40: Advanced 3D in Java

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package { import flash.display.*; import flash.events.Event; import flash.geom.*; public class ch40ex1 extends Sprite { protected var perspective:PerspectiveProjection; protected var viewMatrix:Matrix3D; protected var modelMatrix:Matrix3D; protected var model:Plot3D; public function ch40ex1() { stage.quality = StageQuality.MEDIUM; //center coordinates this.x = stage.stageWidth/2; this.y = stage.stageHeight/2; //set up "camera" perspective projection perspective = new PerspectiveProjection(); perspective.fieldOfView = 50; perspective.projectionCenter = new Point(0, 0); //set up view projection //move camera up, back and angle down before projecting viewMatrix = new Matrix3D(); viewMatrix.appendTranslation(0, 4, 11); viewMatrix.appendRotation(26, Vector3D.X_AXIS); viewMatrix.append(perspective.toMatrix3D()); //set up model transformation modelMatrix = new Matrix3D(); //set up model model = new Plot3D(); model.plot(new Rectangle(-3, -3, 6, 6), 70); //render loop addEventListener(Event.ENTER_FRAME, onEnterFrame); } protected function onEnterFrame(event:Event):void { //rotate model progressively modelMatrix.appendRotation(2, Vector3D.Y_AXIS); //when you project, first apply the model matrix and then the //view/projection matrix. Precalculate into one matrix here. var concatenatedMatrix:Matrix3D = modelMatrix.clone(); concatenatedMatrix.append(viewMatrix); var v:Vector3D; graphics.clear(); graphics.beginFill(0, 1); for each (v in model.vertices) { v = Utils3D.projectVector(concatenatedMatrix, v); graphics.drawRect(v.x, v.y, 1, 1); }
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} import flash.geom.Rectangle; import flash.geom.Vector3D; class Plot3D { public var vertices:Vector.<Vector3D>; public var tOffset:Number = 0; public function plot(xzBounds:Rectangle, resolution:Number = 200):void { vertices = new Vector.<Vector3D>(); var r:Rectangle = xzBounds; for (var z:Number = r.top; z < r.bottom; z += r.height/resolution) { for (var x:Number = r.left; x < r.right; x += r.width/resolution) { var y:Number = Math.sin(Math.pow(x, 2) + Math.pow(z, 2) + tOffset); vertices.push(new Vector3D(x, y, z)); } } } }
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The Plot3D class is built by evaluating the 3D equation at regular intervals, which become the vertices of the model. The essential part of this example is the for..each loop that s been highlighted. It takes every vertex, translates it into screen space by projecting it with a perspective projection matrix, and draws the point in screen space. The Utils3D class contains two static methods for projecting vectors. The syntax of projectVector() here should be self-evident: it takes a projection matrix and a 3D point and projects the point into 2D space by using that matrix. Another new bit of code is the toMatrix3D() method of PerspectiveProjection. You know that a perspective projection matrix is just another matrix; Flash Player exposes it as a different class so that you can set properties of the perspective projection easily. The toMatrix3D() method just retrieves its internal matrix representation. I threw some bonus ideas into this example. The idea of model and view transformations is ubiquitous. Model transformations move objects around the scene in this case, spin the plot around the Y axis. View transformations move the viewport here, moving your eyes back from the origin so you can see the plot, and up so you can look down onto it rather than viewing it front-on. Combine view transformations with a perspective projection, and you nally have the missing pieces of a real camera that can tilt and move around. Remember that the cool thing about transformation matrices is that their effects are chained by simple concatenation (matrix multiplication), an effect that I take advantage of here, rst to create a viewMatrix that moves the camera and applies perspective, and then to concatenate the model and view matrices into concatenatedMatrix so only one matrix is needed when projecting each vertex. The order you apply matrices is essential; concatenate the modelMatrix after the viewMatrix, and you d be rotating the at, post-projection, image plane instead of rotating and projecting the model. This method is pretty slow. You have to iterate through the vertices, projecting each one and drawing each one individually. But there are ways to project the vertices as a batch, as well as to batch up drawing commands. To take advantage of these, you have to start treating vertex lists a little bit more raw.
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