DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

26 Chapter 1. Curvesintersect C either in a line segment or in a single point (at the top and bottom); that is, by movingthese planes from the bottom of C to the top, we fill in a disk, so C is unknotted. □3.2. Plane Curves. We conclude this chapter with some results on plane curves. First, itmakes sense to consider curvature with a sign. Rather than proceeding as we did in Section 2,instead, given an arclength-parametrized curve α, define N(s) sothat {T(s), N(s)} is a righthandedbasis for R 2 (i.e., one turns counterclockwise from T(s) toN(s)), and then set T ′ (s) =κ(s)N(s), as before. So κ>0 when T is twisting counterclockwise and κ0}, U 2 = {(x, y) ∈ S 1 :x0}, and U 4 = {(x, y) ∈ S 1 : y