This book features plane curvesthe simplest objects in differential geometryto illustrate many deep and inspiring results in the field in an elementary and accessible way. After an introduction to the basic properties of plane curves, the authors introduce a number of complex and beautiful topics, including the rotation number (with a proof of the fundamental theorem of algebra), rotation index, Jordan curve theorem, isoperimetric inequality, convex curves, curves of constant width, and the four-vertex theorem. The last chapter connects the classical with the modern by giving an introduction to the curve-shortening flow that is based on original articles but requires a minimum of previous knowledge.
Over 200 figures and more than 100 exercises illustrate the beauty of plane curves and test the reader's skills. Prerequisites are courses in standard one variable calculus and analytic geometry on the plane.
Plane curves
Winding number
Rotation index
Jordan curve theorem
Isoperimetric inequality
Convex curves
The four-vertex theorem
Curve-shortening flow
Appendix A: The class $\mathcal{C}^\infty$ convergence of the curvature
function under the curve-shortening flow
Appendix B: Answers to selected exercises
Bibliography
Index
Hilario Alencar, Federal University of Alagoas, Maceio, Alagoas, Brazil.
Walcy Santos, Federal University of Rio de Janeiro, Brazil.
Gregorio Silva Neto, Universidade Federal de Alagoas, Maceio, Alagoas, Brazil.
