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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
13.3 Arc Length and Curvature 2
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.
13.3 Arc Length and Curvature 3
NotePlane curves are described in 2
while space curves are defined in 3.
13.3 Arc Length and Curvature 4
Example 1 – pg. 860 #4Find the length of the curve.
( ) cos sin ln cos 0 / 4t t t t t r i j k
13.3 Arc Length and Curvature 5
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)) .
13.3 Arc Length and Curvature 6
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
13.3 Arc Length and Curvature 7
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
13.3 Arc Length and Curvature 8
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.
13.3 Arc Length and Curvature 9
Definition - Curvature
13.3 Arc Length and Curvature 10
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
13.3 Arc Length and Curvature 11
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 12
Theorem - CurvatureWe can always use equation 9 to
compute curvature, but the below theorem is easier to apply.
13.3 Arc Length and Curvature 13
Example 3Use Theorem 10 to find the
curvature.
2( ) 1t t t t r i j k
13.3 Arc Length and Curvature 14
Definition – Unit Normal
As you can see, N is perpendicular to T(t).
'( )( )
'( )
tt
tT
NT
13.3 Arc Length and Curvature 15
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
13.3 Arc Length and Curvature 16
VisualizationThe TNB Frame
13.3 Arc Length and Curvature 17
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
13.3 Arc Length and Curvature 18
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/.
13.3 Arc Length and Curvature 19
VisualizationOsculating Circle
13.3 Arc Length and Curvature 20
Summary of Formulas
13.3 Arc Length and Curvature 21
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
13.3 Arc Length and Curvature 22
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
13.3 Arc Length and Curvature 23
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
13.3 Arc Length and Curvature 24
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.
13.3 Arc Length and Curvature 25
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
13.3 Arc Length and Curvature 26
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 27
Demonstrations
Feel free to explore these demonstrations below.
TBN FrameCircle of Curvature