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viewMatrix = new Matrix3D(); viewMatrix.appendTranslation(0, 4, 11); viewMatrix.appendRotation(26, Vector3D.X_AXIS); viewMatrix.append(perspective.toMatrix3D()); modelMatrix = new Matrix3D(); model = new Plot3D(); model.plot(new Rectangle(-3, -3, 6, 6), 70); addEventListener(Event.ENTER_FRAME, onEnterFrame); } protected function onEnterFrame(event:Event):void { modelMatrix.appendRotation(2, Vector3D.Y_AXIS); var concatenatedMatrix:Matrix3D = modelMatrix.clone(); concatenatedMatrix.append(viewMatrix); Utils3D.projectVectors( concatenatedMatrix, model.vertices, projectedPoints, texturePoints); graphics.clear(); graphics.lineStyle(0, 1); if (!drawCommands) { drawCommands = new Vector.<int>(); for (var i:int = 0; i < projectedPoints.length; i+=2) { drawCommands.push(((i/2) % model.resolution == 0) GraphicsPathCommand.MOVE_TO : GraphicsPathCommand.LINE_TO); } } graphics.drawPath(drawCommands, projectedPoints); } } } import flash.geom.Rectangle; import flash.geom.Vector3D; class Plot3D { public var vertices:Vector.<Number>; public var tOffset:Number = 0; public var resolution:Number; public function plot(xzBounds:Rectangle, resolution:Number = 200):void { vertices = new Vector.<Number>(); this.resolution = resolution; 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(x, y, z); } } } }

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In the example, after projectVectors() lls up projectedPoints with the screen-space 2D points of the model, you batch up the drawing commands as well with drawPath(). If you remember from 35, drawPath() takes a list of commands and a list of points. You have the list of

Degrees of freedom for the denominator (v2)

n = 1

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points you d like to draw, so all that remains is a set of drawing commands; here I ve constructed one that connects all vertices at the same z value with a line. Finally, you pulled up all reusable Vectors to be instance variables, so that they don t have to be created anew every frame, an easy and effective optimization. Figure 40-1 shows the results for both Example 40-1 and Example 40-2.

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private int getNeighborReading(int nid) C for (int i = 0; i < neighborIDs.1ength; i++) if (neighborIDsCi1 == nid) return targethadings [il ; return - 1 ; private void log(String msg) m-guinessage = new GUIHessage (m-mystate . m y I D 0 , GUIMessage . GUIM-NODEOUTPUT , m-guiMessage.setNodeOutput(msg); m-GUI.send(m-guiMessage);

FIGURE 40-1

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The optimizations in Example 40-2 should result in more uid animation and reduced CPU utilization. Always batch up drawing commands and vector projection when you can.

(a) O.C. curves for different values of n for the two-sided normal test for a level of significance

Your 3D engine is well underway. You can create a set of vertices and draw them as points or a rudimentary kind of wireframe. The next stage is drawing something solid. To move from a network of points to a solid surface, you have to stop thinking about points and edges and start thinking about faces. Vertices, edges, and faces are de ned by the same data. Connect the points with lines, and you get edges. Fill in the triangles de ned by three points, and you get faces. Figure 40-2 should help visualize this. You have all the points you need but you need to help Flash Player tell which points de ne which triangles. Connecting the wrong vertices to form faces is a sure way to make a model useless. No matter how you ll them, you draw solid faces (triangles) with the drawing API s drawTriangles() method or its equivalent command object version, GraphicsTrianglePath. I ll just cover the imperative method; the objective one is nearly identical. The method, de ned in Graphics, looks like this:

n = 1

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function drawTriangles(vertices:Vector.<Number>, indices:Vector.<int> = null, uvtData:Vector.<Number> = null, culling:String = "none"):void

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The vertices are the projected, 2D vertices, in a at, xyxy order. You already have this data. The one missing piece, indices, tells Flash Player which points to connect into faces. Imagine that all vertices are assigned a number as they are inserted into the vertices list. You connect the dots by number: a single triangle is de ned by three entries in the indices list, each referencing a vertex by its order. The indices are also stored at and repeating, in sets of three. In this manner, you can de ne exactly how the triangles are constructed by the vertices. Figure 40-3 illustrates a sequence of vertices and the indices necessary to create triangles out of them.

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