
        default:
          alert("Error in switch statement."); // Although unnecessary, defensive programming techniques such as this are highly recommended.
      } // switch              
    } // handlePointer    
    
    /*----------------------------------------------------------------------------------------------------------------------------------------------------------*/
    
    function handleResetButton() {
      window.location.hash = adjusted_RE_MAX() + "," + RE_MIN + "," + IM_MAX + "," + IM_MIN + "," + 1; // // This triggers the handleHashChange event handler which, among other things, is responsible for drawing the Mandelbrot set.
    } // handleResetButton
    
    /*----------------------------------------------------------------------------------------------------------------------------------------------------------*/
    
    function handleLightenButton() {
    /* 
      This multiplies by a value (factor) such that black (0) stays black and the lightest gray value in the image becomes white (255). Thus, clicking the 
      Lighten button removes much of the mathematical meaning of the (proper) grayscale but can make "dark" images more visible.
    */
      var grayscaleFactor = 255 / globals.maxPixelGrayscaleValue; // For the canvas element, 255 is white, 0 is black.

      window.location.hash = globals.ReMax + "," + globals.ReMin + "," + globals.ImMax + "," + globals.ImMin + "," + grayscaleFactor; // This invokes handleHashChange which, among other things, is responsibile for drawing the Mandelbrot set.
    } // handleResetButton
    
    /*----------------------------------------------------------------------------------------------------------------------------------------------------------*/

    function handleSaveButton() {
      document.getElementById('filenameForm').style.visibility = "visible";
      document.getElementById('filename').focus(); // Place the cursor in the filename text input box.
    } // handleResetButton 
 
    /*----------------------------------------------------------------------------------------------------------------------------------------------------------*/

    function handleFormSubmit(evt) {
      alert("handleFormSubmit fired.");    
      evt.preventDefault(); // Do not refresh the page when the submit button is clicked.
      document.getElementById('filenameForm').style.visibility = "hidden";
    } // handleFormSubmit
  </script>
</body>

</html>

The main difference between Mandelbrot 4 and Mandelbrot 5 is that the handleHashChange event handler (which is called whenever the page's URL changes) triggers the drawing of the Mandelbrot set:


function handleHashChange() {
  var hashValues = getHashValues(); // This function examines window.location.hash but doesn't change it.
  
  if (hashValues) {
    globals.ReMax = hashValues.ReMax;
    globals.ReMin = hashValues.ReMin;
    globals.ImMax = hashValues.ImMax;
    globals.ImMin = hashValues.ImMin;
    globals.grayscaleFactor = hashValues.grayscaleFactor;
  }
  else {
    globals.ReMax = adjusted_RE_MAX();
    globals.ReMin = RE_MIN;
    globals.ImMax = IM_MAX;
    globals.ImMin = IM_MIN;     
    globals.grayscaleFactor = 1; // Multiplying any value by 1 has no effect.
  } // if-else
  
  drawMandelbrot(globals.ReMax, globals.ReMin, globals.ImMax, globals.ImMin, globals.grayscaleFactor);
} // handelHashChange

Thus, to display a new image of the Mandelbrot set, we need only change the page's URL hash string by updating window.location.hash. This initially happens when the handleLoad event handler directly calls the handleHashChange event handler (see last line):


function handleLoad() {          
  var canvas = document.getElementsByTagName('canvas')[0];
  var canvasWidth = canvas.width;
  var canvasHeight = canvas.height;      
  var ctx = canvas.getContext('2d');
  
  document.getElementsByTagName('table')[0].width = canvasWidth;
  document.getElementById('messageBox').innerHTML = DEFAULT_MESSAGE;            

  globals.canvas = canvas;
  globals.canvas.context = ctx;
  globals.canvas.context.imageDataObject = ctx.createImageData(canvasWidth, canvasHeight);
  
  globals.staticZoomBoxWidth = STATIC_ZOOM_BOX_FACTOR * canvasWidth; 
  globals.staticZoomBoxHeight = STATIC_ZOOM_BOX_FACTOR * canvasHeight; 
  
  globals.pointer = {};
  globals.pointer.down = false;  
             
  window.addEventListener('hashchange', handleHashChange, false); // This event handler executes whenever the URL hash string changes.
  
  canvas.addEventListener('mousedown', handlePointer, false);
  canvas.addEventListener('mousemove', handlePointer, false);
  canvas.addEventListener('mouseup', handlePointer, false);    
        
  document.getElementById('resetButton').addEventListener('click', handleResetButton, false);
  document.getElementById('lightenButton').addEventListener('click', handleLightenButton, false);    
  document.getElementById('saveButton').addEventListener('click', handleSaveButton, false);        
  document.getElementById('filenameForm').addEventListener('submit', handleFormSubmit, false);    
  
  ctx.fillStyle = "rgba(255, 0, 0, 0.3)";

  handleHashChange(); // On page load, simulate a page URL change to draw the initial Mandelbrot set.
} // handleLoad

