Differential Geometry of Curves and Surfaces (Relié)

This textbook is well-suited to a variety of readers. For those seeking an elementary text, the prerequisites are minimal, there are plenty of examples and intermediate steps within proofs, and the more excursive applications and advanced topics are clearly identified as optional. For readers bound for graduate school in math or physics, this is a clear, concise, rigorous development of the topic including the deep global theorems.
For the benefit of all readers, the author employs an arsenal of techniques to render the difficult abstract ideas herein more understandable and engaging. For example, over 30o color illustrations bring the mathematics to life, instantly clarifying concepts in ways that grayscale could not. Applications abound ! The study of conformal and equiareal functions is grounded in its application to cartography.
Evolutes, involutes and cycloids are introduced through Christiaan Huygens' fascinating story : in attempting to solve the famous longitude problem with a mathematically-improved pendulum clock, he invented mathematics that would later be applied to optics and gears. Clairaut's Theorem is presented as a conservation law for angular momentum. Green's Theorem makes possible a drafting tool called a planimeter.
Foucault's Pendulum helps one visualize a parallel vector field along a latitude of the earth. Even better, a south-pointing chariot helps one visualize a parallel vector field along any curve in any surface. Throughout this book, applications, metaphors and visualizations are tools that motivate and clarify the rigorous mathematical content, but never replace it. Differential geometry represents a beautiful interwoven collection of visual insights, made rigorous with a bit of calculus and linear algebra.
This book makes it as easy as possible for readers to internalize the proofs and the intuitions that support them.