x | | x | | 2 | = + | y | | y | | 3 | in Java

Integrating EAN-13 Supplement 5 in Java x | | x | | 2 | = + | y | | y | | 3 |
| x | | x | | 2 | = + | y | | y | | 3 |
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Sure, basically, it s the same so far. Instead of two values, x and y, you have a one-column matrix that contains those values. Matrices let you deal with whole systems of values at once. This is just a warm-up, though. To scale up by a factor of 2 horizontally and 3 vertically, I would use these equations:
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You can t use matrix addition for this, obviously, and you can t simply multiply [2 3] by [x y] the sizes are incompatible for matrix multiplication. But if you use a 2 2 matrix instead, you can do matrix multiplication:
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| x | | a b | | x | | ax + by | = = | y | | c d | | y | | cx + dy |
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Part VIII: Graphics Programming and Animation
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Or, in linear equations,
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To scale up x and y, you can ignore y in x s equation and x in y s equation. You can make their contributions nil by using 0 for the b and c terms in the matrix:
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| x | | a 0 | | x | | ax + 0y | | ax | = = = | y | | 0 d | | y | | 0x + dy | | dy |
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So if the computer wanted to scale up by a factor of 2 horizontally and 3 vertically, it might use the matrix
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| 2 0 | | 0 3 |
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and multiply it by the coordinates:
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| x | | 2 0 | | x | | 2x | = = | y | | 0 3 | | y | | 3y |
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This is a transformation matrix! In general, the form
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is a transformation matrix that scales up by sx horizontally and sy vertically. When a computer graphics algorithm is processing millions of points per second, it can do so with terri c speed by thinking of translations as matrices. The most twisted series of skews, scales, rotations, and translations is as simple as a single translation: they all boil down to multiplying a transformation matrix with the point s coordinate vector. When we simple humans look at a rotation matrix, it s dif cult for us to see a rotation. It seems pretty arbitrary compared to the sprite.rotation = 90 statement you could otherwise use. But when a computer looks at a matrix, it sees a tight package of data that it can crank through a million multiplications in the blink of an eye. It may not look semantically meaningful, but the CPU cares not for such things it needs to process numbers. Using transformation matrices and the DisplayObject API, programmers have access to both perspectives on transformation. Let s focus on the math again. You ve derived (or guessed at) a 2 2 scale transformation matrix. The real transformation matrices Flash Player uses internally are 3 3 matrices. With this dimension and a padding value in the coordinate vector to match you can perform translations within the same matrix multiplication. Here s the multiplication decomposed:
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| x | | a c tx | | x | | ax + cy + tx | | y | = | b d ty | | y | = | bx + dy + ty | | 1 | | 0 0 1 | | 1 | | 1 |
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Or, in linear equations,
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x = ax + cy + tx y = bx + dy + ty
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34: Geometric and Color Transformations
I ve renamed a lot of the coef cients in this transformation matrix to match how the Matrix class names them. Can you see how the addition of 1 and tx and ty enables translation in the matrix multiplication The translation elements of the matrix, tx and ty, are only ever multiplied by the new row 1, so they remain independent of x and y.
Kinds of Af ne Transform and Their Matrices
With this general form for a 2D af ne transform, this section examines the kinds of transforms mentioned, without necessarily deriving them.
The Identity Matrix
An identity matrix is a square matrix that, when multiplied by any compatible matrix, leaves it unchanged. Likewise, the identity matrix is a no-transform transform, but I thought it would be good to look at it anyway.
| x | | 1 0 0 | | x | | x | | y | = | 0 1 0 | | y | = | y | | 1 | | 0 0 1 | | 1 | | 1 |
As with all the examples, I encourage you to follow along and perform the matrix multiplication yourself. I promise it will help you gain insight into why the transformation matrices are constructed the way they are. The identity matrix here leaves x = x and y = y. Nothing goes anywhere.