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package { import com.actionscriptbible.Example; import flash.geom.Matrix3D; public class ch34ex5 extends Example { public function ch34ex5() { //create a default (identity) matrix var m:Matrix3D = new Matrix3D(); prettyPrintMatrix(m); //identity matrix var v:Vector.<Number> = m.rawData; v[4] = 9.2; m.rawData = v;
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prettyPrintMatrix(m); //9.2 appears on top of 2nd column m.identity(); //reset matrix m.appendTranslation(16, 19, 21); prettyPrintMatrix(m); //the translation factors appear in column 4 } protected function prettyPrintMatrix(m:Matrix3D):void { var str:String = ""; for (var col:int = 0; col < 4; col++) { var line:String = "|"; for (var row:int = 0; row < 4; row++) { line += "\t" + m.rawData[row*4+col].toPrecision(3) + "\t"; } str += line + "|\n"; } trace(str); } } }
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These methods enable some matrix operations on a Matrix3D instance:
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identity():void Replaces the matrix with a 4 4 identity matrix. transpose():void Replaces the matrix with its transpose. Transposing a matrix swaps
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its columns and rows. It s easiest to think of this as ipping a matrix (as you d write it down on paper) across the top-left-to-bottom-right diagonal. For a matrix M, the transpose is written MT . Also, (MT )T = M.
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| 1 2 3 |T | 1 4 7 | | 4 5 6 | = | 2 5 8 | | 7 8 9 | | 3 6 9 |
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invert():Boolean Replaces the matrix with its inverse. Inverting a matrix is discussed earlier in Matrices. Not all matrices are invertible, but all transformation matrices are. The method aborts and returns false if the matrix is not invertible. determinant:Number A read-only property that calculates the determinant of the matrix. If this is zero, the matrix is not invertible.
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Unlike 2D transformations, in 3D you have the option of applying the transformation to the transformation matrix before all the existing transformations or after all the existing transformations. Recall that the concat() method of Matrix multiplied its argument to the contents of the matrix, with the argument matrix rst and the subject matrix (the object whose concat() method you call) second. Matrix3D, on the other hand, does not have a concat() method but instead has prepend() and append() methods that perform the multiplication with the argument matrix on either the left or
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34: Geometric and Color Transformations
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the right hand of the multiplication. Additionally, each transformation matrix method has two avors. One immediately prepends the matrix and the other immediately appends it, as follows:
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prependTranslation(x:Number, y:Number, z:Number):void, appendTranslation(x:Number, y:Number, z:Number):void Composes a
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translation with the existing transformation. The arguments determine the offsets in each axis.
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prependScale(xScale:Number, yScale:Number, zScale:Number):void, appendScale(xScale:Number, yScale:Number, zScale:Number):void
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Composes a scale with the existing transformation. The arguments determine the scale factor in each direction (1 being no change).
prependRotation(degrees:Number, axis:Vector3D, pivotPoint:Vector3D = null):void, appendRotation(degrees:Number, axis:Vector3D, pivotPoint:Vector3D = null):void Composes a rotation with the existing trans-
formation. The arguments describe the rotation as a rotation around a certain vector, which need not be one of the coordinate space s axes. When passed a pivotPoint in the object s own coordinate space, this position is used for the center of rotation. There s another convenient way to set the rotation of an object using its transformation matrix. The
pointAt() method sets the contents of the Matrix3D to a transformation that reorients its subject
to point at a certain target. This can shortcut potentially painful calculations, and it s quite convenient for games. Example 34-6 creates a vector pointing in a random direction, uses Vector3D methods you just learned to set its magnitude to the radius of a sphere. This algorithm creates a random point on the surface of a sphere, because the surface of a sphere is the union of all points the same distance from its center. Then pointAt() is used to reorient an arrow to point at the randomly placed target. EXAMPLE 34-6
Matrix3D s pointAt() Method
package { import flash.display.*; import; import flash.geom.Matrix3D; import flash.geom.Vector3D; [SWF(width="500",height="500",backgroundColor="#000000")] public class ch34ex6 extends Sprite { protected const targetSphereRadius:Number = 180; protected var holder:Sprite; protected var target:Shape; protected var arrow:Shape; public function ch34ex6() { holder = new Sprite(); holder.x = stage.stageWidth/2; holder.y = stage.stageHeight/2; addChild(holder); makeArrow(); makeTarget();