Curves I: Curvature and Torsion. Table of contents

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1 Math 48 Fall 07 Curves I: Curvature and Torsion Disclaimer. As we have a textbook, this lecture note is for guidance and sulement only. It should not be relied on when rearing for exams. In this lecture we study how a curve curves. We will show that the curving of a general curve can be characterized by two numbers, the curvature and the torsion. The required textbook sections are:.,.. The examles in this note are mostly dierent from examles in the textbook. Please read the textbook carefully and try your hands on the exercises. During this lease don't hesitate to contact me if you have any questions. Table of contents Curves I: Curvature and Torsion Curvature Curvature for arc length arametrized curves Alternative characterization of the curvature (otional) Examles Torsion The osculating lane Denition of torsion Examles Curvature and torsion in general arametrization Formulas

2 Dierential Geometry of Curves & Surfaces. Curvature Curvature measures how quickly a curve turns, or more recisely how quickly the unit tangent vector turns... Curvature for arc length arametrized curves Consider a curve (s): (; ) 7! R. Then the unit tangent vector of (s) is given by T (s) := _(s). Consequently, how quickly T (s) turns can be characterized by the number which call the curvature of the curve. (s) := T_(s) = k(s)k () As k_(s)k =, we have (s)?_(s). This leads to the following denition. Definition. Let (s) be a curve arametrized by arc length. Then its curvature is dened as (s) := k(s)k. We further denote by N(s) the unit vector (s)/k(s)k and call it the normal vector at s. We also denote the unit tangent vector _(s) by T (s). Formulas for curves in arc length arametrization. Curvature. Tangent and Normal vectors. (s 0 ) = k(s 0 )k: () T (s 0 ) = _(s 0 ); N(s 0 ) = (s 0) k(s 0 )k : ().. Alternative characterization of the curvature (otional) Consider a curve (s): (; ) 7! R. Let = (s 0 ) for some s 0 (; ). We try to understand how quickly (s) turns aways from the tangent line at x(s 0 ). The equation for the tangent line is (s 0 ) + t _(s 0 ). The distance from a oint (s) to the tangent line is := k((s) (s 0 )) _(s 0 )k (4) Note that here we have used the fact that _(s 0 ) is a unit vector. Now recall Taylor exansion: (s) (s 0 ) = _(s 0 ) (s s 0 ) + (s 0) (s s 0 ) + R(s; s 0 ) (5) where lim s!s0 kr(s; s 0 )k (s s 0 ) = 0. Substituting (5) into (4), we see that = (s s 0) ((s 0 ) _(s 0 )) + R(s; s 0 ) _(s 0 ) (6)

3 Math 48 Fall 07 and consequently lim s!0 (s s 0 ) / = k(s 0) _(s 0 )k (7) Exercise. Prove (7). Thus we see that the quantity k(s 0 ) _(s 0 )k=(s 0 ) measures how the curve curves at the oint (s 0 ). Exercise. Prove that k(s 0 ) _(s 0 )k = k(s 0 )k = (s 0 ). Exercise. (7) can be derived slightly dierently as follows. i. Find T such that (s) [(s 0 ) + T _(s 0 )]?_(s 0 ). ii. Then = k(s) [(s 0 ) + T _(s 0 )]k. iii. Calculate the limit lim.. Examles (s s 0) /. Examle. For the unit circle, the curvature is constantly. For a circle with radius R, the curvature is constantly /R. sin t. We Examle. (Shifrin06) Let (t) = calculate cos t + sin t; cos t; cos t Tangent vector: _(t) = sin t + cos t; sin t; sin t cos t k_(t)k = so we are already in arc length arametrization. We have k(t)k =. So the curvature of this curve is constantly. Examle 4. (Shifrin06) Let (t) = e t ; e t ; t. We calculate Tangent vector: and therefore k_(t)k = e t + e t. Solve we obtain S(t) = e t e t. Solve t as a function of s: _(t) = e t ; e t ; (8) (9) S 0 (t) = e t + e t (0) e t = s + s + 4 : ()

4 Dierential Geometry of Curves & Surfaces Thus (s) is given by s + s + 4 s ; + 4 s!! s + s ; + 4 ln : () We calculate _(s) = + s s ; s + 4 s + 4 ; s + 4! : () To make sure our calculation is correct, we check k_(s)k = : (4) Finally we calculate (s) = 8 (s + 4) / ; 8 (s + 4) / ; s (s + 4) / (5) which gives (s) = k(s)k = s + 64 s : (6) + 4. Torsion Torsion measures how quickly a curve twists... The osculating lane Motivation. Consider a oint on a sace curve. We have seen that to measure how quickly it curves, we should measure the rate of change for the unit tangent vector. Similarly, to measure how quickly it twists, we should measure the change rate of the tangent lane. The osculating lane. Let (s) be a sace curve. Its osculating lane at (s 0 ) is the lane assing (s 0 ) that is sanned by the unit tangent vector T (s 0 ) := _(s 0 ) and the unit normal vector N(s 0 ) := (s 0) k(s 0 )k. We see that the osculating lane contains the tangent line. The unit normal vector of the osculating lane is then given by B(s) := T (s) N(s) (7) which we call the unit binormal vector of the curve (s). Exercise 4. Prove that T (s) = N(s) B(s), N(s) = B(s) T (s). Among all the lanes containing the tangent line, the osculating lane is the one that ts the curve best. See Exercise 5 below. 4

