ABSTRACT

Geometry always has two parts to it: one part is the description of the objects that can be generated; the other investigates how these objects can be changed or transformed. Any object formed by several vectors may be mapped to an arbitrarily bizarre curved or distorted object—here, we are interested in those maps that map 2D vectors to 2D vectors and are “benign” in some well-defined sense. All these maps may be described using the tools of matrix operations, or linear maps. Matrices are introduced and we emphasize that many of their properties can be studied by just considering the humble 2x2 case, which corresponds to 2D linear maps. The fundamental linear maps described include scaling, reflection, rotation, shear, and projection. We go on to discuss combining linear maps. A free-form deformation application illustrates how linear maps can be used to create an interesting shape. The determinant is introduced, and it is used as a defining feature of the fundamental linear maps, along with the action ellipse. The chapter ends with a list of matrix rules. Sketches and figures illustrate the concepts.