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.

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.

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.

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.

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.

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.

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.