Solid Analytic Geometry

This slim volume is a concise introduction to the basic topics of solid analytic geometry. The content is sufficient in quantity and velocity for a one-semester course for undergraduates. There is a more rigorous consideration of general theory, as opposed to application cases and contrived exercises, than I see in modern texts aimed at the same level. Basically, the author moves from the general to the specific, a trend uncommon in comparable modern texts. For instance, cylinders are introduced: “A cylinder is a surface consisting all of the points on all the lines which are parallel to a given line and which pass through a fixed plane curve in a plane not parallel to the given line.” There is something a tad awkward about these introductions. Yet, I appreciate the approach of beginning verbally before giving the mathematical definition and defining generally instead of building up from simpler examples. This more easily admits of, say, an elliptic or even hyperbolic cylinder, the latter of which would strike many students as contrary to initial definitions and examples and even unsettling.

The economical overview going from vectors to the fundamentals of projective geometry is a good launching point for students considering a transition to higher mathematics, especially future engineers. The basic structure of a section is definitions, possibly some theorems and lemmas, an “illustrative example” (solved exercise), and concluding exercises. While the many exercises do not have solutions, they closely follow the subject matter, making the material enlightening for even the unguided reader with a decent grounding in Euclidean geometry and trigonometry. Dating from 1949, some terminology sounds dated, such as a sheaf of planes referred to as a “pencil”, etc. There are few obstacles in this way, as few fundamentals have drifted in nomenclature on the main subject matter of quadric surfaces as generalizations of conic sections. Enough matrix theory is introduced to support rotations of axes and other transformations.

Tom Schulte is an R&D senior software architect at Plex Systems in Michigan currently focused on API governance.