Geometry, Perspective Drawing, and Mechanisms

Author Don Row hopes that this “blend of theory and application to the everyday world will catch the reader’s imagination” and it does. Four chapters develop geometry based on combinatorial geometry. Four chapters clarify perspective drawing and scene analysis, three investigate engineering mechanisms, and two deal with spherical polyhedra. The text concludes with a chapter on the unifying matroid abstraction. Each section has examples and exercises. The preface contains suggestions for several interesting courses.

The first chapter introduces combinatorial figures, allowing readers to develop intuition about figures with several points and lines. Chapter 2 formalizes the idea of a combinatorial geometry and relates the figures of the first chapter to certain combinatorial geometries. Combinatorial geometries are not restricted to being planar. Chapter 3 looks at planar geometries including projective planes where each pair of coplanar lines intersect. Appropriately the next chapter covers non-planar geometries and projective spaces.

The next section begins the applications with an introduction to perspective with helpful diagrams, reproductions, and practical drawing methods. There are interesting challenges for the reader; for example, to make a 2-point perspective drawing of a small staircase. Chapter 6 covers binocular vision and shows how to construct stereograms, even including a Mathematica program and steps for making a viewing box. (Other sections refer to The Geometer’s Sketchpad, which is not essential but adds a bit extra for those willing to try it.). The next chapter deals with the converse problem of scene analysis, determining whether a planar figure is a perspective drawing of a non-planar figure.

The engineering applications section starts with planar bar-and-joint mechanisms. Examples include a garden sprinkler, an auto hood hinge, a bicycle, a plow, a steam engine and a crane. The following chapter looks at non-planar hinged panel mechanisms. To analyze larger and more complicated mechanisms, Chapter 11 develops graph structures and discusses their rigidity.

Many mathematicians, mathematics students, and others will enjoy this material. Instructors of a geometry course should definitely consider this text. Among Row’s suggestions for such a course are:

• geometry and perspective drawing,
• or geometry and mechanisms,
• or geometry, graphs, and matroids.

The first two choices introduce fascinating applications, while the last illustrates the common structure of geometries, graphs, and matroids.

While this text does not follow the traditional Euclidean approach to geometry, it does provide ample opportunity for creative assignments, guiding teachers to design class projects involving perspective, binocular vision, or mechanisms that are likely to engage students who otherwise find geometry quite abstruse. This novel approach deserves serious consideration.

Art Gittleman (artg@csulb.edu) is Professor of Computer Science at California State University Long Beach.