Lecture 12 Curves in 3 - Space

Some Definitions and Notations. In this section α : [0, L] → R will denote a curve in R that is parametrized by arclength and is of class C, k ≥ 2. We recall that the unit tangent to α at s is defined by −→t (s) := α′(s). (That it is a unit vector is just the definition of α being parametrized by arclength.) The curvature of α is the non-negative real-valued function k : [0, L] → R defined by k(s) := ‖α′′(s)‖, and we call α a Frenet curve if its curvature function is strictly positive. For a Frenet curve, we define its normal vector −→n (s)at s by −→n (s) := α ′′(s) k(s) . Since −→ t (s) has constant length one, it is orthogonal to its derivative −→t (s) = α′′(s), hence: 12.0.1 First Frenet Equation. If α : [0, L] → R is a Frenet curve then its normal vector −→n (s) is a unit vector that is orthogonal to −→t (s), and −→t (s) = k(s)−→n (s).