Problems related to the visibility and observation of objects and regions form the core focus of this book. The problems considered range from single stationary observers and multiple stationary observers to multiple moving observers. Aside from the first two chapters, each chapter considers a single class of problems, initially concentrating on theory that gives the mathematical underpinnings of the solutions. With the exception of the last chapter, this is followed by material on numerical algorithms for the problems. Each chapter (except for the first) concludes with a set of exercises.

The introductory chapter briefly describes a number of applications for the book’s material. Each area considered is enhanced with references to the literature that the interested reader can pursue. A helpful feature here is that the references are indicated in the section title. The second chapter contains various mathematical preliminaries, defines notions of visibility, and discusses more complex objects. Of particular note here is the discussion of outward normal intersection points and their effects on visibility.

The remaining chapters consider the various types of problems mentioned earlier in more detail. Chapter 3, “Static Optimal Visibility Problems,” as the name suggests, considers optimizing visibility problems for static observers, taking the case of a single observer first and then that of multiple observers. In the case of multiple observers, the author distinguishes the case of simply connected objects from non-simply connected ones, such as a torus. The case of limited angle of view is considered here as well. The chapter concludes with a discussion of computational algorithms and their costs, followed by numerical examples.

Chapter 4, “Visibility-Based Optimal Path Planning,” considers the problem of finding the optimal path required to view the whole of an object when a fixed observer cannot see the whole object from any single location, whether a surface or a 3D object. Think here of a Mars rover, which is restricted to the surface of Mars and views the surface from a mast that extends upwards so that its viewpoint is a fixed distance above the surface. Another example is that of observing the interior of a building with walls such as one finds in a gallery.

In chapter 5, “Visibility-Based Optimal Motion Planning,” the observer is further constrained by Newton’s laws and friction in its search for an optimal path. Thus, for example, the Mars rover must avoid excessive inclines, whether up or down.

Chapter 6, “Multiple Observer Cooperative and Non-Cooperative Optimal Motion Planning,” considers the problems that arise when several observers view a common object. In the cooperative case, the observers will wish to avoid occluding one another’s view, whereas in the non-cooperative case one supposes that the observers compete to maximize the amount that they can see. This chapter concludes with a short list of questions for further study.

The level of mathematical sophistication asked of the reader increases as one works though the book. This is to be expected as the problems become more complex. However, the arguments are fairly complete. Each chapter has a dedicated bibliography where the reader can follow up on the material presented. The book provides a wide-ranging survey of the topic of visibility and can be read with profit by both students and practitioners with a strong background in mathematics.