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Chapter 13 – Vector Functions13.3 Arc Length and Curvature

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

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Arc LengthAssuming 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.

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NotePlane curves are described in 2

while space curves are defined in 3.

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Example 1 – pg. 860 #4Find the length of the curve.

( ) cos sin ln cos 0 / 4t t t t t r i j k

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Parameterization in Terms of Arc LengthThe 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)) .

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Example 2 – pg.860 #14Reparametrize the curve with

respect to arc length measured from the point where t = 0 in the direction of increasing t.

2 2( ) cos 2 2 sin 2t tt e t e t r i j k

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RecallIf 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.

'

'

tt

tr

Tr

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VisualizationThe 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.

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Definition - Curvature

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Curvature and the Chain RuleIf we use the Chain rule on

curvature, we will have:

So we have:

/ and where / '( )

/

d d ds d d dtds dt t

dt ds dt ds ds dt

T T T Tr

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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.

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Theorem - CurvatureWe can always use equation 9 to

compute curvature, but the below theorem is easier to apply.

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Example 3Use Theorem 10 to find the

curvature.

2( ) 1t t t t r i j k

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Definition – Unit Normal

As you can see, N is perpendicular to T(t).

'( )( )

'( )

tt

tT

NT

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Definition – Binormal VectorThe Binormal Vector is

perpendicular to both T and N. It is also a unit vector and is defined as:( ) ( ) ( )t t t B T N

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VisualizationThe TNB Frame

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Example 4 – pg.861 # 48Find the vectors T, N, and B at

the given point.

( ) cos ,sin , ln cos (1,0,0)t t t tr

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Other DefinitionsThe 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/.

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VisualizationOsculating Circle

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Summary of Formulas

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In groups, work on the following problemsProblem 1 – pg. 860 #6

Find the arc length of the curve.3/2 2( ) 12 8 3 0 1t t t t t r i j k

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In groups, work on the following problemsProblem 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?

2 2

2 2( ) 1

1 1

tt

t t

r i j

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In groups, work on the following problemsProblem 3

a) Find the unit tangent and unit normal vectors.b) Use formula 9 to find curvature.

( ) 2sin ,5 ,2cost t t tr

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In groups, work on the following problemsProblem 4 – pg. 860 #31

At what point does the curve have a maximum curvature? What happens to the curvature as x.

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In groups, work on the following problemsProblem 5 – pg. 861 #50

Find equations of the normal plane and osculating plane of the curve at the given point.

2 3, , ; (1,1,1)x t y t z t

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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

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Demonstrations

Feel free to explore these demonstrations below.

TBN FrameCircle of Curvature