3 Curvature 3.1 Curvature Now that we re parametrizing curves, it makes sense to wonder how we might measure the extent to which a curve actually curves. That is, how much does our path deviate from being a straight line? When we travel along a straight line, our tangent vector always points along our path; along a curving path, the tangent vector approximates our path, but doesn t exactly match it. Figure 10: Tangent vectors approximating a path. Since the tangent vector of our parametrization gives a straight-line approximation to our curve, a reasonable first attempt at measuring curvature is to ask how quickly our tangent vector is changing. That is, if we have a parametrization r(t) for our curve, we might consider r (t), the speed with which r (t) is changing. To see why this might measure curvature, have a look at Figure 10. Near points where our path makes sharp turns, the tangent vectors change quickly, but along straighter parts the tangent vector doesn t change much. Unfortunately this approach has a big problem our tangent vector r (t) depends on how we parametrize the curve. To see what we mean, consider the following parametrizations: r 1 (t) = cos(t), sin(t), 0 t 2π and r 2 (t) = cos(πt/2), sin(πt/2), 0 t 4. Both of these parametrizations trace out the unit circle, but they do so at different speeds. As a result, their tangent vectors have different lengths, as can be seen in Figure 11. It s unsurprising, then, that these parametrizations give us different acceleration vectors: r 1(t) = cos(t), sin(t) and r 2(t) = π2 cos(πt/2), sin(πt/2). 4 Both parametrizations would tell us that the circle has constant curvature (which seems accurate), but r 1 would call this curvature 1, while r 2 would call it π 2 /4: r 1(t) 1 and r 2(t) π2 4. As a means of standardizing our curvature measurements, we can replace the idea of measuring how quickly the tangent vector is changing with measuring how quickly the unit 17
Figure 11: Tangent vectors for r 1 (in blue) and r 2 (in orange). tangent vector is changing. That is, as long as our tangent vector r (t) is never zero, we can scale it to be unit length and call this vector T(t) our unit tangent vector. We could then measure how quickly this vector changes. Notice that for our parametrizations of the circle we have T 1 (t) = r 1(t) = sin(t), cos(t) r 1(t) and T 2 (t) = r 2(t) r 2(t) = 2 π π 2 sin(πt/2), π 2 cos(πt/2) = sin(πt/2), cos(πt/2). In Figure 11 this corresponds to scaling our orange vectors down to unit length, so that if r 1 (t 1 ) = r 2 (t 2 ) for some parameter values t 1, t 2, then T(t 1 ) = T(t 2 ). Our second attempt at curvature is to measure how quickly this unit tangent vector changes, so we compute T 1(t) = cos(t), sin(t) = 1 and T 2(t) = π 2 cos(πt/2), π 2 sin(πt/2) π = 2. So our new measure of curvature still has the problem that it depends on how we parametrize our curves. The problem with asking how quickly the unit tangent vector changes is that we need to be consistent with the value against which we measure this change. When we compute T 1(t) we re measuring the change in our unit tangent vector against the parameter for r 1, and T 2(t) measures this change against the parameter for r 2. What we really want to know is how T changes with respect to arclength. We want to ask how T will be different one unit down the curve. The following definition is made with this consideration in mind. 18
Curvature. Suppose r(s) is an arclength parametrization for the curve C. The the curvature of C at the point r(s) is given by κ(s) = ds (s), where T(s) is the unit tangent vector to C at r(s). Example 3.1. The parametrization r 1 (t) of the unit circle given earlier is an arclength parametrization, since r 1(t) 1. It follows that the curvature of the circle is T 1(t) 1. So the circle has constant curvature. This is consistent with our intuition, since the circle is always deviating from its tangent vector in the same way. The definition of curvature we finally settled on is on a good theoretical footing, since it prevents us from finding different curvature values by using different parametrizations for our curve. But it has a practical problem: finding arclength parametrizations for curves can be very difficult. We d like to find a way of computing curvature from parametrizations that aren t necessarily arclength parametrizations. Suppose r(t) parametrizes the curve C and that r (t) is never zero, but is not necessarily an arclength parametrization. Then T(t) = r (t) r (t), so the chain rule tells us that dt = ds ds dt = ( d 1 ds dt We then apply the fundamental theorem of calculus to find dt = ds r (t). 0 ) r (u) du. So we can compute the arclength derivative of T according to ds = 1 r (t) dt. Since the curvature is the magnitude of this vector, we have κ(t) = 1 r (t) dt. 19
