J.M. Sullivan, TU Berlin A: Curves Diff Geom I, SS 2013 Intro: smooth curves and surfaces in Euclidean space (usu- We instead focus on regular smooth curves α. Then if ally R3). Usually not as level sets (like x2 + y2 = 1) as in ϕ: J → I is a diffeomorphism (smooth with nonvanishing algebraic geometry, but parametrized (like (cos t, sin t)). derivative, so ϕ−1 is also smooth) then α ◦ ϕ is again smooth Euclidean space Rn 3 x = (x ,..., x ) with scalar product and regular. We are interested in properties invariant under P 1 n √ (inner prod.) ha, bi = a · b = aibi and norm |a| = ha, ai). such smooth reparametrization. This is an equivalence rela- tion. An unparametrized (smooth) curve can be defined as an equivalence class. We study these, but implicitly. A. CURVES For a fixed t0 ∈ I we define the arclength function s(t):= R t |α˙(t)| dt. Here s maps I to an interval J of length len(α). t0 I ⊂ R interval, continuous α: I → Rn is a parametrized If α is a regular smooth curve, then s(t) is smooth, with pos- n curve in R . We write α(t) = α1(t), . , αn(t) . itive derivatives ˙ = |α˙| > 0 equal to the speed. Thus it has a The curve is Ck if it has continuous derivs of order up to smooth inverse function ϕ: J → I. We say β = α ◦ ϕ is the ∞ k.(C0=contin, C1,..., C =smooth). Recall: if I not open, arclength parametrization (or unit-speed parametrization) of differentiability at an end point means one-sided derivatives α. We have β(s) = α(ϕ(s)), so β(s(t)) = α(ϕ(s(t))) = α(t). It exist, or equiv. that there is a diff’ble extension. We deal with follows that β has constant speed 1, and thus that the arclength smooth curves except when noted. of β|[a,b] is b − a. Examples: The arclength parametetrization is hard to write down ex- • α(t) = (a cos t, a sin t, bt) helix in R3 plicitly for examples (integrate a square root, then invert the function). But it always exists, and is often the easiest to • α(t) = (t2, t3) curve in R2 with cusp use for theory. (When considering curves with less smooth- ness, e.g., Ck, there is a general principle that no regular • 2 α(t) = (sin t, sin 2t) figure-8 curve in R parametrization is smoother than the arclength parametriza- • α(t) = (t, t2,..., tn) moment curve in Rn tion.) Although for an arbitrary parameter we have used the name Define simple and closed curves (and simple closed curves). t (thinking of time) and written d/dt with a dot, when we use 1 A smooth (or even just C ) curve α has a velocity vector the arclength parametrization, we’ll call the parameter s and n α˙(t) ∈ R at each point. The fundamental thm of calculus says write d/ds with a prime. Of course, for any function f along R b α α − α α |α | α the curve, the chain rule says a ˙(t) dt = (b) (a). The speed of is ˙(t) . We say is regular if the speed is positive (never vanishes). Then the d f ds d f speed is a (smooth) positive function of t. (The cusped curve = , i.e., f 0 = f˙/s˙ = f˙/|α˙|. ds dt dt above is not regular at t = 0; theR others are regular.) The length of α is len(α) = |α˙(t)| dt. We see I Suppose now that α is a regular smooth unit-speed curve. 0 Z b Z b Then its velocity α is everywhere a unit vector, the (unit) tan- 0 |α˙(t)| dt ≥ α˙(t) dt = α(b) − α(a) . gent vector T(s):= α (s) to the curve. (In terms of an arbitrary a a regular parametrization, we have of course T = α/˙ |α˙|.) That is, a straight line is the shortest path. (To avoid using End of Lecture 8 Apr 2013 vector version of integral inequality, take scalar product with α(b) − α(a).) We should best think of T(s) as a vector based at p = α(s), perhaps as an arrow from p to p + T(s), rather than as a point The length of an arbitrary curve can be defined (Jordan) as n total variation: in R . The tangent line to α at the point p = α(s) is the line {p + tT(s): t ∈ R}. (Note about non-simple curves.) (As n X long as α has a first derivative at s, this line is the limit (as len(α) = TV(α) = sup α(t ) − α(t ) . i i−1 h → 0) of secant lines through p and α(s + h). If α is C1 near t0<···<tn∈I i=1 s, then it is the arbitrary limit of secant lines through α(s + h) This