Category Archives: Math

Gumdrop Torus

By Zevan | August 7, 2009
CLICK HERE TO COPY
Actionscript:
  1. var matrix:Matrix3D = new Matrix3D();
  2.  
  3. var verts:Vector.<Number> = new Vector.<Number>();
  4. var pVerts:Vector.<Number> = new Vector.<Number>();
  5. var uvts:Vector.<Number> = new Vector.<Number>();
  6.  
  7. for (var i:Number = -2; i<2; i+=.04){
  8.     for (var j:Number = -2; j<2; j+=.04){
  9.         for (var k:Number = -2; k<2; k+=.04){
  10. // equation from: http://local.wasp.uwa.edu.au/~pbourke/geometry/gumdrop/
  11.             var yz:Number = j * j + k * k;
  12.              var s:Number = 4 * (Math.pow(i,4) + Math.pow(yz,2))
  13.              + 17 * i * i *(yz) - 20 * (yz + i * i) + 17;
  14.             if (s <0 && s> -0.5){
  15.                 verts.push(i * 60);
  16.                 verts.push(j * 60);
  17.                 verts.push(k * 60);
  18.                 pVerts.push(0), pVerts.push(0);
  19.                 uvts.push(0), uvts.push(0), uvts.push(0);
  20.              }
  21.         }
  22.     }
  23. }
  24.  
  25. var tVerts:Vector.<Number> = new Vector.<Number>();
  26. matrix.appendRotation(90, Vector3D.X_AXIS);
  27. matrix.appendRotation(45, Vector3D.Y_AXIS);
  28. matrix.appendScale(1.7, 1.7, 1.7);
  29. matrix.transformVectors(verts, tVerts);
  30.  
  31. var p:Point = new Point();
  32. var brush:BitmapData=new BitmapData(3,3,true,0x41FFFFFF);
  33. var canvas:BitmapData = new BitmapData(500,500,false, 0x000000);
  34. addChild(new Bitmap(canvas));
  35. var dx:Number=0;
  36. var dy:Number=0;
  37. addEventListener(Event.ENTER_FRAME, onLoop);
  38. function onLoop(evt:Event):void {
  39.     dx += (mouseX - dx)/4;
  40.     dy += (mouseY - dy)/4;
  41.     matrix.identity();
  42.     matrix.appendRotation(dy,Vector3D.X_AXIS);
  43.     matrix.appendRotation(dx,Vector3D.Y_AXIS);
  44.     matrix.appendTranslation(250, 250, 0);
  45.     Utils3D.projectVectors(matrix, tVerts, pVerts, uvts);
  46.     canvas.lock();
  47.     canvas.fillRect(canvas.rect, 0x000000);
  48.     var inc:int = 0;
  49.     for (var i:int = 0; i<pVerts.length; i+=2){
  50.      
  51.         p.x = pVerts[i];
  52.         p.y = pVerts[i+1];
  53.         canvas.copyPixels(brush, brush.rect, p, null, null, true);
  54.     }
  55.     canvas.unlock();
  56. }

Felt like revisiting implicit surface plotting today...

