Curves Summary

Differential Geometry · Axiom Academy

SUMMARY Curves in Differential Geometry Let's review the foundational concepts of parametric curves, their geometric properties, and the invariants that completely determine their shape. Definition: A smooth curve is a map α: I → ℝ³ where I is an interval Position Vector: α(t) = (x(t), y(t), z(t)) traces out the curve in space Velocity Vector: α'(t) is tangent to the curve and indicates direction of motion Why Parametrization Matters: Same geometric curve can have many parametrizations with different speeds Regular Curve: A curve is regular if α'(t) ≠ 0 for all t Geometric Meaning: No stopping, no cusps, no self-intersections at a point Why It's Essential: Regularity ensures the curve has a well-defined tangent line everywhere Example: The curve α(t) = (t², t³) fails regularity at t = 0 (cusp point) Arc Length Function: s(t) = ∫₀ᵗ ||α'(u)|| du measures distance traveled along curve Unit Speed: A curve parametrized by arc length satisfies ||α'(s)|| = 1 Natural Parametrization: Arc length is the "most natural" way to describe a curve geometrically Key Property: Arc length is independent of parametrization—it's a geometric invariant Tangent Vector T: T = α'/||α'|| , unit vector in direction of motion Normal Vector N: N = T'/||T'|| , points toward center of curvature Binormal Vector B: B = T × N , perpendicular to osculating plane Moving Frame: T, N, B forms an orthonormal basis that moves along the curve Definition: κ = ||T'(s)|| = ||α''(s)|| for unit-speed curves

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