The getHashValues function is used to convert the current hash string to usable numeric values:


function getHashValues() {
  var dirtyComplexPlaneExtremaString = (window.location.hash).replace('#', ''); // Remove the leading "#" character from the string.
  var complexPlaneExtremaString = dirtyComplexPlaneExtremaString.split(','); // Returns an array. Assumes the following string form: "ReMax,ReMin,ImMax,ImMin,grayscaleFactor" (note that if grayscaleFactor is 1, the image's grayscale is not effected).
  
  var ReMax = parseFloat( complexPlaneExtremaString[0] ); 
  var ReMin = parseFloat( complexPlaneExtremaString[1] ); 
  var ImMax = parseFloat( complexPlaneExtremaString[2] ); 
  var ImMin = parseFloat( complexPlaneExtremaString[3] );
  var grayscaleFactor = parseFloat( complexPlaneExtremaString[4] );
  
  if ( isNaN(ReMax) || isNaN(ReMin) || isNaN(ImMax) || isNaN(ImMin) || isNaN(grayscaleFactor) ) { 
    return null;
  } // if 
  
  return {ReMax: ReMax, ReMin: ReMin, ImMax: ImMax, ImMin: ImMin, grayscaleFactor: grayscaleFactor};
} // getHashValues

The first step is to remove the leading "#" character from the hash string. Next, assuming the complex plane extrema are of the form "ReMax,ReMin,ImMax,ImMin,grayscaleFactor", we split them into an array and convert them to numeric values using parseFloat.

If any of the expected values are missing or malformed, null is returned. Otherwise, an object literal is returned containing the complex plane extrema.

The handlePointer event handler is nearly the same as before except that:


setExtrema(ReMax, ReMin, ImMax, ImMin); // Must set these globals prior to calling drawMandelbort because drawMandelbort accesses them.
if (window.setImmediate) {
  window.setImmediate(drawMandelbrot);
}
else {
  window.setTimeout(drawMandelbrot, 0);
}

Has been replaced with:


window.location.hash = ReMax + "," + ReMin + "," + ImMax + "," + ImMin + "," + globals.grayscaleFactor;

As your might have guessed, the setImmediate/setTimeout methods have moved to the drawMandelbrot function:


function drawMandelbrot(ReMax, ReMin, ImMax, ImMin, grayscaleFactor) {            
  document.getElementById('messageBox').innerHTML = "Calculating..."; // This isn't displayed until the drawMandelbrot function block exits. 
  document.getElementById('elapsedTime').innerHTML = ""; // Erase the prior run's statistics.           
  
  if (window.setImmediate) {
    window.setImmediate(calculateMandelbrot); // Allow the drawMandelbrot function to immediately terminate, thus printing "Calculating..." to the screen.       
  }
  else {
    window.setTimeout(calculateMandelbrot, 0); // Allow the drawMandelbrot function to immediately terminate, thus printing "Calculating..." to the screen.      
  }           
  
  function calculateMandelbrot() {
    var startTime = new Date(); // Report how long it takes to render this particular region of the Mandelbrot set.             
    
    var canvas = globals.canvas;
    var canvasWidth = canvas.width;
    var canvasHeight = canvas.height;
    var ctx = canvas.context;
    var imageDataObjectData = ctx.imageDataObject.data; // imageDataObjectData is a reference to, not a copy of, ctx.imageDataObject.data
    
    var maxPixelGrayscaleValue = 0; // This will contain the lightest shade of gray in the drawn Mandelbrot image.
    
    var x_coefficient = (ReMax - ReMin) / canvasWidth; // Keep the below loops as computation-free as possible.
    var y_coefficient = (ImMin - ImMax) / canvasHeight; // Keep the below loops as computation-free as possible.

    var iterationSum = 0;
    var currentPixel = 0;          
    for (var y = 0; y < canvasHeight; y++) {
      var c_Im = (y * y_coefficient) + ImMax; // Note that c = c_Re + c_Im*i
      
      for (var x = 0; x < canvasWidth; x++) {
        var c_Re = (x * x_coefficient) + ReMin // Convert the canvas x-coordinate to a complex plane Re-coordinate. c_Re represents the real part of a c value.
        
        var z_Re = 0; // The first z value (Zo) must be 0.
        var z_Im = 0; // The first z value (Zo) must be 0. Note that z = z_Re + z_Im*i
        
        var c_belongsToMandelbrotSet = true; // Assume that the c associated with Zn belongs to the Mandelbrot set (i.e., Zn remains bounded under iteration of Zn+1 = (Zn)^2 + c).
        var exponentialSmoothingSum = 0;
        for (var iterationCount = 1; iterationCount <= MAX_ITERATIONS; iterationCount++) {
          iterationSum++; // Keep track of how many iterations were performed in total so we can report this to the user.
        
