Math 114 – Rimmer 13.3/13.4 Arc Length and Curvature

[Pages:11]Math 114 ? Rimmer 13.3/13.4 Arc Length and Curvature

In the plane (2-d):

b

y = f ( x) : Arc Length = 1+ f ( x)2 dx

z

a

parametric equations : x = f (t ) and y = g (t ) :

t =b

b

Arc Length = f (t )2 + g(t )2 dt

a

t=a

y

b

Arc Length = r(t ) dt

x r(t) = f (t), g (t),h(t)

b

Arc Length = f (t )2 + g(t )2 + h(t )2 dt

a

a

Arc Length Function:

t

s (t ) = r(u) du

a

Math 114 ? Rimmer 13.3/13.4 Arc Length and Curvature

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Math 114 ? Rimmer 13.3/13.4 Arc Length and Curvature

Math 114 ? Rimmer 13.3/13.4 Arc Length and Curvature

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Math 114 ? Rimmer 13.3/13.4 Arc Length and Curvature

Math 114 ? Rimmer 13.3/13.4 Arc Length and Curvature

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t

s (t ) = r(u) du

a

s = distance and t = time

Math 114 ? Rimmer 13.3/13.4 Arc Length and Curvature

r (t ) gives position on the curve as a function of time

like driving and looking at the clock to tell where you are after you have traveled a certain time t

its more natural to look at the odometer or the mile markers to tell where you are after you have traveled a certain distance s

we would like to have t as a function of s so that we could reparametrize in terms of s.

Using the arc length function we can find t as a function of s

giving us r (t (s)) or just r (s)

r (t ) = 3sin t, 4t,3cos t

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

r(t ) = 3cos t, 4, -3sin t

Math 114 ? Rimmer 13.3/13.4 Arc Length and Curvature

( ) r(t ) = 9 cos2 t +16 + 9sin2 t = 9 cos2 t + sin2 t +16 = 9 +16 = 25 = 5

t

t

s = r(u) du = 5du = 5t

0

0

s = 5t t = s 5

r(s) =

3

sin

(

s 5

)

,

4

(

s 5

)

,

3

cos

(

s 5

)

r (5) is the position vector of the point 5 units

of length along the curve from its starting point.

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Math 114 ? Rimmer 13.3/13.4 Arc Length and Curvature

Curvature - a measure how quickly a curve changes direction at a point another way to look at curvature is how sharply the curve bends at a point

Q

P T(t)

T(t)

s

curvature @ P > curvature @ Q

In order to calculate curvature ( ),we need

to use the unit tangent vector T and

T

the arc length parameter s.

Curvature - the magnitude of the rate of change of the unit

tangent vector T with respect to the arc length parameter s.

T(t + t)

= dT = T(s)

ds

r(s) =

3

sin

(

s 5

)

,

4

(

s 5

)

,

3

cos

(

s 5

)

Math 114 ? Rimmer 13.3/13.4 Arc Length and Curvature

( ) ( ) ( ) r s

=

3 5

cos

s 5

,

4 5

,

-3 5

sin

s 5

( ) ( ) ( ) r s =

9 25

cos2

s 5

+

16 25

+

9 25

sin 2

s 5

=

r(s) =1

( ) ( ) 9

25

cos2

s 5

+ sin2

s 5

+

16 25

=

+ 9 16

25 25

T(s) =

r(s) r(s)

=

( ) ( ) 3

5

cos

s 5

,

4 5

,

-3 5

sin

s 5

( ) ( ) ( ) T s

=

-3 25

sin

s 5

,

0,

-3 25

cos

s 5

= T(s) =

( ) ( ) 9

625

sin

2

s 5

+ cos2

s 5

= 3 25

=

dT ds

=

d dr ds ds

=

r ( s )

T

= r(s)

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Curvature in terms of t

dT

= dT ds

=

dt ds

T(t )

=

r (t )

dt

T(t ) = r(t)

Math 114 ? Rimmer 13.3/13.4 Arc Length and Curvature

t

s (t ) = r(u) du

a

Fund. Theorem ds = r(t )

of Calculus dt

r (t ) = 3sin t, 4t,3cos t r(t ) = 3cos t, 4, -3sin t r(t) = 5

3

=

T(t ) r (t )

=

5 5

=

3 25

T(t) =

r (t ) r (t )

=

3 5

cos

t,

4 5

,

-3 5

sin

t

T(t ) =

-3 5

sin

t

,

0,

-3 5

cos

t

T(t ) =

9 25

sin 2

t

+ cos2

t

=

3 5

Curvature in terms of t

scalar

function

( ) r t ( ) T t =

( ) ( ) ( ) ( ) ( ) ( ) r t

r t = r t T t r t = ds T t dt

Math 114 ? Rimmer 13.3/13.4 Arc Length and Curvature

vector

Take a derivative to find r(t ) :

Consider: r(t )?r(t )

r ( t

)

=

d 2s dt 2

T

(t

)

+

ds dt

T

(t

)

=

ds dt

T

?

d 2s dt 2

T

+

ds dt

T

=

ds dt

T

?

d 2s dt 2

T

+

ds dt

T

?

ds dt

T

=

ds dt

d 2s dt 2

[T ? T] +

0

ds dt

2

[T?

T]

r(t )?r(t ) = r(t ) 2 [T? T] r(t )? r(t ) = r(t ) 2 T? T

scalar funct.

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Curvature in terms of t

T ? T = T T sin

T T sin = 1

T? T = T T T? T = T

Math 114 ? Rimmer 13.3/13.4 Arc Length and Curvature

Section 14.2

Show that if r (t ) = c (constant), then r(t ) r (t ) for all t.

T(t ) = 1 so T(t ) T(t ) for all t.

T =1

r(t )? r(t ) = r(t ) 2 T ? T

r(t )? r(t )

=

r(t) 2

r (t ) T r(t )

= r(t) 3

T

r (t )

r(t )? r(t ) = r(t ) 3

r(t )?r(t )

=

r(t) 3

r (t ) = 3sin t, 4t,3cos t r(t ) = 3cos t, 4, -3sin t r(t ) = -3sin t, 0, -3cos t

r(t) = 5

Math 114 ? Rimmer 13.3/13.4 Arc Length and Curvature

i jk

r(t ) ? r(t ) = 3cos t ( ) 4 -3sin t = -12 cos t, - -9 cos2 t + sin2 t ,12sin t

-3sin t 0 -3cos t

r(t )?r(t ) = -12 cost,9,12sin t

r(t )? r(t ) = 144 cos2 t + sin2 t + 81 = 225

r(t)?r(t ) = 15

= r(t )? r(t ) = 15

r(t) 3

125

= 3 25

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Math 114 ? Rimmer 13.3/13.4 Arc Length and Curvature

Let r (t ) be a smooth curve in the xy plane. r (t ) = f (t ), g (t )

Show

= f g - f g

(

f

)2

+

(

g

)2

3/

2

Math 114 ? Rimmer 13.3/13.4 Arc Length and Curvature

r(t )?r(t )

=

r(t) 3

r(t) = f (t), g (t),0 r(t) = f (t), g(t),0 r(t ) = f (t ), g(t ),0

r(t )? r(t ) = 0, 0, f g - f g r(t )? r(t ) = ( f g - f g )2 = f g - f g

r(t) = ( f (t))2 + ( g(t))2

( ) r(t ) 3 = ( f (t ))2 + ( g(t ))2 3/2

r(t )?r(t )

=

=

f g - f g

r(t) 3

(

f

)2

+

(

g)2

3/ 2

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