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3D Pythagoras tree

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makc3d

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forked : 1
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lines : 131
license : see code comments
modified : 2009-03-26 18:15:04

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// Three-dimensional Pythagoras tree
// First implementation ever, to my best knowledge :)
//
// Click anywhere to regenerate.
// Check http://en.wikipedia.org/wiki/Pythagoras_tree for more info on subject.

package {

import flash.display.*
	import flash.events.*

import alternativa.engine3d.controllers.*
	import alternativa.engine3d.core.*
	import alternativa.engine3d.display.*
	import alternativa.engine3d.materials.*
	import alternativa.engine3d.primitives.*
	import alternativa.types.*
     
	[SWF(width=465,height=465,frameRate=30)]
	public class PythagorasTree3D extends Sprite {

// tree elements (center positions, left and up vectors)
		// similar to http://wonderfl.kayac.com/code/18d2f77aeee90b0a3ae76dc5509a60f8bacad276
		private var atA:Point3D = new Point3D, ltA:Point3D = new Point3D, upA:Point3D = new Point3D;
		private var atB:Point3D = new Point3D, ltB:Point3D = new Point3D, upB:Point3D = new Point3D;
		private var atC:Point3D = new Point3D, ltC:Point3D = new Point3D, upC:Point3D = new Point3D;
		private var atD:Point3D = new Point3D, ltD:Point3D = new Point3D, upD:Point3D = new Point3D;
		private var ltE:Point3D = new Point3D (-1, 0, 0), upE:Point3D = new Point3D (0, -1, 0);
		private var matD:Matrix3D = new Matrix3D;
		private var matE:Matrix3D = new Matrix3D;
		private var sA:Number, sB:Number, sC:Number;
		private var rA:Point3D = new Point3D, rB:Point3D = new Point3D, rC:Point3D = new Point3D;
		private function calculateElements (c1:Number, c2:Number):void
		{
			// using c1 and c2 in 0..1 range, select such a, b, c and d that a^2 + b^2 + c^2 = d^2
			// this choise is almost completely free (we only need to make sure that d != 0)
			var a:Number = 0.1 + c1, c:Number = 1.1 - c1; // +
			var b:Number = ((c2 > 0.5) ? (c2 - 0.49) : (c2 - 0.51)) * 2; // + or -
			var d:Number = Math.sqrt (a * a + b * b + c * c); // +

// define corresponding tree elements in some convenient frame
			// we are constrained by Pythagoras theorem, but orientation of planes is arbitrary
			// additionally, we want our tree to coincide with 2D one in boundary case of b = 0
			atA.x = +a/2; atA.y = 0; atA.z = -a/2;
			ltA.x = +a/2; ltA.y = 0; ltA.z =  0;
			upA.x =  0;   upA.y = 0; upA.z = -a/2;

atC.x = -c/2; atC.y = b; atC.z = +c/2;
			ltC.x =  0;   ltC.y = 0; ltC.z = -c/2;
			upC.x = -c/2; upC.y = 0; upC.z =  0;

atD.x = +a/2; atD.y = +b/2; atD.z = +c/2;
			ltD.x = +a/2; ltD.y = -b/2; ltD.z = -c/2;
			upD.x = -c; upD.y = 0; upD.z = -a; upD.normalize (); upD.multiply (ltD.length);
			atD.subtract (upD);

ltB.x = 0; ltB.y = -b/2; ltB.z = 0;
			upB.copy (upD); upB.normalize (); upB.multiply (Math.abs (b/2));
			atB.copy (upB); atB.y = +b/2;

// find transformation that aligns D element with 2x2 plane frame (E)
			// this 2x2 condition is there to use getRotations () method later
			var lxuD:Point3D = Point3D.cross (ltD, upD);
			lxuD.normalize (); lxuD.multiply (ltD.length);
			matD.a = -ltD.x; matD.e = -ltD.y; matD.i = -ltD.z;
			matD.b = -upD.x; matD.f = -upD.y; matD.j = -upD.z;
			matD.c = lxuD.x; matD.g = lxuD.y; matD.k = lxuD.z;

var lxuE:Point3D = Point3D.cross (ltE, upE);
			matE.a = -ltE.x; matE.e = -ltE.y; matE.i = -ltE.z;
			matE.b = -upE.x; matE.f = -upE.y; matE.j = -upE.z;
			matE.c = lxuE.x; matE.g = lxuE.y; matE.k = lxuE.z;

matD.invert (); matD.combine (matE);

