The authors focus their attention on the differential geometry of planar curves with great depth, although phenomenal books on differential geometry already exist. Many interesting and inspiring geometrical and topological results on planar curves are here presented in an elementary form. The topics are chosen to sharpen the reader’s mathematical intuition for asserted geometric concepts and results. A very good selection of examples guides the reader towards a better understanding of each notion.
The book consists of 8 chapters and 2 appendices, one of which contains answers to selected exercises. Chapter 1 starts by studying curves locally and explores the concept of curvature of a planar curve. The authors continue with global properties in Chapter 2. The notion of winding number of a curve and several applications of this notion are discussed in this chapter. Chapter 3 deals with the rotation index of a differentiable curve. The authors prove Jordan’s theorem for regular curves of class \(C^{2}\) in Chapter 4. The isoperimetric inequality for closed curves in the plane is handled in Chapter 5. Chapter 6 contains convex curves in the plane. Necessary conditions to prove the Four-Vertex Theorem are introduced in Chapter 7. The book ends up with an introduction to the curve-shortening flow and some recent results about it in Chapter 8.