is the supremal length of inscribed polygons. (One can and α(s + k).) While velocity depends on parametrization, the show this length is finite over finite intervals if and only if α tangent line and unit tangent vector do not. has a Lipschitz reparametrization (e.g., by arclength). Lips- We are really most interested in properties that are also in- chitz curves have velocity defined a.e., and our integral for- dependent of rigid motion. It is not hard to show that a Eu- mulas for length work fine.) clidean motion of Rn is a rotation A ∈ SO(n) followed by a If J is another interval and ϕ: J → I is an orientation- translation by some vector v ∈ Rn: x 7→ Ax + v. Thus α could preserving homeomorphism, i.e., a strictly increasing surjec- be considered equivalent to Aα + v: I → Rn, t 7→ Aα(t) + v. tion, then α◦ϕ: J → Rn is a parametrized curve with the same Of course, given any two lines in space, there is a rigid mo- image (trace) as α, called a reparametrization of α. (Note: re- tion carrying one to the other. To find Euclidean invariants verse curveα ¯ : − J → Rn,α ¯(t):= α(−t) has the same trace in of curves, we need to take higher derivatives. We define the reverse order – orientation reversing reparam.) curvature vector ~κ := T 0 = α00; its length is the curvature For studying continuous curves, it’s sometimes helpful to κ := |~κ|. allow reparam’s that stop for a while (monotonic but not Recall the Leibniz product rule for the scalar product: if v strictly) – or that remove such a constant interval. and w are vector-valued functions, then (v · w)0 = v0 · w + v · w0. 1 J.M. Sullivan, TU Berlin A: Curves Diff Geom I, SS 2013 In particular, if v ⊥ w (i.e., v · w ≡ 0) then v0 · w = −w0 · v. Rotating orthonormal frame, infinitesimal rotation (speed 0 And if |v| is constant then v ⊥ v. (Geometrically, this is just κ±) given by skew-symmetric matrix. The curvature tells us saying that the tangent plane to a sphere is perpendicular to how fast the tangent vector T turns as we move along the curve the radius vector.) In particular, we have ~κ ⊥ T. at unit speed. Example: the circle α(t) = (r cos t, r sin t) of radius r (parametrized here with constant speed r) has −1 End of Lecture 11 Apr 2013 T = (− sin t, cos t), ~κ = (cos t, sin t), κ ≡ 1/r. r Given regular smooth parametrization α with speed σ := Since T(s) is a unit vector in the plane, it can be expressed s˙ = |α˙|, the velocity is σT, so the acceleration vector is as (cos θ, sin θ) for some θ = θ(s). Although θ is not uniquely determined (but only up to a multiple of 2π) we claim that we · ˙ 2 0 2 α¨ = σT = σ˙ T + σT = σ˙ T + σ T = σ˙ T + σ ~κ. can make a smooth choice of θ along the whole curve. Indeed, 0 if there is such a θ, its derivative is θ = κ±. Picking any θ0 Note the second-order Taylor series for a unit-speed curve R s θ , θ θ θ κ around the point p = α(0) (we assume without further com- such that T(0) = (cos 0 sin 0) define (s):= 0 + 0 ±(s) ds. ment that 0 ∈ I): This lets us prove what is often called the fundamental theo- s2 rem of plane curves (although it really doesn’t seem quite that α(s) = p + sT(0) + ~κ(0) + O(s3). 2 important): Given a smooth function κ± : I → R there exists a smooth unit-speed curve α: I → R with signed curvature κ±; These first terms parametrize a parabola agreeing with α to this curve is unique up to rigid motion. First note that inte- second order (i.e., with the same tangent and curvature vec- grating κ± gives the angle function θ: I → R (uniquely up to a tor). Geometrically, it is nicer to use the osculating circle, the constant of integration), or equivalently gives the tangent vec- unique circle agreeing to second order (a line if ~κ = 0). It has 2 tor T = (cos θ, sin θ) (uniquely up to a rotation). Integrating T radius 1/κ and center p + ~κ/κ . Thus we can also write then gives α (uniquely up to a vector constant of integration, α(s) = p + cos(κs)~κ/κ2 + sin(κs)T/κ + O(s3). that is, up to a translation). (Constant second derivative – parabola – points equivalent by Now suppose α is a closed plane curve, that is, an L- shear.