// transform A, B and C elements
			atA.subtract (atD); atA.transform (matD); ltA.transform (matD); upA.transform (matD);
			atB.subtract (atD); atB.transform (matD); ltB.transform (matD); upB.transform (matD);
			atC.subtract (atD); atC.transform (matD); ltC.transform (matD); upC.transform (matD);

// calculate scales and normalize left/up vectors
			sA = ltA.length; ltA.normalize (); upA.normalize ();
			sB = ltB.length; ltB.normalize (); upB.normalize ();
			sC = ltC.length; ltC.normalize (); upC.normalize ();

// finally, calculate corresponding rotations (re-using matE/lxuE variables)
			lxuE = Point3D.cross (ltA, upA);
			matE.a = -ltA.x; matE.e = -ltA.y; matE.i = -ltA.z;
			matE.b = -upA.x; matE.f = -upA.y; matE.j = -upA.z;
			matE.c = lxuE.x; matE.g = lxuE.y; matE.k = lxuE.z;
			matE.getRotations (rA);

lxuE = Point3D.cross (ltB, upB);
			matE.a = -ltB.x; matE.e = -ltB.y; matE.i = -ltB.z;
			matE.b = -upB.x; matE.f = -upB.y; matE.j = -upB.z;
			matE.c = lxuE.x; matE.g = lxuE.y; matE.k = lxuE.z;
			matE.getRotations (rB);

lxuE = Point3D.cross (ltC, upC);
			matE.a = -ltC.x; matE.e = -ltC.y; matE.i = -ltC.z;
			matE.b = -upC.x; matE.f = -upC.y; matE.j = -upC.z;
			matE.c = lxuE.x; matE.g = lxuE.y; matE.k = lxuE.z;
			matE.getRotations (rC);
		}

private var planes:Array = [], planesDone:Array = [];
		private function step ():void {
			var pD:Plane = Plane (planes.shift ()); planesDone.push (pD);

var pA:Plane = createPlane ();
			pA.rotationX = rA.x;  pA.rotationY = rA.y;  pA.rotationZ = rA.z;
			pA.scaleX    = sA;    pA.scaleY    = sA;    pA.scaleZ    = sA;
			pA.x         = atA.x; pA.y         = atA.y; pA.z         = atA.z;
			pD.addChild (pA); planes.push (pA);

var pB:Plane = createPlane ();
			pB.rotationX = rB.x;  pB.rotationY = rB.y;  pB.rotationZ = rB.z;
			pB.scaleX    = sB;    pB.scaleY    = sB;    pB.scaleZ    = sB;
			pB.x         = atB.x; pB.y         = atB.y; pB.z         = atB.z;
			pD.addChild (pB); planes.push (pB);

var pC:Plane = createPlane ();
			pC.rotationX = rC.x;  pC.rotationY = rC.y;  pC.rotationZ = rC.z;
			pC.scaleX    = sC;    pC.scaleY    = sC;    pC.scaleZ    = sC;
			pC.x         = atC.x; pC.y         = atC.y; pC.z         = atC.z;
			pD.addChild (pC); planes.push (pC);
		}

private function regenerate (c1:Number, c2:Number):void {
			var p:Plane;

// delete all the planes
			for each (p in planesDone) destroyPlane (p);
			planesDone.length = 0; planes.length = 0;

// create base plane
			p = createPlane (); planes.push (p); scene.root.addChild (p);

// do the math
			calculateElements (c1, c2);
		}

private function destroyPlane (p:Plane):void {
			// this is supposed to make alternativa trash GC-friendly
			if (p.parent != null) p.parent.removeChild (p);
			if (p.hasSurface ("back")) p.setMaterialToSurface (null, "back");
			if (p.hasSurface ("front")) p.setMaterialToSurface (null, "front");
			p.moveAllFacesToSurface (null, true);
		}

private function createPlane ():Plane {
			var p:Plane = new Plane (2, 2);
			p.cloneMaterialToAllSurfaces (new FillMaterial (0xFFFFFF * Math.random ()));
			return p;
		}

private var scene:Scene3D;
		private var tripod:Object3D;
		private var view:View;

