Book contents
- Frontmatter
- Contents
- Preface
- Notation
- 1 Euclidean geometry
- 2 Curve theory
- 3 Classical surface theory
- 4 The inner geometry of surfaces
- 5 Geometry and analysis
- 6 Geometry and topology
- Appendix A Hints for solutions to (most) exercises
- Appendix B Formulary
- Appendix C List of symbols
- References
- Index
- Plate section
2 - Curve theory
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Notation
- 1 Euclidean geometry
- 2 Curve theory
- 3 Classical surface theory
- 4 The inner geometry of surfaces
- 5 Geometry and analysis
- 6 Geometry and topology
- Appendix A Hints for solutions to (most) exercises
- Appendix B Formulary
- Appendix C List of symbols
- References
- Index
- Plate section
Summary
We analyse curves in n-dimensional space with a special focus on plane curves and space curves. Length, curvature and torsion are introduced. We prove Hopf's Umlaufsatz for simple closed curves, characterise convex curves and derive the four-vertex theorem. The isoperimetric inequality, which compares the length of a simple closed plane curve with the enclosed area, is proved using the Fourier series. We show that for a given curvature and torsion the resulting space curve is unique up to a Euclidean motion. We investigate how much a space curve needs to curve if it is closed and make the result even stronger in the case that the space curve is knotted.
Curves in ℝn
We now want to use the tools of differentiation and integration to describe curves in n-dimensional space. We usually graphically imagine a curve as a bent line in space. Mathematically we express this as follows:
Definition 2.1.1 Let I ⊂ ℝ be an interval. A parametrised curve is a map c : I → ℝn that can be differentiated infinitely often. A parametrised curve is said to be regular if its velocity vector does not vanish anywhere; ċ(t) ≠ 0 for all t ∈ I.
The interval I from the definition may be open, closed or half-open; furthermore, I can be bounded or unbounded. The condition ċ(t) ≠ 0 ensures that the point c(t) on the curve moves at t ∈ I. In particular, this excludes the constant map c(t) = c0.
- Type
- Chapter
- Information
- Elementary Differential Geometry , pp. 22 - 80Publisher: Cambridge University PressPrint publication year: 2010