Differential Geometry of Curves in Space

Differential Geometry of Curves

Parametric Curves

A curve in ℝ³: r(t) = (x(t), y(t), z(t)), t ∈ [a,b].

Arc length: s = ∫_{a}^{b} |r'(t)| dt = ∫ √(x'²+y'²+z'²) dt.

Natural parameter: parametrization by arc length s. Then |r'(s)| = 1.

Frenet Frame

Tangent vector: τ = dr/ds.

Curvature: κ = |dτ/ds| — a measure of how the direction of the tangent changes.

Normal vector: ν = (dτ/ds)/κ.

Binormal vector: β = τ × ν.

{τ, ν, β} — Frenet frame — a right-handed orthonormal triple at each point of the curve.

Torsion: χ = −dβ/ds · ν — a measure of the curve "departing" from the osculating plane (the plane of τ, ν).

Frenet Formulas

dτ/ds = κν, dν/ds = −κτ + χβ, dβ/ds = −χν.

Theorem: a curve is determined (up to a movement) by the pair of functions κ(s) > 0 and χ(s) (the natural equations).

Examples

Circle: κ = 1/R = const, χ = 0. A plane curve ⟺ χ = 0.

Helix (spiral): κ and χ are constants. x = R cos t, y = R sin t, z = pt. Κ = R/(R²+p²), χ = p/(R²+p²).