public function PythagorasTree3D () {
			scene = new Scene3D; scene.root = new Object3D;
			view = new View; view.camera = new Camera3D;
			view.camera.z = -6; view.camera.rotationX = -0.4; view.camera.y = -6;
			tripod = new Object3D; scene.root.addChild (tripod); tripod.addChild (view.camera);
			view.width = view.height = 465; addChild (view);

regenerate (0.5, 0.6);

var s:Sprite = new Sprite; s.buttonMode = s.useHandCursor = true;
			s.graphics.beginFill (0, 0); s.graphics.drawRect (0, 0, 465, 465);
			addChild (s); s.addEventListener (MouseEvent.CLICK, onClick);

addEventListener (Event.ENTER_FRAME, onEnterFrame);
		}

private function onClick (e:MouseEvent):void {
			regenerate (mouseX / 465.0, mouseY / 465.0);
		}

private function onEnterFrame (e:Event):void {
			if (planesDone.length < 50) step ();
			tripod.rotationY += 0.1; scene.calculate ();
		}
	}
}

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3D Pythagoras tree 3D Pythagoras tree (anaglyph) [diff(174)]

makc3d

// forked from 3D Pythagoras tree - anaglyph version
// Three-dimensional Pythagoras tree
// First implementation ever, to my best knowledge :)
//
// Click anywhere to regenerate.
// Check http://en.wikipedia.org/wiki/Pythagoras_tree for more info on subject.
package {
import flash.display.*
import flash.events.*
import flash.geom.ColorTransform;
import alternativa.engine3d.*
import alternativa.engine3d.controllers.*
import alternativa.engine3d.core.*
import alternativa.engine3d.display.*
import alternativa.engine3d.materials.*
import alternativa.engine3d.primitives.*
import alternativa.types.*
use namespace alternativa3d;
[SWF(width=465,height=465,frameRate=30,backgroundColor=0)]
public class PythagorasTree3D extends Sprite {
// tree elements (center positions, left and up vectors)
// similar to http://wonderfl.kayac.com/code/18d2f77aeee90b0a3ae76dc5509a60f8bacad276
private var atA:Point3D = new Point3D, ltA:Point3D = new Point3D, upA:Point3D = new Point3D;
private var atB:Point3D = new Point3D, ltB:Point3D = new Point3D, upB:Point3D = new Point3D;
private var atC:Point3D = new Point3D, ltC:Point3D = new Point3D, upC:Point3D = new Point3D;
private var atD:Point3D = new Point3D, ltD:Point3D = new Point3D, upD:Point3D = new Point3D;
private var ltE:Point3D = new Point3D (-1, 0, 0), upE:Point3D = new Point3D (0, -1, 0);
private var matD:Matrix3D = new Matrix3D;
private var matE:Matrix3D = new Matrix3D;
private var sA:Number, sB:Number, sC:Number;
private var rA:Point3D = new Point3D, rB:Point3D = new Point3D, rC:Point3D = new Point3D;
private function calculateElements (c1:Number, c2:Number):void
{
// using c1 and c2 in 0..1 range, select such a, b, c and d that a^2 + b^2 + c^2 = d^2
// this choise is almost completely free (we only need to make sure that d != 0)
var a:Number = 0.1 + c1, c:Number = 1.1 - c1; // +
var b:Number = ((c2 > 0.5) ? (c2 - 0.499999) : (c2 - 0.500001)) * 2; // + or -
var d:Number = Math.sqrt (a * a + b * b + c * c); // +
// define corresponding tree elements in some convenient frame
// we are constrained by Pythagoras theorem, but orientation of planes is arbitrary
// additionally, we want our tree to coincide with 2D one in boundary case of b = 0
atA.x = +a/2; atA.y = 0; atA.z = -a/2;
ltA.x = +a/2; ltA.y = 0; ltA.z = 0;
upA.x = 0; upA.y = 0; upA.z = -a/2;
