3. Geometric Representations and Transformations

This chapter provides important background material that will be needed for Part II. Formulating and solving motion planning problems require defining and manipulating complicated geometric models of a system of bodies in space. Section 3.1 introduces geometric modeling, which focuses mainly on semi-algebraic modeling because it is an important part of Chapter 6. If your interest is mainly in Chapter 5, then understanding semi-algebraic models is not critical. Sections 3.2 and 3.3 describe how to transform a single body and a chain of bodies, respectively. This will enable the robot to ``move.'' These sections are essential for understanding all of Part II and many sections beyond. It is expected that many readers will already have some or all of this background (especially Section 3.2, but it is included for completeness). Section 3.4 extends the framework for transforming chains of bodies to transforming trees of bodies, which allows modeling of complicated systems, such as humanoid robots and flexible organic molecules. Finally, Section 3.5 briefly covers transformations that do not assume each body is rigid.