Section 10.3 Notes - Completed Section 10.3 Arc Length and Curvature Last year in Calculus you deined the length of a smooth plane curve given by parametric equations x = f(t) and y = g(t) for t on [a, b] as the limit of the lengths of inscribed polygons. If f ' and g' were continuous you used the formula: Naturally, in 3D if C is a smooth space curve given by r (t) = <f(t), g(t), h(t)> on an interval [a, b], then the arc length of C on the interval is: Section 10.3 Notes - Completed Find the arc length of the following curve on the interval [-10, 10]. Find the arc length of the following curve on the interval [1, e]. Section 10.3 Notes - Completed Find the arc length of the following curve from (1, 0, 0) to (1, 0, 2π). Arc Length Function Suppose that the curve C is a curve given by a vector function r(t) = f (t)i + g(t)j + h(t)k on the interval [a, b] where r' is continuous and C is traversed exactly once as t increases from a to b. The arc length function, s(t), gives the length of the part of C between r(a) and r(b). Note that: Section 10.3 Notes - Completed Re-parametrize the helix below with respect to arc length starting from (1, 0, 0) in the direction of increasing t. For the last example, calculate Note: This is always true for any smooth curve C given by r(s) - where s is the arc length parameter. In fact if l is ANY parameter for the vector-valued function r(l) such that , then l MUST be the arc length parameter. Section 10.3 Notes - Completed Curvature The curvature, at a given point, of a space curve is deSined as the rate of change of the unit tangent vector with respect to arc length. In other words, the CURVATURE at a point tells you how fast the curve is turning at that point. Consider a car driving along a curvy road. The tighter the curve, the more difSicult the driving is. In math, the curvature describes this "tightness". If the curvature is zero, then the curve looks like a line near this point. While if the curvature is a large number, then the curve has a sharp bend. Section 10.3 Notes - Completed Recall the unit tangent vector, T(t), that indicates the direction of the curve. T changes direction very slowly when C is fairly straight, but it changes direction more quickly when C bends or twists more sharply. T(t) ΔT T(t) T(t + Δt) Δs T(t + Δt) ΔT T(t) Δs T(t) De=inition of Curvature Let C be a smooth curve (in the plane or in space) given by r(s), where s is the arc length parameter. The curvature, κ, at s is given by Remember: Curvature is the rate of change of the unit tangent vector with respect to the arc length Section 10.3 Notes - Completed Find the curvature, κ, of the curve where s is the arc length parameter. This manner of Sinding the curvature is only useful if the curve is already written in terms of s. Here are some formulas (that I will provide on the test) for Sinding the curvature when the smooth curve C is written in terms of t or in rectangular coordinates (for plane curves). Section 10.3 Notes - Completed Find the curvature, κ, of the curve given by the vector function below. Suppose N is normal to a curve at a point P and suppose that we draw a circle with center on the line N which passes through PQ where Q is another point on the curve. Section 10.3 Notes - Completed If we let Q get closer and closer to P, the result will be a sequence of circles which converge to a circle of radius R called the osculating circle at point P. The osculating circle is practically the same as a small section of the curve. The osculating circle, or circle of curvature, shares the same tangent with the curve at Point P. The radius is called the radius of curvature, and the center of the circle is called the center of curvature. Section 10.3 Notes - Completed Find the curvature of the circle parametrized by: How is the curvature we found related to the radius of the circle? Section 10.3 Notes - Completed Find the point of the curve where the curvature is at a maximum. y = ex Tangent, Normal and Binormal Vectors Let C be a smooth curve represented by r on an open interval I. The unit tangent vector T(t) at t is deined to be Section 10.3 Notes - Completed A Vector Orthogonal to T(t) This implies that and so we can see that T(t) and T '(t) are orthogonal Principal Normal Vector Let C be a smooth curve represented by r on an open interval I with T '(t) ≠0. The principal unit normal vector N(t) at t is defined to be Section 10.3 Notes - Completed EXAMPLE: Find the unit normal N to the helix Unit Normal Vectors - ALMOST ALWAYS REQUIRE THE QUOTIENT RULE Unfortunately, the unit normal vector is very cumbersome to find Section 10.3 Notes - Completed EXAMPLE: Find the unit normal vector for the vector valued function when t =1 Binormal Vector The binormal vector The binormal vector is perpendicular to the unit tangent and the unit normal, and is a unit vector itself Thus, B(t), N(t), and T(t) are mutually orthogonal unit vectors Section 10.3 Notes - Completed Planes Osculating Plane - The plane determined by T and N at a point P on the curve r(t) Normal Plane - The plane determined by N and B at a point P on the curve. This contains all lines orthogonal to the tangent vector Find T, N, and B of at Section 10.3 Notes - Completed Find the equations of the normal plane and osculating planes at the same point. Section 10.3 Homework: p. 714 #s 3, 9, 14, 17, 24, 37, 42, 49, 55 For extra practice try: p. 714#s 1, 5, 8, 11, 16, 19, 22, 27, 29, 41, 51, 60.