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Chapter 13 – Vector Functions 13.3 Arc Length and Curvature Objectives: Find vector, parametric, and general forms of equations of lines and planes. Find distances and angles between lines and planes 13.3 Arc Length and Curvature 1 Arc Length Assuming that the space curve is traversed exactly once on [a,b], and the component functions are differentiable on [a,b], then arc length is given by the integrals below. 13.3 Arc Length and Curvature 2 Note Plane curves are described in 2 while space curves are defined in 13.3 Arc Length and Curvature 3. 3 Example 1 – pg. 860 #4 Find the length of the curve. r(t ) cos ti sin tj ln cos tk 0t / 4 13.3 Arc Length and Curvature 4 Parameterization in Terms of Arc Length The relation below allows us to use distance along the curve as the parameter. This parameterization does not depend on coordinate system. Replace r(t) with r(t(s)) . 13.3 Arc Length and Curvature 5 Example 2 – pg.860 #14 Reparametrize the curve with respect to arc length measured from the point where t = 0 in the direction of increasing t. r(t ) e cos 2ti 2j e sin 2tk 2t 2t 13.3 Arc Length and Curvature 6 Recall If C is a smooth curve defined by the vector r, recall that the unit tangent is given by T t r ' t r ' t and indicates the direction of the curve. 13.3 Arc Length and Curvature 7 Visualization The Unit Tangent Vector T(t) changes direction very slowly when C is fairly straight, but it changes direction more quickly when C bends or twists more sharply. 13.3 Arc Length and Curvature 8 Definition - Curvature 13.3 Arc Length and Curvature 9 Curvature and the Chain Rule If we use the Chain rule on curvature, we will have: dT dT ds dt ds dt and dT dT / dt ds ds / dt where ds / dt r '(t ) So we have: 13.3 Arc Length and Curvature 10 Note: Small circles have large curvature. Large circles have small curvature. Curvature of a straight line is always 0 because the tangent vector is constant. 13.3 Arc Length and Curvature 11 Theorem - Curvature We can always use equation 9 to compute curvature, but the below theorem is easier to apply. 13.3 Arc Length and Curvature 12 Example 3 Use Theorem 10 to find the curvature. r (t ) ti tj 1 t k 2 13.3 Arc Length and Curvature 13 Definition – Unit Normal T '(t ) N(t ) T '(t ) As you can see, N is perpendicular to T(t). 13.3 Arc Length and Curvature 14 Definition – Binormal Vector The Binormal Vector is perpendicular to both T and N. It is also a unit vector and is defined as: B(t ) T(t ) N(t ) 13.3 Arc Length and Curvature 15 Visualization The TNB Frame 13.3 Arc Length and Curvature 16 Example 4 – pg.861 # 48 Find the vectors T, N, and B at the given point. r(t ) cos t,sin t,ln cos t (1,0,0) 13.3 Arc Length and Curvature 17 Other Definitions The normal plane is determined by the vectors N and B at a point P on the curve C. It consists of all lines that are orthogonal to the tangent vector. The osculating plane of C and P is determined by the vectors T and N. An osculating circle is a circle that lies in the oculating place of C at P, has the same tangent as C at P, lies on the concave side of C (towards N), and has radius =1/. 13.3 Arc Length and Curvature 18 Visualization Osculating Circle 13.3 Arc Length and Curvature 19 Summary of Formulas 13.3 Arc Length and Curvature 20 In groups, work on the following problems Problem 1 – pg. 860 #6 Find the arc length of the curve. r(t ) 12ti 8t j 3t k 3/2 2 0 t 1 13.3 Arc Length and Curvature 21 In groups, work on the following problems Problem 2 – page 860 #16 Reparametrize the curve below with respect to arc length measured from the point (1,0) in the direction of increasing t. Express the reparametrization in its simplest form. What can you conclude about the curve? 2t 2 r (t ) 2 1 i 2 j t 1 t 1 13.3 Arc Length and Curvature 22 In groups, work on the following problems Problem 3 a) Find the unit tangent and unit normal vectors. b) Use formula 9 to find curvature. r(t ) 2sin t,5t,2cos t 13.3 Arc Length and Curvature 23 In groups, work on the following problems Problem 4 – pg. 860 #31 At what point does the curve have a maximum curvature? What happens to the curvature as x. 13.3 Arc Length and Curvature 24 In groups, work on the following problems Problem 5 – pg. 861 #50 Find equations of the normal plane and osculating plane of the curve at the given point. x t, y t , z t ; 2 3 (1,1,1) 13.3 Arc Length and Curvature 25 More Examples The video examples below are from section 13.3 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦ Example 1 ◦ Example 3 ◦ Example 7 13.3 Arc Length and Curvature 26 Demonstrations Feel free to explore these demonstrations below. TBN Frame Circle of Curvature 13.3 Arc Length and Curvature 27