5 Math 48 Fall 07.. Denition of torsion How quickly the osculating lane turns is clearly characterized by how quickly the unit binormal vector turns. We calculate B _ (s) = T _ (s) N(s) + T (s) N _ (s) = T (s) N _ (s): (8) Now notice that kn(s)k = =) N_ (s) N(s) = 0. As T (s) N(s) = 0 too, we see that B_ (s) k N(s). Consequently there is a scalar function (s) such that B_ (s) = (s) N(s): (9) We call (s) the torsion of the curve (s). Formula for (s). We calculate N _ (s) = d (s) dt (s) Thus = (s) (s) _(s) (s): (0) (s) (s) = T (s) N_ (s) N(s) T (s) (s) = (s) _(s) (s) = N(s) (s) = Formula for torsion (arc length arametrization). _(s) T (s) (s) (s) N(s) (_(s) (s)) (s) : () (s) (s) = (_(s) (s)) (s) (s) () Remark 5. Alternative denition of torsion. Distance to osculating lane. We easily see that = k((s) (s 0 )) B(s 0 )k () Exercise 5. Prove that, among all lanes assing (s 0 ), the osculating lane is the only one satisfying = 0: (4) (s s 0 ) lim s!s 0 Taylor exansion of x(s) x(s 0 ) to order three: (s) (s 0 ) = _(s 0 ) (s s 0 ) + (s 0) kr(s; s where lim 0 )k s!s0 = 0. (s s 0 ). There is no articular reason for the negative sign. (s s 0 ) + (s 0) 6 (s s 0 ) + R(s; s 0 ) (5) 5

6 Dierential Geometry of Curves & Surfaces Substituting (5) into () we see that.. Examles lim s!s 0 (s s 0 ) /6 = j (s 0) B(s 0 )j: (6) Exercise 6. Show that j (s 0 ) B(s 0 )j = (_(s) (s)) (s) (s) : (7) Comare to (). Discuss ossible reasons for the dierence. Do you think () is a more reasonable denition for torsion? Why? Examle 6. We calculate the torsion of the curve (t) = cos t + sin t; cos t; cos t sin t. Note that we have seen there that t is already the arc length arameter. _(t) = sin t + cos t; sin t; sin t cos t ; (8) (t) = cos t sin t; cos t; cos t + sin t = x(t): (9) Thus clearly (_(t) (t)) (t) = 0 =) (t) = 0: (0). Curvature and torsion in general arametrization.. Formulas The key idea is that ; ; T ; N ; B should be indeendent of arametrization. In other words, if (t) and (s) are two arametrizations of the same curve, and = (t 0 ) = (s 0 ), then we must have (t 0 ) = (s 0 ); (t 0 ) = (s 0 ), and so on. We will try to obtain the formulas intuitively here. For rigorous derivation, lease see. of the textbook. In the following let (t) be a curve not necessarily in arc length arametrization. T is the unit tangent vector. So we must have T (t) = _(t) k_(t)k : () B. Since N clearly lies in the lane sanned by _ and, B k _. Consequently we must have _(t) (t) B(t) = k_(t) (t)k : () N. We have (_(t) (t)) _(t) N(t) = B(t) T (t) = k_(t) (t)k k_(t)k : () 6

7 Math 48 Fall 07. We have = dt ds N = k_(t)k T_ N which gives (t) [(_(t) (t)) _(t)] k_(t) (t)k (t) = = : (4) k_(t)k k_(t) (t)k k_(t)k. We have = db N which gives ds = (_(t) (t)) [(_(t) (t)) _(t)] k_(t)k k_(t) (t)k (_(t) (t)) (t) = k_(t) (t)k (_(t) (t)) (t) = : (5) k_(t) (t)k Exercise 7. In the above we have used the vector identity (u v) w = (v w) u + (u w) v (6) for u; v; w R. Prove this identity and identify where it is used. Warning DG of Curves: Formulas for general arametrization = = k_ k k_k : (7) (_ ) k_ k : (8) T = _ k_ k ; (9) B = _ k_ k ; (40) N = B T : (4) In exams, the formulas (7) and (8) will be rovided, but (840) will not be rovided. 7

Lecture 8: The First Fundamental Form Table of contents

Math 38 Fall 2016 Lecture 8: The First Fundamental Form Disclaimer. As we have a textbook, this lecture note is for guidance and sulement only. It should not be relied on when rearing for exams. In this