atC.x = -c/2; atC.y = b; atC.z = +c/2;
ltC.x = 0; ltC.y = 0; ltC.z = -c/2;
upC.x = -c/2; upC.y = 0; upC.z = 0;
atD.x = +a/2; atD.y = +b/2; atD.z = +c/2;
ltD.x = +a/2; ltD.y = -b/2; ltD.z = -c/2;
upD.x = -c; upD.y = 0; upD.z = -a; upD.normalize (); upD.multiply (ltD.length);
atD.subtract (upD);
ltB.x = 0; ltB.y = -b/2; ltB.z = 0;
upB.copy (upD); upB.normalize (); upB.multiply (Math.abs (b/2));
atB.copy (upB); atB.y = +b/2;
// find transformation that aligns D element with 2x2 plane frame (E)
// this 2x2 condition is there to use getRotations () method later
var lxuD:Point3D = Point3D.cross (ltD, upD);
lxuD.normalize (); lxuD.multiply (ltD.length);
matD.a = -ltD.x; matD.e = -ltD.y; matD.i = -ltD.z;
matD.b = -upD.x; matD.f = -upD.y; matD.j = -upD.z;
matD.c = lxuD.x; matD.g = lxuD.y; matD.k = lxuD.z;
var lxuE:Point3D = Point3D.cross (ltE, upE);
matE.a = -ltE.x; matE.e = -ltE.y; matE.i = -ltE.z;
matE.b = -upE.x; matE.f = -upE.y; matE.j = -upE.z;
matE.c = lxuE.x; matE.g = lxuE.y; matE.k = lxuE.z;
matD.invert (); matD.combine (matE);
// transform A, B and C elements
atA.subtract (atD); atA.transform (matD); ltA.transform (matD); upA.transform (matD);
atB.subtract (atD); atB.transform (matD); ltB.transform (matD); upB.transform (matD);
atC.subtract (atD); atC.transform (matD); ltC.transform (matD); upC.transform (matD);
// calculate scales and normalize left/up vectors
sA = ltA.length; ltA.normalize (); upA.normalize ();
sB = ltB.length; ltB.normalize (); upB.normalize ();
sC = ltC.length; ltC.normalize (); upC.normalize ();
// finally, calculate corresponding rotations (re-using matE/lxuE variables)
lxuE = Point3D.cross (ltA, upA);
matE.a = -ltA.x; matE.e = -ltA.y; matE.i = -ltA.z;
matE.b = -upA.x; matE.f = -upA.y; matE.j = -upA.z;
matE.c = lxuE.x; matE.g = lxuE.y; matE.k = lxuE.z;
matE.getRotations (rA);
lxuE = Point3D.cross (ltB, upB);
matE.a = -ltB.x; matE.e = -ltB.y; matE.i = -ltB.z;
matE.b = -upB.x; matE.f = -upB.y; matE.j = -upB.z;
matE.c = lxuE.x; matE.g = lxuE.y; matE.k = lxuE.z;
matE.getRotations (rB);
lxuE = Point3D.cross (ltC, upC);
matE.a = -ltC.x; matE.e = -ltC.y; matE.i = -ltC.z;
matE.b = -upC.x; matE.f = -upC.y; matE.j = -upC.z;
matE.c = lxuE.x; matE.g = lxuE.y; matE.k = lxuE.z;
matE.getRotations (rC);
}
private var planes:Array = [], planesDone:Array = [];
private function step ():void {
var pD:Plane = Plane (planes.shift ());
if (pD == null) return;
planesDone.push (pD);
var scale:Number =
pD.transformation.a * pD.transformation.a +
pD.transformation.e * pD.transformation.e +
pD.transformation.i * pD.transformation.i;
if (scale < 0.001) return;
var pA:Plane = createPlane ();
pA.rotationX = rA.x; pA.rotationY = rA.y; pA.rotationZ = rA.z;
pA.scaleX = sA; pA.scaleY = sA; pA.scaleZ = sA;
pA.x = atA.x; pA.y = atA.y; pA.z = atA.z;
pD.addChild (pA); planes.push (pA);
var pB:Plane = createPlane ();
pB.rotationX = rB.x; pB.rotationY = rB.y; pB.rotationZ = rB.z;
pB.scaleX = sB; pB.scaleY = sB; pB.scaleZ = sB;
pB.x = atB.x; pB.y = atB.y; pB.z = atB.z;
pD.addChild (pB); planes.push (pB);
var pC:Plane = createPlane ();
pC.rotationX = rC.x; pC.rotationY = rC.y; pC.rotationZ = rC.z;
pC.scaleX = sC; pC.scaleY = sC; pC.scaleZ = sC;
pC.x = atC.x; pC.y = atC.y; pC.z = atC.z;
pD.addChild (pC); planes.push (pC);
}
private function regenerate (c1:Number, c2:Number):void {