          var z_Re_squared = z_Re * z_Re; // A small speed optimization.
          var z_Im_squared = z_Im * z_Im; // A small speed optimization.
          
          exponentialSmoothingSum += Math.exp( -(z_Re_squared + z_Im_squared) ); // Technically, this should be e^(-|z|). However, avoiding the expensive square root operation doesn't really affect the resulting image.              
          if (exponentialSmoothingSum >= 255) { // Don't cycle through the gray colors.
            exponentialSmoothingSum = 255;
          }
  
          if (z_Re_squared + z_Im_squared > 4) { // Checks if |z^2| is greater than 2. This approach avoids the expensive square root operation.
            c_belongsToMandelbrotSet = false; // This complex c value is not part of the Mandelbrot set (because it will always tend towards infinity under iteration).
            break; // Immediately check the next c value to see if it belongs to the Mandelbrot set or not.
          } // if
          
          // The next two lines perform Zn+1 = (Zn)^2 + c (recall that (x + yi)^2 = x^2 - y^2 + 2xyi, thus the real part is x^2 - y^2 and the imaginary part is 2xyi).
          z_Im = (2 * z_Re * z_Im) + c_Im; // We must calculate the next value of z_Im first because it depends on the current value of z_Re (not the next value of z_Re).
          z_Re = z_Re_squared - z_Im_squared + c_Re; // Calculate the next value of z_Re.
        } // for   
        
        if (c_belongsToMandelbrotSet) { // This complex c value is probably part of the Mandelbrot set because Zn did not tend toward infinity within MAX_ITERATIONS iterations.
          imageDataObjectData[currentPixel++] = 0; // Red. Note that there are 255 possible shades of red, green, blue, and alpha (i.e., opacity).
          imageDataObjectData[currentPixel++] = 0; // Green.
          imageDataObjectData[currentPixel++] = 0; // Blue.
          imageDataObjectData[currentPixel++] = 255; // Alpha (i.e., 0% transparency).
        } 
        else { // This complex c value is definitely not part of the Mandelbrot set because Zn would tend toward infinity under iteration (i.e., |Zn| > 2).
          var pixelGrayscaleValue = 255 - exponentialSmoothingSum % 256; // Force the value of exponentialSmoothingSum to be between 0 and 255 inclusively. Note that all values for red, green, and blue are identical when using a grayscale.
          var adjustedPixelGrayscaleValue = pixelGrayscaleValue * grayscaleFactor; // Avoids doing this more than once.
          
          imageDataObjectData[currentPixel++] = adjustedPixelGrayscaleValue; // Because we mod by 256, the value of exponentialSmoothingSum will always be between 0 and 255.
          imageDataObjectData[currentPixel++] = adjustedPixelGrayscaleValue; // If exponentialSmoothingSum is 255 (its maximum possible value), then 255 % 256 = 255.
          imageDataObjectData[currentPixel++] = adjustedPixelGrayscaleValue; // When exponentialSmoothingSum is 255, we have 255 - 255 = 0, so the shade values for RGB are all set to 0 (that is, the c-value pixel is rendered black - indicating that this particular c-value very slowly tends towards infinity).
          imageDataObjectData[currentPixel++] = 255; // Always draw the c-value pixels with no transparency.
          
          if (pixelGrayscaleValue > maxPixelGrayscaleValue) {
            maxPixelGrayscaleValue = pixelGrayscaleValue; // Determine the lightest shade of gray in case the user clicks the Lighten button.
          } // if
        } // if-else
      } // for
    } // for        
    
    globals.maxPixelGrayscaleValue = maxPixelGrayscaleValue; // Store the lightest shade of gray in case the user clicks the Lighten button.      
    ctx.putImageData(ctx.imageDataObject, 0, 0); // Render our carefully constructed canvas image data array to the canvas.
        
    var elapsedMilliseconds = (new Date()) - startTime;
    document.getElementById('elapsedTime').innerHTML = iterationSum.format() + " iterations in " + (elapsedMilliseconds / 1000).toFixed(2) + " seconds"; // Note that the UI element is not updated until after this block terminates (which is the desired behavior).            
    document.getElementById('messageBox').innerHTML = DEFAULT_MESSAGE; // Erase the "Calculating..." message and replace it with the default message.        
  } // calculateMandelbrot
} // drawMandelbrot

