Regular Curves

Differential Geometry · Axiom Academy

Understanding regularity conditions and parametrizations in differential geometry 1. Definition of Regular Curve A parametrized curve α : I → ℝⁿ is called regular if its velocity vector is never zero. The vector α'(t) is called the velocity vector or tangent vector . When it's never zero, the curve has a well-defined tangent direction at every point. The regularity condition ensures that the curve moves smoothly without any pathological behavior. When α'(t) = 0, the curve can have cusps , corners , or stopping points . The tangent direction becomes undefined Geometric quantities (curvature, arc length) may fail The curve may trace back on itself Regular curves guarantee that: Arc length parametrization exists The Frenet-Serret frame can be constructed 3. Examples: Regular vs Non-Regular Since ||α'(t)|| = 1 ≠ 0, this is regular! At t = 0, we have α'(0) = (0, 0), so this is NOT regular. 4. Reparametrization of Curves The same geometric curve can be traced by different parametrizations. A reparametrization is a change of parameter that traces the same curve, possibly at different speeds. is a reparametrization of α. The new velocity is given by the chain rule: 5. Equivalent Parametrizations Two parametrizations are equivalent if they trace the same geometric curve, possibly in different directions or at different speeds. Positive equivalence: h'(s) > 0 (same orientation) Negative equivalence: h'(s) < 0 (opposite orientation) α(t) = (cos t, sin t) - counterclockwise

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