var p:Plane;
// delete all the planes
for each (p in planesDone) destroyPlane (p);
planesDone.length = 0; planes.length = 0;
// create base plane
p = createPlane (); planes.push (p); scene.root.addChild (p);
// do the math
calculateElements (c1, c2);
// mark location
loc.x = stage.stageWidth * c1;
loc.y = stage.stageHeight * c2;
}
private function destroyPlane (p:Plane):void {
// this is supposed to make alternativa trash GC-friendly
if (p.parent != null) p.parent.removeChild (p);
if (p.hasSurface ("back")) p.setMaterialToSurface (null, "back");
if (p.hasSurface ("front")) p.setMaterialToSurface (null, "front");
p.moveAllFacesToSurface (null, true);
}
private function createPlane ():Plane {
var p:Plane = new Plane (2, 2);
p.cloneMaterialToAllSurfaces (new FillMaterial (0x00FF));
return p;
}
private function interpolateColor (fromColor:uint, toColor:uint, progress:Number):uint {
var q:Number = 1-progress;
var fromR:uint = (fromColor >> 16) & 0xFF;
var fromG:uint = (fromColor >> 8) & 0xFF;
var fromB:uint = fromColor & 0xFF;
var toR:uint = (toColor >> 16) & 0xFF;
var toG:uint = (toColor >> 8) & 0xFF;
var toB:uint = toColor & 0xFF;
var resultR:uint = fromR*q + toR*progress;
var resultG:uint = fromG*q + toG*progress;
var resultB:uint = fromB*q + toB*progress;
var resultColor:uint = resultR << 16 | resultG << 8 | resultB;
return resultColor;
}
private function lightPlane (p:Plane):void {
var face:Face = p.faces.peek () as Face;
var dot:Number = (face.globalNormal.x + face.globalNormal.y - face.globalNormal.z) / Math.sqrt (3);
if (dot < -1) dot = -1; if (dot > 1) dot = 1;
var lum:Number = 0.6 + 0.4 * Math.abs (dot);
var color1:uint = 0x40302 * int (63 * lum); // yellow-ish
var color2:uint = 0x20400 * int (63 * lum); // green-ish
var m:Matrix3D = p.transformation;
var prog:Number = Math.sqrt (m.a * m.a + m.e * m.e + m.i * m.i);
var color:uint = interpolateColor (color2, color1, prog);
var mat1:FillMaterial = FillMaterial (Surface (p.surfaces ["front"]).material); mat1.color = color;
var mat2:FillMaterial = FillMaterial (Surface (p.surfaces ["back"]).material); mat2.color = color;
}
private var scene:Scene3D;
private var tripod:Object3D;
private var viewL:View;
private var viewR:View;
private var loc:Shape;
public function PythagorasTree3D () {
stage.quality = "best";
scene = new Scene3D; scene.root = new Object3D;
viewL = new View; viewL.camera = new Camera3D;
viewL.camera.z = -6; viewL.camera.rotationX = -0.6; viewL.camera.y = -9;
tripod = new Object3D; scene.root.addChild (tripod); tripod.addChild (viewL.camera);
viewL.width = 500; viewL.height = 465; addChild (viewL); viewL.x = -26;
viewL.transform.colorTransform = new ColorTransform (1, 0, 0);
viewR = new View; viewR.camera = new Camera3D;
viewR.camera.x = 0.8; viewL.camera.addChild (viewR.camera);
viewR.width = 500; viewR.height = 465; addChild (viewR);
viewR.transform.colorTransform = new ColorTransform (0, 0.9, 1);
viewR.blendMode = "add";
var s:Sprite = new Sprite; s.buttonMode = s.useHandCursor = true;
s.graphics.beginFill (0, 0); s.graphics.drawRect (0, 0, 465, 465);
addChild (s); s.addEventListener (MouseEvent.CLICK, onClick);
addEventListener (Event.ENTER_FRAME, onEnterFrame);
loc = new Shape;
loc.graphics.lineStyle ();
loc.graphics.beginFill (0xFFFFFF);
loc.graphics.drawCircle (0, 0, 2);
