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The Matrix3D class stores a 4 4 matrix. Although you can store any 4 4 matrix in it, it s designed for 3D transformation matrices. Unlike the Matrix class, in which you could set the elements of the matrix directly through instance properties, the Matrix3D class has too many elements (16) for this to be very manageable. Instead, you can access its contents directly through a Vector of 16 Numbers (Vector.<Number>). Just like 2D transformation matrices, however, it s much easier to use the convenience methods of Matrix3D to construct and concatenate the proper matrices for you. You can pass these 16 values to Matrix3D s constructor or set them with the rawData property. The values are in column-major order that is, they are stored from top to bottom and then left to right. If you don t pass a Vector of values to Matrix3D s constructor, a 4 4 identity matrix is created. The rawData property returns a copy of the matrix s elements; to change them you have to assign a whole new Vector to the rawData property. (You can t modify it inline.) The default Matrix3D constructed when you pass nothing to the constructor is a 4 4 identity matrix. Of course, you can restore the matrix to the identity matrix at any time by calling identity() on it. All this is shown in Example 34-5. EXAMPLE 34-5



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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;

YO= (0x241fb061,0x6b119143,Oxd4021163,0x4f73aca9)

EXAMPLE 34-5





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Using Equation (22.5), rx = for white noise and N = 1 25, we obtain that the theoretical correlation coefficient between features in Equations (22.1) and (22.2) is also nonzero: rRI = 0 13, where is the Dirac function, N is the number of periods related to the resonant frequency on the signal duration T . Using Equation (22.21), we find the theoretical effectiveness gain GPSD = 1 31. One can see that the simulation results match the theoretical results. We consider the experimental fatigue crack diagnostics of objects using the forced oscillation method. The nonlinear equations of a cracked object motion under white Gaussian noise stationary excitation can be written as follows [2,41]: x+2 x+2 where =

(continued)

Y = 1

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:

(Ox47dc28e3,0~424f e3bf,Oxb4498cd8,0~75b4ddef) (Oxlb72328c,Ox4a05f 4c8,0x39a9974f,0x72750024)

identity():void Replaces the matrix with a 4 4 identity matrix. transpose():void Replaces the matrix with its transpose. Transposing a matrix swaps

kS kC c c , k and kC are the stiffnesses at tension = C = S = m C m S 2 kS m 2 kC m and compression, S and C are the damping ratios at tension and compression. At compression, the crack is closed and the object behaves like a continuum; therefore, the stiffness is the same as that of the object without a crack, i.e. kC = k. At tension, the crack is opened and the k , object is discontinuous; therefore, the stiffness decreases with the quantity k = kC kS k = k k is the stiffness ratio. A relative crack size characterizes the stiffness ratio [2,41,42]. The basis for using this method lies in the fact that the level of the object nonlinearity changes with the crack size [2,41,42]. We consider the two-class diagnostics of the object stiffness ratio: k = kj for class Hj , j = 0 1. This consideration is generic because it is independent of the correlation between the stiffness ratio and relative crack size. We employ the following recognition (diagnostic) features:

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.

b. Describe a practical attack that, could be used t o recover a plaintext message consisting of multiple blocks, provided that you know the underlying message consists of English text, represented as ASCII. 23. The purpose of this problem is to explore the subkey recovery in the Akelarre attack. Show that in (4.23) it is possible to recover L and either K O K3, K4, K7 and K1 @ Ks, K2 @ Ktj , (4.40) or

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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