Example 3.2. We ve already computed T 2(t) = π/2 from our parametrization r 2 (t) of the unit circle. Since r 2(t) = π/2, we see that κ(t) = 1 r 2(t) T 2(t) = π/2 π/2 = 1, and this agrees with our computation of curvature from the parametrization r 1 (t). Before computing curvature for a curve other than the circle, we ll come up with one more formula for curvature that starts from an arbitrary 1 parametrization r(t). First we need a lemma. Lemma 3.1. Let T(t) be a parametrization of the unit tangent vector to a curve C. Then T(t) is perpendicular to T (t). Proof. Let f(t) = T(t) 2. Because T(t) always has unit length, f is the constant function 1. At the same time, f (t) = d dt (T(t) T(t)) = 2T(t) T (t). Because f is constant, f is always 0, so 2T(t) T (t) = 0 for all t, meaning that T(t) is perpendicular to T (t) for all t. This observation may seem irrelevant, but it will be helpful in the computation that follows. Suppose r(t) is a parametrization for the curve C so that r (t) is never zero. Then for all t, r (t) = r (t) T(t). From this we can compute r (t): r (t) = d dt ( r (t) )T(t) + r (t) T (t). Now (for no immediately apparent reason) we ll compute the cross product 2 r (t) r (t): ( ) d r (t) r (t) = ( r (t) T(t)) dt ( r (t) )T(t) + r (t) T (t) = r (t) d dt ( r (t) )(T(t) T(t)) + r (t) 2 (T(t) T (t)) = r (t) 2 (T(t) T (t)). The first term on the second-to-last line vanishes because we re crossing a vector with itself. We now have that r (t) r (t) = r (t) 2 T(t) T (t) = r (t) 2 sin θ T(t) T (t), 1 Not quite arbitrary: we do require that r (t) not vanish. 2 Computing the cross product assumes that r (t) and r (t) are vectors in three dimensions. In the case that we re working in two dimensions, we simply assume the z-coordinate to be 0. 20
where θ is the angle between T(t) and T (t). But our lemma says that this angle must always be π/2. Because sin(π/2) = 1 and T(t) = 1, we see that r (t) r (t) = r (t) 2 T (t). This is very nearly the curvature formula we need only divide by r (t) 3. So we see that if our curve is parametrized by r(t) and r (t) never vanishes, then κ(t) = r (t) r (t) r (t) 3. Example 3.3. Let C be the ellipse defined by x 2 25 + y2 4 = 1. Find the extreme values for the curvature of C. (Solution) Since C has major and minor axes of length 5 and 2, respectively, we can parametrize C by r(t) = 5 cos(t), 2 sin(t), 0 t 2π. Then so r (t) = 5 sin(t), 2 cos(t) and r (t) = 5 cos(t), 2 sin(t), i j k r (t) r (t) = 5 sin(t) 2 cos(t) 0 5 cos(t) 2 sin(t) 0 = 0, 0, 10 sin2 (t) + 10 cos 2 (t) = 0, 0, 10. Then r (t) r (t) = 10. We also have so r (t) = 25 sin 2 (t) + 4 cos 2 (t) = κ(t) = r (t) r (t) r (t) 3 = 21 sin 2 (t) + 4, 10 (21 sin 2 (t) + 4) 3/2. We re interested in minimizing and maximizing κ(t), so we compute κ 630 cos(t) sin(t) (t) = (21 sin 2 (t) + 4). 5/2 Then κ (t) will vanish whenever either cos(t) or sin(t) is 0, so we have critical points at t = 0, π/2, π, and 3π/2. We calculate the curvature at each of these points and find t 0 π/2 π 3π/2 κ(t) 5/4 2/25 5/4 2/25 21
Figure 12: Unit tangent vectors for an ellipse. So C achieves its maximum curvature of 5/4 at r(0) = (5, 0) and r(π) = ( 5, 0) and achieves its minimum curvature of 2/25 at r(π/2) = (0, 2) and r(3π/2) = (0, 2). Notice that this fits with the plot of C in Figure 12. The path is curving most sharply at the two horizontal edges, where our unit tangent vectors are changing rapidly. At the vertical extremes of our path, the curve is much straighter, and this is reflected in its smaller curvature. References [1] Colin Adams Jon Rogawski. Calculus. W.H. Freeman and Company, New York, NY, 2015. 22