Have a look at the swf over at wonderfl.net

Also posted in 3D, BitmapData | Tagged , , | Leave a comment

Quick Pixel Sphere

By Zevan | July 26, 2009
CLICK HERE TO COPY
Actionscript:
  1. var pointNum:int = 20000;
  2. var radius:int = 150;
  3.  
  4. var canvas:BitmapData = new BitmapData(400,400,false, 0x000000);
  5. addChild(new Bitmap(canvas));
  6. var verts:Vector.<Number>  = new Vector.<Number>();
  7. var pVerts:Vector.<Number> = new Vector.<Number>();
  8. var uv:Vector.<Number> = new Vector.<Number>();
  9.  
  10. for (var i:int = 0; i<pointNum; i+=3){
  11.     var xp:Number = Math.random() * 400 - 200;
  12.     var yp:Number = Math.random() * 400 - 200;
  13.     var zp:Number = Math.random() * 400 - 200;
  14.     var dist:Number = Math.sqrt(xp * xp + yp * yp + zp * zp);
  15.     // normalize and scale x,y,z
  16.     verts[i] = xp / dist * radius;
  17.     verts[i+1] = yp / dist * radius;
  18.     verts[i+2] = zp / dist * radius;
  19. }
  20.  
  21. var m:Matrix3D = new Matrix3D();
  22. var dx:Number = 0, dy:Number = 0;
  23. addEventListener(Event.ENTER_FRAME, onLoop);
  24. function onLoop(evt:Event):void {
  25.        m.identity();
  26.        dx += (mouseX - dx) / 4;
  27.        dy += (mouseY - dy) / 4;
  28.        m.appendRotation(dx, Vector3D.X_AXIS);
  29.        m.appendRotation(dy, Vector3D.Y_AXIS);
  30.        m.appendTranslation(200,200,0);
  31.        Utils3D.projectVectors(m, verts, pVerts, uv);
  32.        canvas.fillRect(canvas.rect, 0x000000);
  33.        for (var i:int = 0; i<pVerts.length; i+=2){
  34.          canvas.setPixel(pVerts[i], pVerts[i + 1], 0xFFFFFF);
  35.        }
  36. }

This snippet shows a quick way to randomly place a bunch of xyz coordinates on the surface of a sphere. I saw this trick in an OpenGL book a few years back - dug around my books but couldn't find it... If I find it I'll update this post.

The trick is achieved by normalizing the vector defined by each 3D coordinate...


Have a look at the swf...

