# Chapter 13 – Vector Functions

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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

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

Note Plane curves are described in 2 while space curves are defined in 3. 13.3 Arc Length and Curvature

Example 1 – pg. 860 #4 Find the length of the curve.
13.3 Arc Length and Curvature

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

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. 13.3 Arc Length and Curvature

Recall If C is a smooth curve defined by the vector r, recall that the unit tangent is given by and indicates the direction of the curve. 13.3 Arc Length and Curvature

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

Definition - Curvature
13.3 Arc Length and Curvature

Curvature and the Chain Rule
If we use the Chain rule on curvature, we will have: So we have: 13.3 Arc Length and Curvature

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

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

Example 3 Use Theorem 10 to find the curvature.
13.3 Arc Length and Curvature

Definition – Unit Normal
As you can see, N is perpendicular to T(t). 13.3 Arc Length and Curvature

Definition – Binormal Vector
The Binormal Vector is perpendicular to both T and N. It is also a unit vector and is defined as: 13.3 Arc Length and Curvature

Visualization The TNB Frame 13.3 Arc Length and Curvature

Example 4 – pg.861 # 48 Find the vectors T, N, and B at the given point. 13.3 Arc Length and Curvature

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

Visualization Osculating Circle 13.3 Arc Length and Curvature

Summary of Formulas 13.3 Arc Length and Curvature

In groups, work on the following problems
Problem 1 – pg. 860 #6 Find the arc length of the curve. 13.3 Arc Length and Curvature

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? 13.3 Arc Length and Curvature

In groups, work on the following problems
a) Find the unit tangent and unit normal vectors. b) Use formula 9 to find curvature. 13.3 Arc Length and Curvature

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

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. 13.3 Arc Length and Curvature

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

Demonstrations Feel free to explore these demonstrations below.
TBN Frame Circle of Curvature 13.3 Arc Length and Curvature