Appendix A Geometry of Curves “Arc, amplitude, and curvature sustain a similar relation to each other as time, motion and velocity, or as volume, mass and density.” Carl Friedrich Gauss The rest of this lecture notes is about geometry of curves and surfaces in R2 and R3. It will not be covered during lectures in MATH 4033 and is not essential to the course. However, it is recommended for readers who want to acquire workable knowledge on Differential Geometry. A.1. Curvature and Torsion A.1.1. Regular Curves. A curve in the Euclidean space Rn is regarded as a function r(t) from an interval I to Rn. The interval I can be finite, infinite, open, closed n n or half-open. Denote the coordinates of R by (x1, x2, ... , xn), then a curve r(t) in R can be written in coordinate form as: r(t) = (x1(t), x2(t),..., xn(t)). One easy way to make sense of a curve is to regard it as the trajectory of a particle. At any time t, the functions x1(t), x2(t), ... , xn(t) give the coordinates of the particle in n R . Assuming all xi(t), where 1 ≤ i ≤ n, are at least twice differentiable, then the first derivative r0(t) represents the velocity of the particle, its magnitude jr0(t)j is the speed of the particle, and the second derivative r00(t) represents the acceleration of the particle. As a course on Differential Manifolds/Geometry, we will mostly study those curves which are infinitely differentiable (i.e. C¥). For some technical purposes as we will explain later, we only study those C¥ curves r(t) whose velocity r0(t) is never zero. We call those curves: Definition A.1 (Regular Curves). A regular curve is a C¥ function r(t) : I ! Rn such that r0(t) 6= 0 for any t 2 I. Example A.2. The curve r(t) = (cos(et), sin(et)), where t 2 (−¥, ¥), is a regular curve since r0(t) = −et sin(et), et cos(et) and jr0(t)j = et 6= 0 for any t. 151 152 A. Geometry of Curves 2 2 However, er(t) = (cos t , sin t ), where t 2 (−¥, ¥), is not a regular curve since 0 2 2 0 er (t) = (−2t sin t , 2t cos t ) and so er (0) = 0. Although both curves r(t) and er(t) represent the unit circle centered at the origin in R2, one is regular but another is not. Therefore, the term regular refers to the parametrization rather than the trajectory. A.1.2. Arc-Length Parametrization. From Calculus, the arc-length of a curve r(t) from t = t0 to t = t1 is given by: Z t 1 0 r (t) dt. t0 Now suppose the curve r(t) starts at t = 0 (call it the initial time). Then the following quantity: Z t 0 s(t) := r (t) dt 0 measures the distance traveled by the particle after t unit time since its initial time. Given a curve r(t) = (cos(et − 1), sin(et − 1)), we have r0(t) = (−et sin(et − 1), et cos(et − 1)), 0 r (t) = et 6= 0 for any t 2 (−¥, ¥). Therefore, r(t) is a regular curve. By an easy computation, one can show s(t) = et − 1 and so, regarding t as a function of s, we have t(s) = log(s + 1). By substituting t = log(s) into r(t), we get: r(t(s)) = r(log(s + 1)) = cos(elog(s+1) − 1), sin(elog(s+1) − 1) = (cos s, sin s). The curve r(t(s)) is ultimately a function of s. With abuse of notations, we denote r(t(s)) simply by r(s). Then, this r(s) has the same trajectory as r(t) and both curves at C¥. The difference is that the former travels at a unit speed. The curve r(s) is a reparametrization of r(t), and is often called an arc-length parametrization of the curve. However, if we attempt to do find a reparametrization on a non-regular curve 2 2 say er(t) = (cos(t ), sin(t )), in a similar way as the above, we can see that such the reparametrization obtained will not be smooth. To see this, we first compute ( Z t Z t t2 if t ≥ 0; ( ) = r0( ) = j j = s t e t dt 2 t dt 2 0 0 −t if t < 0. Therefore, regarding t as a function of s, we have (p s if s ≥ 0; t(s) = p − −s if s < 0. Then, ( (cos(s), sin(s)) if s ≥ 0; r(s) := r(t(s)) = e e (cos(−s), sin(−s)) if s < 0, or in short, er(s) = (cos(s), sin jsj), which is not differentiable at s = 0. It turns out the reason why the reparametrization by s works well for r(t) but not for er(t) is that the former is regular but the later is not. In general, one can always reparametrize a regular curve by its arc-length s. Let’s state it as a theorem: A.1. Curvature and Torsion 153 Theorem A.3. Given any regular curve r(t) : I ! Rn, one can always reparametrize it by arc-length. Precisely, let t0 2 I be a fixed number and consider the following function of t: Z t 0 s(t) := r (t) dt. t0 Then, t can be regarded as a C¥ function of s, and the reparametrized curve r(s) := r(t(s)) is d r( ) = a regular curve such that ds s 1 for any s. Proof. The Fundamental Theorem of Calculus shows Z t ds d 0 0 = r (t) dt = r (t) > 0. dt dt t0 We have jr0(t)j > 0 since r(t) is a regular curve. Now s(t) is a strictly increasing function of t, so one can regard t as a function of s by the Inverse Function Theorem. Since s(t) is C¥ (because r(t) is C¥ and jr0(t)j 6= 0), by the Inverse Function Theorem t(s) is C¥ too. d r( ) = To verify that ds s 1, we use the chain rule: d dr dt r(s) = · ds dt ds 1 = r0(t) · ds dt d 0 1 r(s) = r (t) · = 1. ds jr0(t)j Exercise A.1. Determine whether each of the following is a regular curve. If so, reparametrize the curve by arc-length: (a) r(t) = (cos t, sin t, t), t 2 (−¥, ¥) (b) r(t) = (t − sin t, 1 − cos t), t 2 (−¥, ¥) A.1.3. Definition of Curvature. Curvature is quantity that measures the sharpness of a curve, and is closely related to the acceleration. Imagine you are driving a car along a curved road. On a sharp turn, the force exerted on your body is proportional to the acceleration according to the Newton’s Second Law. Therefore, given a parametric curve r(t), the magnitude of the acceleration jr00(t)j somewhat reflects the sharpness of the path – the sharper the turn, the larger the jr00(t)j. However, the magnitude jr00(t)j is not only affected by the sharpness of the curve, but also on how fast you drive. In order to give a fair and standardized measurement of sharpness, we need to get an arc-length parametrization r(s) so that the “car” travels at unit speed. Definition A.4 (Curvature). Let r(s) : I ! Rn be an arc-length parametrization of a path g in Rn. The curvature of g is a function k : I ! R defined by: 00 k(s) = r (s) . Remark A.5. Since an arc-length parametrization is required in the definition, we talk about curvature for only for regular curves. 154 A. Geometry of Curves Another way (which is less physical) to understand curvature is to regard r00(s) as d T(s) where T(s) := r0(s) is the unit tangent vector at r(s). The curvature k(s) is then ds d T( ) T( ) given by ds s which measures how fast the unit tangents s move or turn along the curve (see Figure A.1). Figure A.1. curvature measures how fast the unit tangents move Example A.6. The circle of radius R centered at the origin (0, 0) on the xy-plane can be parametrized by r(t) = (R cos t, R sin t). It can be easily verified that jr0(t)j = R and so r(t) is not an arc-length parametrization. To find an arc-length parametrization, we let: Z t Z t 0 s(t) = r (t) dt = R dt = Rt. 0 0 s Therefore, t(s) = R as a function of s and so an arc-length parametrization of the circle is: s s r(s) := r(t(s)) = R cos , R sin . R R To find its curvature, we compute: d s s r0(s) = R cos , R sin ds R R s s = − sin , cos R R 1 s 1 s r00(s) = − cos , − sin R R R R 00 1 k(s) = r (s) = . R 1 Thus the curvature of the circle is given by R , i.e. the larger the circle, the smaller the curvature. Exercise A.2. Find an arc-length parametrization of the helix: r(t) = (a cos t, a sin t, bt) where a and b are positive constants. Hence compute its curvature. A.1. Curvature and Torsion 155 Exercise A.3.