4 Transforming vectors and graphics

 

This chapter covers

  • Transforming and drawing 3D objects by applying mathematical functions
  • Creating computer animations using transformations to vector graphics
  • Identifying linear transformations, which preserve lines and polygons
  • Computing the effects of linear transformations on vectors and 3D models

With the techniques from the last two chapters and a little creativity, you can render any 2D or 3D figure you can think of. Whole objects, characters, and worlds can be built from line segments and polygons defined by vectors. But, there’s still one thing standing in between you and your first feature-length, computer-animated film or life-like action video game−you need to be able to draw objects that change over time.

Animation works the same way for computer graphics as it does for film: you render static images and then display dozens of them every second. When we see that many snapshots of a moving object, it looks like the image is continuously changing. In chapters 2 and 3, we looked at a few mathematical operations that take in existing vectors and transform them geometrically to output new ones. By chaining together sequences of small transformations, we can create the illusion of continuous motion.

4.1 Transforming 3D objects

 
 

4.1.1 Drawing a transformed object

 
 

4.1.2 Composing vector transformations

 
 
 
 

4.1.3 Rotating an object about an axis

 

4.1.4 Inventing your own geometric transformations

 
 

4.1.5 Exercises

 
 

4.2 Linear transformations

 
 
 
 

4.2.1 Preserving vector arithmetic

 
 
 

4.2.2 Picturing linear transformations

 
 
 

4.2.3 Why linear transformations?

 
 

4.2.4 Computing linear transformations

 
 
 

4.2.5 Exercises

 
 
 

Summary

 
 
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