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Matrix Functions

The Movie Toolbox provides a number of functions that allow you to work with transformation matrices. This chapter describes those functions.

The Transformation Matrix

The Movie Toolbox makes extensive use of transformation matrices to define graphical operations that are performed on movies when they are displayed. A transformation matrix defines how to map points from one coordinate space into another coordinate space. By modifying the contents of a transformation matrix, you can perform several standard graphical display operations, including translation, rotation, and scaling. The Movie Toolbox provides a set of functions that make it easy for you to manipulate translation matrices. This section provides an introduction to matrix operations in a graphical environment.

The matrix used to accomplish two-dimensional transformations is described mathematically by a 3-by-3 matrix. Figure 10-1 shows a sample 3-by-3 matrix. Note that QuickTime assumes that the values of the matrix elements u and v are always 0.0, and the value of matrix element w is always 1.0.

Figure 10-1  A point transformed by a 3-by-3 matrix

During display operations, the contents of a 3-by-3 matrix transform a point (x,y) into a point (x',y') by means of the following equations:

x' = ax + cy + t(x)

y' = bx + dy + t(y)

For example, the matrix shown in Figure 10-2 performs no transformation. It is referred to as the identity matrix.

Figure 10-2  The identity matrix

Using the formulas discussed earlier, you can see that this matrix would generate a new point (x',y') that is the same as the old point (x,y):

x' = 1x + 0y + 0

y' = 0x + 1y + 0

x' = y and y' = y

To move an image by a specified displacement, you perform a translation operation. This operation modifies the x and y coordinates of each point by a specified amount. The matrix shown in Figure 10-3 describes a translation operation.

Figure 10-3  A matrix that describes a translation operation

You can stretch or shrink an image by performing a scaling operation. This operation modifies the x and y coordinates by some factor. The magnitude of the x and y factors governs whether the new image is larger or smaller than the original. In addition, by making the x factor negative, you can flip the image about the x-axis; similarly, you can flip the image horizontally, about the y-axis, by making the y factor negative. The matrix shown in Figure 10-4 describes a scaling operation.

Figure 10-4  A matrix that describes a scaling operation

Finally, you can rotate an image by a specified angle by performing a rotation operation. You specify the magnitude and direction of the rotation by specifying factors for both x and y. The matrix shown in Figure 10-5 rotates an image counterclockwise by an angle q.

Figure 10-5  A matrix that describes a rotation operation

You can combine matrices that define different transformations into a single matrix. The resulting matrix retains the attributes of both transformations. For example, you can both scale and translate an image by defining a matrix similar to that shown in Figure 10-6.

Figure 10-6  A matrix that describes a scaling and translation operation

You combine two matrices by concatenating them. Mathematically, the two matrices are combined by matrix multiplication. Note that the order in which you concatenate matrices is important; matrix operations are not commutative.

Transformation matrices used by the Movie Toolbox contain the following data types:

[0] [0] Fixed       [1] [0] Fixed       [2] [0] Fract

[0] [1] Fixed       [1] [1] Fixed       [2] [1] Fract

[0] [2] Fixed       [1] [2] Fixed       [2] [2] Fract

Each cell in this table represents the data type of the corresponding element of a 3-by-3 matrix. All of the elements in the first two columns of a matrix are represented by Fixed values. Values in the third column are represented as Fract values. The Fract data type specifies a 32-bit, fixed-point value that contains 2 integer bits and 30 fractional bits. This data type is useful for accurately representing numbers in the range from -2 to 2.

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