The primary change to the drawMandelbrot function is its ability to lighten an image based on the value of its grayscaleFactor parameter:


function drawMandelbrot(ReMax, ReMin, ImMax, ImMin, grayscaleFactor) {
  .
  .
  .        
        if (c_belongsToMandelbrotSet) {
          imageDataObjectData[currentPixel++] = 0;
          imageDataObjectData[currentPixel++] = 0;
          imageDataObjectData[currentPixel++] = 0;
          imageDataObjectData[currentPixel++] = 255;
        } 
        else {
          var pixelGrayscaleValue = 255 - exponentialSmoothingSum % 256;green, and blue are identical when using a grayscale.
          var adjustedPixelGrayscaleValue = pixelGrayscaleValue * grayscaleFactor; 
          
          imageDataObjectData[currentPixel++] = adjustedPixelGrayscaleValue; // Because we mod by 256, the value of exponentialSmoothingSum will always be between 0 and 255.
          imageDataObjectData[currentPixel++] = adjustedPixelGrayscaleValue; // If exponentialSmoothingSum is 255 (its maximum possible value), then 255 % 256 = 255.
          imageDataObjectData[currentPixel++] = adjustedPixelGrayscaleValue; // When exponentialSmoothingSum is 255, we have 255 - 255 = 0, so the shade values for RGB are all set to 0 (that is, the c-value pixel is rendered black - indicating that this particular c-value very slowly tends towards infinity).
          imageDataObjectData[currentPixel++] = 255;
          
          if (pixelGrayscaleValue > maxPixelGrayscaleValue) {
            maxPixelGrayscaleValue = pixelGrayscaleValue; // Determine the lightest shade of gray in case the user clicks the Lighten button.
          }
  .
  .
  . 
      
    globals.maxPixelGrayscaleValue = maxPixelGrayscaleValue; // Store the lightest shade of gray in case the user clicks the Lighten button.      
    ctx.putImageData(ctx.imageDataObject, 0, 0); // Render our carefully constructed canvas image data array to the canvas.
  .
  .
  . 


As you can see in the line var adjustedPixelGrayscaleValue = pixelGrayscaleValue * grayscaleFactor, the grayscaleFactor parameter is multiplied with the current pixel's grayscale value (pixelGrayscaleValue). This adjusted grayscale value (adjustedPixelGrayscaleValue) is then used to set the imageDataObjectData array as shown in the previous code example.

As an image is being calculated, the lightest (maximum) grayscale value is determined and saved to a global variable:


globals.maxPixelGrayscaleValue = maxPixelGrayscaleValue;

When the Lighten button is clicked, the lightest shade of gray for the current image is already known, and a lighter version of the image is drawn as follows:


function handleLightenButton() {
  var grayscaleFactor = 255 / globals.maxPixelGrayscaleValue; // For the canvas element, 255 is white, 0 is black.

  window.location.hash = globals.ReMax + "," + globals.ReMin + "," + globals.ImMax + "," + globals.ImMin + "," + grayscaleFactor; // This invokes handleHashChange which, among other things, is responsibile for drawing the Mandelbrot set.
}

In other words, when handleLightenButton executes the line:


window.location.hash = globals.ReMax + "," + globals.ReMin + "," + globals.ImMax + "," + globals.ImMin + "," + grayscaleFactor

the value (grayscaleFactor) passed to the drawMandelbrot function (via the URL hash string) is such that black (0) stays black and the lightest (maximum) grayscale value in the image becomes white (255).

Consider the following dark image along with its URL:

A dark region of the Mandelbrot set (before the Lighten button is pressed)

http://samples.msdn.microsoft.com/Workshop/samples/mandelbrot/mandelbrotExplorer.html#-0.7521294801871914,-0.7521614447995544,0.03795943464882932,0.037938124907254026,1

The trailing "1" in the URL is the grayscaleFactor for this image (a value of 1 has no lightening effect in that multiplying any real value be 1 has no effect).

When the Lighten button is clicked, the lightest grayscale value is used to produce a grayscaleFactor of approximately 3.0386:

The dark region of the Mandelbrot set after the Lighten button has been pressed

http://samples.msdn.microsoft.com/Workshop/samples/mandelbrot/mandelbrotExplorer.html#-0.7521294801871914,-0.7521614447995544,0.03795943464882932,0.037938124907254026,3.03862433470554

This value (3.0386) is multiplied with the grayscale value of every pixel in the first image, resulting in the lightened second image.

As an aside, because grayscaleFactor = 255 / globals.maxPixelGrayscaleValue, it follows that the lightest (maximum) grayscale value in the dark image is about 84:

The grayScaleFactor equation

Thus, the lightest grayscale value becomes white in the lightened image as follows:

Equation of 84 times 3.0386 (which is about 255)

Be aware that the black pixels (0) remain black in that zero multiplied by any real number is always zero.

Now that we are able to produce quality images of the Mandelbrot set, the next logical feature to a add is the ability to save images locally. This is discussed in Saving canvas images locally.

Related topics

Saving canvas images locally

 

 

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