Plane Curves: Local Properties
Parametrizations
Position, Velocity, and Acceleration
Curvature
Osculating Circles, Evolutes, and Involutes
Natural Equations
Plane Curves: Global Properties
Basic Properties
Rotation Index
Isoperimetric Inequality
Curvature, Convexity, and the Four-Vertex Theorem
Curves in Space: Local Properties
Definitions, Examples, and Differentiation
Curvature, Torsion, and the Frenet Frame
Osculating Plane and Osculating Sphere
Natural Equations
Curves in Space: Global Properties
Basic Properties
Indicatrices and Total Curvature
Knots and Links
Regular Surfaces
Parametrized Surfaces
Tangent Planes and Regular Surfaces
Change of Coordinates
The Tangent Space and the Normal Vector
Orientable Surfaces
The First and Second Fundamental Forms
The First Fundamental Form
Map Projections (Optional)
The Gauss Map
The Second Fundamental Form
Normal and Principal Curvatures
Gaussian and Mean Curvature
Developable Surfaces and Minimal Surfaces
The Fundamental Equations of Surfaces
Gauss’s Equations and the Christoffel Symbols
Codazzi Equations and the Theorema Egregium
The Fundamental Theorem of Surface Theory
The Gauss–Bonnet Theorem and Geometry of Geodesics
Curvatures and Torsion
Gauss–Bonnet Theorem, Local Form
Gauss–Bonnet Theorem, Global Form
Geodesics
Geodesic Coordinates
Applications to Plane, Spherical and Elliptic Geometry
Hyperbolic Geometry
Curves and Surfaces in n-Dimensional Euclidean Space
Curves in n-Dimensional Euclidean Space
Surfaces in Rn
Appendix: Tensor Notation