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Above, you uniformly scale up a vector <x, y, z> by a factor of 2 by using a 3 3 scale transformation matrix. You should be familiar with the rules for matrix multiplication by now. Recall also that the identity matrix I is a square matrix with 1s down the diagonal. Following is the 3 3 identity matrix.

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You might notice that this is the same as scaling by a factor of 1, which should, of course, have no effect! When using 3D transformation matrices, it s important that all the matrices are invertible that is, for matrix M there exists a matrix M 1 such that M M 1 = I. The matrix M 1 is the inverse of M. You can check whether a matrix is invertible by calculating its determinant. Matrices with nonzero determinants are invertible. You won t have to know how to calculate a determinant to understand the concepts in this chapter; you can nd out more in the recommended references. The inverse of a transformation matrix reverses the effects of that transformation. Unlike 3D transformation matrices, the 2D transformation matrices in ActionScript 3.0 are nonsquare, so they can t have inverses. A nal note on matrices and vectors. When you break up a matrix, its rows and columns are vectors, and it s not uncommon to look at the rows or columns that make up a matrix. For example, the second row (or the second column, as the matrix is diagonal) of the previous identity matrix is the vector <0, 1, 0>.

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Angular orientation can be measured in many ways. Orientation describes the rotation in three dimensions of an object. You can also talk about angular displacement instead of orientation, just as vectors are interchangeable with points. In any case, Flash Player gives you access to three ways of describing orientations: Euler angles These de ne an orientation as a series of rotations around the coordinate space s axes. To achieve a certain direction, you rotate around the x-axis, then the y, then the z. You may be familiar with roll, pitch, and yaw. Those are Euler angles. Euler angles are famously easy to understand but infamously bad for interpolation and for gimbal lock, where two axes become coplanar during the rotation, eliminating a degree of freedom. Axis angles These describe orientation as a scalar angular rotation around some vector, which is not necessarily an axis vector. The vector shows which direction the object is facing, and the rotation about it illustrates which way is facing up. This form is also easy to understand and can simplify some calculations like movement in the direction of an object. Quaternions These are notoriously weird representations of a rotation by a four-dimensional number, which is actually a kind of complex number in which there are three imaginary parts and one real part rather than one imaginary part and one real part. It can be written as the 4D vector <x, y, z, w>. Quaternions are great at interpolating orientations smoothly and are easy to concatenate.

approx.: 2.0

Recall that for 2D af ne transformations, an extra column was added onto the transformation matrix to enable translations (and extend linear transformations to af ne transformations). Similarly, you can perform af ne transformations in three dimensions by adding a fourth dimension to the matrix:

6. Trudy wants to attack a block cipher that has a 64-bit key and 64-bit blocks. Each time she attacks this cipher, she can conduct a chosen plaintext attack and the cipher is used in ECB mode.

| | | | x y z 1 | | | | | = | | | a d g 0 b e h 0 c f i 0 tx ty tz 1 | | | | | | | | x y z 1 | | | |

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