classical differential geometry

Curves in ${ℝ}^2$

• inflexion point

• singular points of plane curve

• isocline

• curvature (plane curve)

• circle of curvature

• curvature determines the curve

• curvature of Nielsen’s spiral

• osculating curve

• orthogonal curves

• isogonal trajectory

• parallel curves

• properties of parallel curves

• evolute

• evolute of cycloid

• Serret-Frenet equations in ${ℝ}^2$ (http://planetmath.org/SerretFrenetEquationsInMathbbR2)

• famous curves in the plane

• arc-parametrizations

• envelope

• determining envelope

• catacaustic

Curves in ${ℝ}^3$

• Serret-Frenet equations in ${ℝ}^3$ (http://planetmath.org/SerretFrenetFormulas)

• space curve

• level curve

• curvature (http://planetmath.org/CurvatureOfACurve) and torsion (http://planetmath.org/Torsion) of a space curve

• moving trihedron

Surfaces in ${ℝ}^3$

• level curve, level surface

• surface of revolution

• surface normal

• normal section

• normal curvatures

• Meusnier’s theorem

• mean curvature at surface point

• first fundamental form (http://planetmath.org/FirstFundamentalForm)

• second fundamental form

• sphere map and shape operator

• Gaussian curvature and mean curvature

• geodesic

• Gauss-Bonnet theorem

• standard connection in ${ℝ}^3$

• Gauss equation

The space ${ℝ}^3$

• ortho-normal frame fields in ${ℝ}^3$ (or non constant ortho-normal triples of vector fields)

• rate of rotation of an o.f.f.

• euclidean spin connection

• ${ℝ}^3$ Cartan structural equations I, II

Variational calculus

• calculus of variations

• classical isoperimetric problem

• least surface of revolution

• brachistochrone curve

• equation of catenary via calculus of variations