addChild (loc); loc.addEventListener (MouseEvent.CLICK, onClick);
regenerate (0.5, 0.5);
stage.addEventListener (KeyboardEvent.KEY_UP, onKeyUp);
}
private function onClick (e:MouseEvent):void {
regenerate (mouseX / stage.stageWidth, mouseY / stage.stageHeight);
if (e.shiftKey)
trace ("\t\t\tnew Point ("
+ (mouseX / stage.stageWidth).toFixed (5)
+ ", "
+ (mouseY / stage.stageHeight).toFixed (5)
+ "),");
}
private function onEnterFrame (e:Event):void {
// add 3 more planes
var lastPlaneDone:int = planesDone.length;
var grow:Boolean = (planesDone.length < 15000);
if (grow) {
for (var s:int = 0; s < 90; s++) step ();
}
scene.calculate ();
// apply lighting after global normals were set
if (grow || (lastPlaneDone < planesDone.length)) {
for (var p:int = lastPlaneDone; p < planesDone.length; p++)
lightPlane (planesDone [p]);
for (var i:int = Math.max (0, planes.length - 9*3); i < planes.length; i++)
lightPlane (planes [i]);
}
}
private var dae:XML =
<COLLADA version="1.4.0" xmlns="http://www.collada.org/2005/11/COLLADASchema">
<asset>
<unit meter="0.01" name="centimeter"/>
<up_axis>Z_UP</up_axis>
</asset>
<library_geometries>
<geometry id="Plane-Geometry" name="Plane-Geometry">
<mesh>
<source id="Plane-Geometry-Position">
<float_array count="36" id="Plane-Geometry-Position-array">1.00000 1.00000 0.00000 1.00000 -1.00000 0.00000 -1.00000 -1.00000 0.00000 -1.00000 1.00000 0.00000</float_array>
<technique_common>
<accessor count="4" source="#Plane-Geometry-Position-array" stride="3">
<param type="float" name="X"></param>
<param type="float" name="Y"></param>
<param type="float" name="Z"></param>
</accessor>
</technique_common>
</source>
<source id="Plane-Geometry-Normals">
<float_array count="3" id="Plane-Geometry-Normals-array">0.00000 0.00000 1.00000</float_array>
<technique_common>
<accessor count="1" source="#Plane-Geometry-Normals-array" stride="3">
<param type="float" name="X"></param>
<param type="float" name="Y"></param>
<param type="float" name="Z"></param>
</accessor>
</technique_common>
</source>
<vertices id="Plane-Geometry-Vertex">
<input semantic="POSITION" source="#Plane-Geometry-Position"/>
</vertices>
<polygons count="1">
<input offset="0" semantic="VERTEX" source="#Plane-Geometry-Vertex"/>
<input offset="1" semantic="NORMAL" source="#Plane-Geometry-Normals"/>
<p>0 0 1 0 2 0 3 0</p>
</polygons>
</mesh>
</geometry>
</library_geometries>
<library_visual_scenes>
<visual_scene id="Scene" name="Scene" />
</library_visual_scenes>
<scene>
<instance_visual_scene url="#Scene"/>
</scene>
</COLLADA>;
private function onKeyUp (e:KeyboardEvent):void {
// export collada 1.4
var scene:XML = <visual_scene id="Scene" name="Scene" />;
for (var i:int = 0; i < planesDone.length; i++) {
var pi:Plane = Plane (planesDone [i]);
var mi:Matrix3D = pi.transformation;
var ni:XML = new XML (
"<node layer=\"L1\" id=\"Plane" + i + "\" name=\"Plane" + i + "\">" +
"<matrix>" +
mi.a + " " + mi.b + " " + mi.c + " " + mi.d + " " +
mi.e + " " + mi.f + " " + mi.g + " " + mi.h + " " +
mi.i + " " + mi.j + " " + mi.k + " " + mi.l + " " +
"0.0 0.0 0.0 1.0" +
"</matrix>" +
"<instance_geometry url=\"#Plane-Geometry\"/>" +
"</node>"
);
scene.appendChild (ni);
}
// dump to flash log
dae.children() [2].setChildren (scene);
trace ("<?xml version=\"1.0\" encoding=\"utf-8\"?>")
trace (dae.toString ());
}
}
}