DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

§2. Local Theory: Frenet Frame 17The Frenet formulas actually characterize the local picture of a space curve.Proposition 2.6 (Local canonical form). Let α beasmooth (C 3 or better) arclength-parametrizedcurve. If α(0) = 0, then for s near 0, wehave() ( )α(s) = s − κ2 0κ06 s3 + ... T(0) +2 s2 + κ′ (0κ0 τ)06 s3 + ... N(0) +6 s3 + ... B(0).(Here κ 0 , τ 0 , and κ ′ 0 denote, respectively, the values of κ, τ, and κ′ at 0, and lims→0.../s 3 =0.)Proof. Using Taylor’s Theorem, we writeα(s) =α(0) + sα ′ (0) + 1 2 s2 α ′′ (0) + 1 6 s3 α ′′′ (0) + ...,where lims→0.../s 3 =0. Now,α(0) = 0, α ′ (0) = T(0), and α ′′ (0) = T ′ (0) = κ 0 N(0). Differentiatingagain, we have α ′′′ (0) = (κN) ′ (0) = κ ′ 0 N(0) + κ 0(−κ 0 T(0) + τ 0 B(0)). Substituting, we obtainas required.α(s) =sT(0) + 1 2 s2 κ 0 N(0) + 1 ( 6 s3 −κ 2 0T(0) + κ ′ 0N(0) + κ 0 τ 0 B(0) ) + ...() ( )= s − κ2 0κ06 s3 + ... T(0) +2 s2 + κ′ (0κ0 τ)06 s3 + ... N(0) +6 s3 + ... B(0),□We now introduce three fundamental planes at P = α(0):(i) the osculating plane, spanned by T(0) and N(0),(ii) the rectifying plane, spanned by T(0) and B(0), and(iii) the normal plane, spanned by N(0) and B(0).We see that, locally, the projections of α into these respective planes look like(i) (u, (κ 0 /2)u 2 + ...)(ii) (u, (κ 0 τ 0 /6)u 3 + ...), and(iii) (u 2 ,( √2τ03 √ κ 0)u 3 + ...),where limu→0.../u 3 =0.Thus, the projections of α into these planes look locally as shown in Figure2.4. The osculating (“kissing”) plane is the plane that comes closest to containing α near P (seeNBBTTNosculating plane rectifying plane normal planeFigure 2.4