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Transformation Matrices
Pages: 1, 2, 3, 4, 5

Combining transformations

Applying more than one transformation matrix can induce multiple actions. In addition, individual transformation matrices can be combined (via a matrix multiply) to form a complex action. Following is the result of combining a shear matrix with a reflection matrix.

Try a few combinations of your own -- see what you can come up with.

Translation

Currently in our plots, the house is centered on the origin of the plot (0,0). Using vector operations, you can move the house away from the center via translation. This is accomplished by simply adding an offset vector to each of the columns in the matrix. Adding a vector of [5,5] will move the drawing as shown:

Using translation

Give the combination of translation and transformation a try. What happens if you rotate a translated image?

End game

Hopefully this only helped whet your appetite for using NumPy and learning more about linear algebra. While not shown here, extending these techniques to more dimensions (three being most common) can allow more interesting applications. In three dimensions, solid objects can be modeled and rotations and reflections can be generated around multiple axes, creating complex object motions. Such actions are fundamental to graphics drawing, modeling packages, and 3D gaming. While the mathematics of linear algebra may be abstract, they are used in common applications. NumPy helps implement these actions clearly, making this difficult topic easier to understand. If you want to play with more complex examples, consult almost any elementary text on linear algebra or books on graphics or image manipulation.

Looking ahead

Next month we will look past the basic NumPy functions to the additional modules provided in the standard package. Using these modules, more complex programs can be written for a wide variety of applications.

Eric Hagemann specializes in fast algorithms for crunching numbers on all varieties of computers from embedded to mainframe.


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