ABSTRACT
We are going to ascend now from the flat world of two dimensions into the real world of three dimensions. When working in three dimensions, we shall insist predominantly on coordinate-free methods. Even in two dimensions we adopted coordinate-free techniques to describe shapes because it is easier to represent geometry without troubling about coordinates. For example, it is simpler to describe a bump using a turtle program rather than trying to delineate the coordinates of all the vertices of the bump. Similarly, we invoke affine transformations-translation, rotation, scaling, and shear-to move and reshape geometry without worrying about the entries-the coordinates-of the corresponding matrices. Coordinates are useful for computations, but conceptually we prefer to work at a higher level of abstraction. Turtle programs and affine transformations were our entry to coordinate-free methods in two dimensions.