Also posted in 3D, BitmapData, matrix, setPixel | Tagged , , | 4 Comments

Utils3D.projectVectors() Lathe

By Zevan | July 6, 2009
CLICK HERE TO COPY
Actionscript:
  1. [SWF(width = 500, height = 500, backgroundColor = 0x000000)]
  2. var halfWidth:Number = stage.stageWidth / 2;
  3. var halfHeight:Number = stage.stageHeight / 2;
  4. var loc:Vector.<Number>;
  5.  
  6. graphics.lineStyle(0, 0xFF0000);
  7. graphics.moveTo(halfWidth, 0);
  8. graphics.lineTo(halfWidth, stage.stageHeight);
  9. graphics.moveTo(0, halfHeight);
  10. graphics.lineTo(stage.stageWidth, halfHeight);
  11.  
  12. var line:Shape = Shape(addChild(new Shape()));
  13. line.x = halfWidth;
  14.  
  15. var idle:Function = function(){};
  16. var currentMode:Function = idle;
  17. stage.addEventListener(MouseEvent.MOUSE_DOWN, onDown);
  18. stage.addEventListener(MouseEvent.MOUSE_UP, onUp);
  19. addEventListener(Event.ENTER_FRAME, onLoop);
  20. function onDown(evt:MouseEvent):void{
  21.     if (contains(frame)){
  22.         removeChild(frame);
  23.         currentMode = idle;
  24.         line.graphics.clear();
  25.         return;
  26.     }
  27.     loc = new Vector.<Number>();
  28.     line.graphics.lineStyle(0,0xFFFFFF);
  29.     line.x = halfWidth
  30.     line.y = halfHeight;
  31.     line.graphics.moveTo(line.mouseX, line.mouseY);
  32.     currentMode = captureLocs;
  33.     canvas.fillRect(canvas.rect, 0x000000);
  34. }
  35. function onUp(evt:MouseEvent):void{
  36.     if (currentMode == idle) return;
  37.     setupLathe();
  38.     currentMode = showLathe;
  39. }
  40.  
  41. function onLoop(evt:Event):void{
  42.     currentMode();
  43. }
  44. function captureLocs():void{
  45.     loc.push(line.mouseX);
  46.     loc.push(line.mouseY);
  47.     loc.push(0);
  48.     line.graphics.lineTo(line.mouseX, line.mouseY);
  49. }
  50. /**
  51.  -- Lathe Stuff:
  52. */
  53. var canvas:BitmapData = new BitmapData(stage.stageWidth, stage.stageHeight,false, 0x000000);
  54. var frame:Bitmap = new Bitmap(canvas);
  55. var dx:Number=0;
  56. var dy:Number=0;
  57.  var matrix:Matrix3D = new Matrix3D();
  58. var pVerts:Vector.<Number>;
  59. var uvts:Vector.<Number>;
  60. function setupLathe():void{
  61.     addChild(frame);
  62.     pVerts =  new Vector.<Number>();
  63.     uvts = new Vector.<Number>();
  64.     var nVerts:Vector.<Number> = new Vector.<Number>();
  65.     var tVerts:Vector.<Number> = new Vector.<Number>();
  66.     matrix.identity();
  67.     var step:Number = 2;
  68.     for (var i:int = 0; i <360; i+=step){
  69.         matrix.appendRotation(step,Vector3D.Y_AXIS);
  70.         matrix.transformVectors(loc, tVerts);
  71.         nVerts = nVerts.concat(tVerts);
  72.     }
  73.     loc = nVerts.concat();
  74. }
  75. function showLathe():void{
  76.     dx += (mouseX - dx)/4;
  77.     dy += (mouseY - dy)/4;
  78.     matrix.identity();
  79.     matrix.appendRotation(dy,Vector3D.X_AXIS);
  80.     matrix.appendRotation(dx,Vector3D.Y_AXIS);
  81.     matrix.appendTranslation(halfWidth, halfHeight, 0);
  82.     Utils3D.projectVectors(matrix, loc, pVerts, uvts);
  83.     canvas.lock();
  84.     canvas.fillRect(canvas.rect, 0x000000);
  85.     var leng:int = pVerts.length;
  86.     for (var i:int = 0; i<leng; i+=2){
  87.         canvas.setPixel( pVerts[i], pVerts[i + 1], 0xFFFFFF);
  88.     }
  89.     canvas.unlock();
  90. }

This snippet allows you to create 3D lathe shapes by drawing a 2D line. This is done using Utils3D.projectVectors() and Matrix.transformVectors().

Have a look at the swf here...

Also posted in 3D, BitmapData, Vector, graphics algorithms, matrix, setPixel | Tagged , , | 3 Comments

2x mod 1 map

By Zevan | June 28, 2009
CLICK HERE TO COPY
Actionscript:
  1. [SWF(width=800, height=600)]
  2. var xn1:Number;
  3. var xn:Number = Math.random() * Math.random() * .2;
  4. var inc:int = 0;
  5. var xp:Number = 10;
  6. var yp:Number = 10;
  7. var count:int = 1;
  8. scaleX = scaleY = 2;
  9. graphics.lineStyle(0,0x00000);
  10. addEventListener(Event.ENTER_FRAME, onLoop);
  11. function onLoop(evt:Event):void {
  12.    
  13.      xn1 = 2 * xn % 1;
  14.      xn = xn1;
  15.      if (inc == 0){
  16.           graphics.moveTo(xp + inc, yp + 30 - xn1 * 30);
  17.      }else{
  18.          graphics.lineTo(xp + inc, yp + 30 - xn1 * 30);
  19.      }
  20.      inc++
  21.      if (inc == 50){
  22.          inc = 0;
  23.          xp = 10 + count % 6 * 60;
  24.          yp = 10 + int(count / 6) * 60;
  25.          xn = Math.random() * Math.random() * .2;
  26.          trace(xn);
  27.          count++;
  28.      }
  29. }

This snippet plots 2x mod 1 maps with random starting values for xn. More info over at wikipedia and mathworld.

Also posted in misc, motion | Tagged , , | Leave a comment